{ "stats":{ "node_count":141, "edge_count":324, "reference_count":142 }, "nodes":[{ "href":"macdonaldE.htm#nonSymmetricElementary", "year":"2019", "id":"nonSymmetricElementary", "category":"Elementary", "page":"macdonaldE", "name":"Non-symmetric elementary polynomials", "space":"All", "symbol":"$\\macdonaldE_\\lambda(\\xvec;1,0)$" },{ "href":"pPartitions.htm#pPartitionsFlagged", "year":"2019", "id":"pPartitionsFlagged", "category":"QuasiElementary", "page":"pPartitions", "name":"Flagged $(P,w)$-partition generating functions", "space":"All", "symbol":"$\\pPartition_{(P,w),\\rho}(\\xvec)$" },{ "href":"schubertVariations.htm#schubertAffine", "year":"2006", "id":"schubertAffine", "category":"Schubert", "page":"schubertVariations", "name":"Affine Schubert polynomials", "space":"All", "symbol":"$\\widetilde{\\schubert}_w$" },{ "href":"schubertVariations.htm#schubertBackStableDouble", "year":"2018", "id":"schubertBackStableDouble", "category":"Schubert", "page":"schubertVariations", "name":"Back stable double Schubert polynomials", "space":"All", "symbol":"$\\overleftarrow{\\schubert}_\\omega(\\xvec;\\avec)$" },{ "href":"grothendieck.htm#grothendieckDouble", "year":"1994", "id":"grothendieckDouble", "category":"Schubert", "page":"grothendieck", "name":"Double Grothendieck polynomials", "space":"All", "symbol":"$\\grothendieck_{\\omega}(x|b)$" },{ "href":"schubertVariations.htm#schubertDouble", "year":"1982", "id":"schubertDouble", "category":"Schubert", "page":"schubertVariations", "name":"Double Schubert polynomials", "space":"All", "symbol":"$\\schubert_\\omega(\\xvec;\\yvec)$" },{ "href":"schubertVariations.htm#schubertEnriched", "year":"2021", "id":"schubertEnriched", "category":"Schubert", "page":"schubertVariations", "name":"Enriched Schubert polynomials", "space":"All", "symbol":"" },{ "href":"grothendieck.htm#grothendieckFactorial", "year":"2019", "id":"grothendieckFactorial", "category":"Schubert", "page":"grothendieck", "name":"Flagged factorial Grothendieck polynomials", "space":"All", "symbol":"$\\grothendieckStable_{\\lambda,f}(\\xvec|b)$" },{ "href":"grothendieck.htm#flaggedRefinedSkewStableGrothendieck", "year":"2024", "id":"flaggedRefinedSkewStableGrothendieck", "category":"Schubert", "page":"grothendieck", "name":"Flagged refined skew stable Grothendieck polynomials", "space":"All", "symbol":"$G_{\\lambda/\\mu}(X_\\Phi;t)$" },{ "href":"grothendieck.htm#grothendieck", "year":"1990", "id":"grothendieck", "category":"Schubert", "page":"grothendieck", "name":"Grothendieck polynomials", "space":"All", "symbol":"$\\grothendieck_\\omega(\\xvec)$" },{ "href":"schubertVariations.htm#schubertInvolution", "year":"2018", "id":"schubertInvolution", "category":"Schubert", "page":"schubertVariations", "name":"Involution Schubert polynomials", "space":"All", "symbol":"$\\ischubert_y(x_1,\\dotsc,x_n)$" },{ "href":"schubertVariations.htm#schubertQuantum", "year":"1997", "id":"schubertQuantum", "category":"Schubert", "page":"schubertVariations", "name":"Quantum Schubert polynomials", "space":"All", "symbol":"$\\schubert^q_\\omega(\\xvec)$" },{ "href":"schubert.htm#schubert", "year":"1982", "id":"schubert", "category":"Schubert", "page":"schubert", "name":"Schubert polynomials", "space":"All", "symbol":"" },{ "href":"schubertVariations.htm#schubertSkew", "year":"2003", "id":"schubertSkew", "category":"Schubert", "page":"schubertVariations", "name":"Skew Schubert polynomials", "space":"All", "symbol":"" },{ "href":"schubertVariations.htm#schubertBCD", "year":"1996", "id":"schubertBCD", "category":"Schubert", "page":"schubertVariations", "name":"Type B/C/D Schubert polynomials", "space":"All", "symbol":"$\\schubert^B_\\omega(\\xvec)$" },{ "href":"schubertVariations.htm#schubertUniversal", "year":"1999", "id":"schubertUniversal", "category":"Schubert", "page":"schubertVariations", "name":"Universal Schubert polynomials", "space":"All", "symbol":"$\\schubert_w(c;d)$" },{ "href":"key.htm#completeFlaggedHomogeneous", "year":"2023", "id":"completeFlaggedHomogeneous", "category":"Schur", "page":"key", "name":"Complete flagged homogeneous polynomials", "space":"All", "symbol":"$h_a(\\xvec)$" },{ "href":"macdonaldEperm.htm#tAtom", "year":"2011", "id":"tAtom", "category":"Schur", "page":"macdonaldEperm", "name":"Demazure $t$-atoms", "space":"All", "symbol":"$\\atom_\\alpha(\\xvec;t)$" },{ "href":"key.htm#atom", "year":"1990", "id":"atom", "category":"Schur", "page":"key", "name":"Demazure atoms", "space":"All", "symbol":"$\\atom_{\\lambda,\\sigma}(z)$" },{ "href":"llt.htm#flaggedLLT", "year":"2025", "id":"flaggedLLT", "category":"Schur", "page":"llt", "name":"Flagged LLT polynomials", "space":"All", "symbol":"$G_{\\nu,\\sigma}(\\xvec;q)$" },{ "href":"assaf.htm#forestPolynomials", "year":"2023", "id":"forestPolynomials", "category":"Schur", "page":"assaf", "name":"Forest polynomials", "space":"All", "symbol":"$P_F(\\xvec)$" },{ "href":"assaf.htm#fundamentalParticle", "year":"2020", "id":"fundamentalParticle", "category":"Schur", "page":"assaf", "name":"Fundamental particle basis", "space":"All", "symbol":"$L_\\alpha(\\xvec)$" },{ "href":"assaf.htm#slideF", "year":"2016", "id":"slideF", "category":"Schur", "page":"assaf", "name":"Fundamental slide polynomials", "space":"All", "symbol":"$\\slideF_\\alpha(\\xvec)$" },{ "href":"generalizedDemazureAtoms.htm#generalizedDemazureAtoms", "year":"2012", "id":"generalizedDemazureAtoms", "category":"Schur", "page":"generalizedDemazureAtoms", "name":"Generalized Demazure atoms", "space":"All", "symbol":"" },{ "href":"key.htm#key", "year":"1974", "id":"key", "category":"Schur", "page":"key", "name":"Key polynomials", "space":"All", "symbol":"$\\key_{\\lambda,\\sigma}(z)$" },{ "href":"kohnert.htm#kohnert", "year":"2019", "id":"kohnert", "category":"Schur", "page":"kohnert", "name":"Kohnert polynomials", "space":"All", "symbol":"$\\kohnert_{D}(z)$" },{ "href":"lascoux.htm#lascouxAtom", "year":"2004", "id":"lascouxAtom", "category":"Schur", "page":"lascoux", "name":"Lascoux atom", "space":"All", "symbol":"$\\lascouxAtom^{(\\beta)}_{\\alpha}(z)$" },{ "href":"lascoux.htm#lascoux", "year":"2001", "id":"lascoux", "category":"Schur", "page":"lascoux", "name":"Lascoux polynomials", "space":"All", "symbol":"$\\lascoux^{(\\beta)}_{\\alpha}(z)$" },{ "href":"assaf.htm#lock", "year":"2019", "id":"lock", "category":"Schur", "page":"assaf", "name":"Lock polynomials", "space":"All", "symbol":"$\\lock_\\alpha(\\xvec)$" },{ "href":"macdonaldE.htm#macdonaldE", "year":"1995", "id":"macdonaldE", "category":"Schur", "page":"macdonaldE", "name":"Macdonald E polynomials", "space":"All", "symbol":"$\\macdonaldE_\\mu(\\xvec;q,t)$" },{ "href":"assaf.htm#slideM", "year":"2016", "id":"slideM", "category":"Schur", "page":"assaf", "name":"Monomial slide polynomials", "space":"All", "symbol":"$\\slideM_\\alpha(\\xvec)$" },{ "href":"superSymmetricSchur.htm#multiSchur", "year":"2023", "id":"multiSchur", "category":"Schur", "page":"superSymmetricSchur", "name":"Multi-Schur functions", "space":"All", "symbol":"$S_{\\lambda/\\mu}^{\\alpha/\\beta}(\\xvec)$" },{ "href":"macdonaldE.htm#macdonaldEspec", "year":"1995", "id":"macdonaldEspec", "category":"Schur", "page":"macdonaldE", "name":"Non-symmetric q-Whittaker polynomials", "space":"All", "symbol":"$\\macdonaldE_\\mu(\\xvec;q,0)$" },{ "href":"macdonaldEperm.htm#macdonaldEperm", "year":"2011", "id":"macdonaldEperm", "category":"Schur", "page":"macdonaldEperm", "name":"Permuted basement Macdonald E polynomials", "space":"All", "symbol":"$\\macdonaldE^{\\sigma}_\\mu(\\xvec;q,t)$" },{ "href":"macdonaldEperm.htm#permutedBasementTAtom", "year":"2019", "id":"permutedBasementTAtom", "category":"Schur", "page":"macdonaldEperm", "name":"Permuted-basement $t$-atoms", "space":"All", "symbol":"$\\atom^\\sigma_\\alpha(\\xvec;t)$" },{ "href":"macdonaldEperm.htm#permutedBasementAtom", "year":"2019", "id":"permutedBasementAtom", "category":"Schur", "page":"macdonaldEperm", "name":"Permuted-basement atoms", "space":"All", "symbol":"$\\atom^\\sigma_\\alpha(\\xvec)$" },{ "href":"kohnert.htm#quasiKey", "year":"2018", "id":"quasiKey", "category":"Schur", "page":"kohnert", "name":"Quasi-key polynomials", "space":"All", "symbol":"$Q_a(\\xvec)$" },{ "href":"schurFlagged.htm#schurFlagged", "year":"1982", "id":"schurFlagged", "category":"Schur", "page":"schurFlagged", "name":"Schur polynomials (flagged)", "space":"All", "symbol":"$\\schurS_{\\lambda,b}(x_1,\\dotsc,x_n)$" },{ "href":"key.htm#keySkew", "year":"2019", "id":"keySkew", "category":"Schur", "page":"key", "name":"Skew key polynomials", "space":"All", "symbol":"" },{ "href":"whittaker.htm#qWhittaker", "year":"1995", "id":"qWhittaker", "category":"Schur", "page":"whittaker", "name":"q-Whittaker functions", "space":"All", "symbol":"$\\qWhittaker_\\mu(\\xvec;q)$" },{ "href":"loopSchur.htm#schurLoop", "year":"2012", "id":"schurLoop", "category":"Schur", "page":"loopSchur", "name":"Loop Schur polynomials", "space":"LSym", "symbol":"$\\schurS^{(r)}_{\\lambda/\\mu}(\\xvec)$" },{ "href":"nonCommutativeFunctions.htm#nonComHomogeneous", "year":"1995", "id":"nonComHomogeneous", "category":"Elementary", "page":"nonCommutativeFunctions", "name":"Noncommutative complete homogeneous functions", "space":"NSym", "symbol":"$\\completeH_\\alpha$" },{ "href":"nonCommutativeFunctions.htm#nonComPowerPhi", "year":"1995", "id":"nonComPowerPhi", "category":"Elementary", "page":"nonCommutativeFunctions", "name":"Noncommutative power sums (Phi)", "space":"NSym", "symbol":"$\\Phi_\\alpha$" },{ "href":"nonCommutativeFunctions.htm#nonComPowerPsi", "year":"1995", "id":"nonComPowerPsi", "category":"Elementary", "page":"nonCommutativeFunctions", "name":"Noncommutative power sums (Psi)", "space":"NSym", "symbol":"$\\Psi_\\alpha$" },{ "href":"key.htm#almostSymmetricSchur", "year":"2024", "id":"almostSymmetricSchur", "category":"Schur", "page":"key", "name":"Almost symmetric Schur functions", "space":"Other", "symbol":"$s_{(\\mu|\\lambda)}$" },{ "href":"schurMisc.htm#schurOrthogonal", "year":"1971?", "id":"schurOrthogonal", "category":"Schur", "page":"schurMisc", "name":"Orthogonal Schur polynomials", "space":"Other", "symbol":"$\\schurOr_\\lambda(\\xvec)$" },{ "href":"schurMisc.htm#schurSymplectic", "year":"1971?", "id":"schurSymplectic", "category":"Schur", "page":"schurMisc", "name":"Symplectic Schur polynomials", "space":"Other", "symbol":"$\\schurSp_{\\lambda/\\mu}(\\xvec)$" },{ "href":"peakQuasisymmetric.htm#peakQSym", "year":"1997", "id":"peakQSym", "category":"Quasi", "page":"peakQuasisymmetric", "name":"Peak quasisymmetric functions", "space":"PQSym", "symbol":"$\\peakQSym_\\Lambda(x)$" },{ "href":"qsymSchur.htm#peakYqSchur", "year":"2025", "id":"peakYqSchur", "category":"Schur", "page":"qsymSchur", "name":"Peak Young quasisymmetric Schur functions", "space":"PQSym", "symbol":"$\\widetilde{S}_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#qSchurQ", "year":"2015", "id":"qSchurQ", "category":"Schur", "page":"qsymSchur", "name":"Quasisymmetric Schur $Q$-functions", "space":"PQSym", "symbol":"$\\widetilde{Q}_\\alpha(\\xvec)$" },{ "href":"gessel.htm#gessel", "year":"1984", "id":"gessel", "category":"QuasiElementary", "page":"gessel", "name":"Gessel quasisymmetric functions", "space":"QSym", "symbol":"$\\gessel_\\alpha(x)$" },{ "href":"standardQuasiSymmetricFunctions.htm#qmonom", "year":"1984", "id":"qmonom", "category":"QuasiElementary", "page":"standardQuasiSymmetricFunctions", "name":"Monomial quasisymmetric functions", "space":"QSym", "symbol":"$\\qmonom_\\alpha(\\xvec)$" },{ "href":"pPartitions.htm#pPartition", "year":"1984", "id":"pPartition", "category":"QuasiElementary", "page":"pPartitions", "name":"P-partitions", "space":"QSym", "symbol":"$\\pPartition_P(x)$" },{ "href":"standardQuasiSymmetricFunctions.htm#qPhi", "year":"2017", "id":"qPhi", "category":"QuasiElementary", "page":"standardQuasiSymmetricFunctions", "name":"Powersum quasisymmetric functions (Phi)", "space":"QSym", "symbol":"$\\qPhi_\\alpha(\\xvec)$" },{ "href":"standardQuasiSymmetricFunctions.htm#qPsi", "year":"2017", "id":"qPsi", "category":"QuasiElementary", "page":"standardQuasiSymmetricFunctions", "name":"Powersum quasisymmetric functions (Psi)", "space":"QSym", "symbol":"$\\qPsi_\\alpha(\\xvec)$" },{ "href":"standardQuasiSymmetricFunctions.htm#qRho", "year":"2021", "id":"qRho", "category":"QuasiElementary", "page":"standardQuasiSymmetricFunctions", "name":"Powersum quasisymmetric functions (p)", "space":"QSym", "symbol":"$\\qRho_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#backwardExtSchur", "year":"2024", "id":"backwardExtSchur", "category":"Schur", "page":"qsymSchur", "name":"Backward extended Schur functions", "space":"QSym", "symbol":"" },{ "href":"qsymSchur.htm#diSchur", "year":"2014", "id":"diSchur", "category":"Schur", "page":"qsymSchur", "name":"Dual immaculate Schur functions", "space":"QSym", "symbol":"$\\diSchur_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#extSchur", "year":"2019", "id":"extSchur", "category":"Schur", "page":"qsymSchur", "name":"Extended Schur functions", "space":"QSym", "symbol":"$\\extSchur_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#flippedExtSchur", "year":"2024", "id":"flippedExtSchur", "category":"Schur", "page":"qsymSchur", "name":"Flipped extended Schur functions", "space":"QSym", "symbol":"" },{ "href":"kohnert.htm#kohnertQuasi", "year":"2019", "id":"kohnertQuasi", "category":"Schur", "page":"kohnert", "name":"Kohnert quasisymmetric functions", "space":"QSym", "symbol":"$\\kohnert_D(\\xvec)$" },{ "href":"macdonaldEperm.htm#macdonaldEQuasi", "year":"2019", "id":"macdonaldEQuasi", "category":"Schur", "page":"macdonaldEperm", "name":"Quasisymmetric Macdonald E polynomials", "space":"QSym", "symbol":"$\\macdonaldEQuasi_\\alpha(\\xvec;q,t)$" },{ "href":"macdonaldP.htm#macdonaldJQuasi", "year":"2022", "id":"macdonaldJQuasi", "category":"Schur", "page":"macdonaldP", "name":"Quasisymmetric Macdonald J polynomials", "space":"QSym", "symbol":"$G_{\\alpha}(\\xvec;q,t)$" },{ "href":"qsymSchur.htm#qSchur", "year":"2011", "id":"qSchur", "category":"Schur", "page":"qsymSchur", "name":"Quasisymmetric Schur functions", "space":"QSym", "symbol":"$\\qSchur_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#rsyqSchur", "year":"2013", "id":"rsyqSchur", "category":"Schur", "page":"qsymSchur", "name":"Row-strict Young quasisymmetric Schur functions", "space":"QSym", "symbol":"$\\rsyqSchur_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#rsdiSchur", "year":"2022", "id":"rsdiSchur", "category":"Schur", "page":"qsymSchur", "name":"Row-strict dual immaculate Schur functions", "space":"QSym", "symbol":"$\\rsdiSchur_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#rsExtSchur", "year":"2023", "id":"rsExtSchur", "category":"Schur", "page":"qsymSchur", "name":"Row-strict extended Schur functions", "space":"QSym", "symbol":"" },{ "href":"qsymSchur.htm#rsqSchur", "year":"2013", "id":"rsqSchur", "category":"Schur", "page":"qsymSchur", "name":"Row-strict quasisymmetric Schur functions", "space":"QSym", "symbol":"$\\rsqSchur_\\alpha(\\xvec)$" },{ "href":"qsymSchur.htm#yqSchur", "year":"2013", "id":"yqSchur", "category":"Schur", "page":"qsymSchur", "name":"Young quasisymmetric Schur functions", "space":"QSym", "symbol":"$\\yqSchur_\\alpha(\\xvec)$" },{ "href":"jackShifted.htm#jackShifted", "year":"1994", "id":"jackShifted", "category":"Schur", "page":"jackShifted", "name":"Shifted Jack polynomials", "space":"ShiftedSym", "symbol":"$\\jackShifted_\\lambda(\\xvec)$" },{ "href":"schurShifted.htm#schurShifted", "year":"1996", "id":"schurShifted", "category":"Schur", "page":"schurShifted", "name":"Shifted Schur polynomials", "space":"ShiftedSym", "symbol":"$\\schurShifted_\\mu(\\xvec) $" },{ "href":"thomPolynomials.htm#thomPolynomial", "year":"1956", "id":"thomPolynomial", "category":"Other", "page":"thomPolynomials", "name":"Thom polynomials", "space":"SuperSym", "symbol":"$\\mathcal{T}^{\\eta}$" },{ "href":"superSymmetricSchur.htm#hookSchur", "year":"1983", "id":"hookSchur", "category":"Schur", "page":"superSymmetricSchur", "name":"Hook Schur polynomials", "space":"SuperSym", "symbol":"$\\schurHook_\\lambda(\\xvec/\\yvec)$" },{ "href":"superSymmetricSchur.htm#littlewoodSchur", "year":"2018", "id":"littlewoodSchur", "category":"Schur", "page":"superSymmetricSchur", "name":"Littlewood--Schur functions", "space":"SuperSym", "symbol":"$\\schurS_{\\lambda/\\mu}(\\xvec/\\yvec)$" },{ "href":"chromaticQuasisymmetric.htm#chromaticQuasisymmetric", "year":"2012", "id":"chromaticQuasisymmetric", "category":"", "page":"chromaticQuasisymmetric", "name":"Chromatic quasisymmetric functions", "space":"Sym", "symbol":"" },{ "href":"cylindricSchur.htm#schurCylindric", "year":"2005", "id":"schurCylindric", "category":"", "page":"cylindricSchur", "name":"Cylindric Schur polynomials", "space":"Sym", "symbol":"" },{ "href":"kromaticSymmetric.htm#kromaticSymmetric", "year":"2023", "id":"kromaticSymmetric", "category":"", "page":"kromaticSymmetric", "name":"Kromatic symmetric functions", "space":"Sym", "symbol":"" },{ "href":"schur.htm#schurS", "year":"1841", "id":"schurS", "category":"", "page":"schur", "name":"Schur polynomials", "space":"Sym", "symbol":"" },{ "href":"cylindricSchur.htm#schurToric", "year":"2005", "id":"schurToric", "category":"", "page":"cylindricSchur", "name":"Toric Schur polynomials", "space":"Sym", "symbol":"" },{ "href":"tutteSymmetric.htm#tutteSymmetric", "year":"1998", "id":"tutteSymmetric", "category":"", "page":"tutteSymmetric", "name":"Tutte symmetric functions", "space":"Sym", "symbol":"" },{ "href":"standardSymmetricFunctions.htm#augmentedMonomial", "year":"2015", "id":"augmentedMonomial", "category":"Elementary", "page":"standardSymmetricFunctions", "name":"Augmented monomial symmetric functions", "space":"Sym", "symbol":"$\\widetilde{\\monomial}_\\lambda(\\xvec)$" },{ "href":"standardSymmetricFunctions.htm#completeH", "year":"1700", "id":"completeH", "category":"Elementary", "page":"standardSymmetricFunctions", "name":"Complete homogeneous symmetric functions", "space":"Sym", "symbol":"$\\completeH_\\lambda(\\xvec)$" },{ "href":"standardSymmetricFunctions.htm#elementaryE", "year":"1700", "id":"elementaryE", "category":"Elementary", "page":"standardSymmetricFunctions", "name":"Elementary symmetric functions", "space":"Sym", "symbol":"$\\elementaryE_\\lambda(\\xvec)$" },{ "href":"standardSymmetricFunctions.htm#forgotten", "year":"1900", "id":"forgotten", "category":"Elementary", "page":"standardSymmetricFunctions", "name":"Forgotten symmetric functions", "space":"Sym", "symbol":"$\\forgotten_\\lambda(\\xvec)$" },{ "href":"standardSymmetricFunctions.htm#monomial", "year":"1700", "id":"monomial", "category":"Elementary", "page":"standardSymmetricFunctions", "name":"Monomial symmetric functions", "space":"Sym", "symbol":"$\\monomial_\\lambda(\\xvec)$" },{ "href":"standardSymmetricFunctions.htm#powerSum", "year":"1700", "id":"powerSum", "category":"Elementary", "page":"standardSymmetricFunctions", "name":"Power-sum symmetric functions", "space":"Sym", "symbol":"$\\powerSum_\\lambda(\\xvec)$" },{ "href":"booleanProductPolynomials.htm#booleanProduct", "year":"2018", "id":"booleanProduct", "category":"Other", "page":"booleanProductPolynomials", "name":"Boolean product polynomials", "space":"Sym", "symbol":"" },{ "href":"weightedChromatic.htm#completeMultipartite", "year":"2021", "id":"completeMultipartite", "category":"Other", "page":"weightedChromatic", "name":"Complete multipartite basis", "space":"Sym", "symbol":"$r_\\lambda$" },{ "href":"cycleIndexPolynomial.htm#cycleIndexPolynomial", "year":"1937", "id":"cycleIndexPolynomial", "category":"Other", "page":"cycleIndexPolynomial", "name":"Cycle index polynomials", "space":"Sym", "symbol":"$\\cyc_G(\\xvec)$" },{ "href":"eulerianSymmetric.htm#eulerianSymmetric", "year":"2010", "id":"eulerianSymmetric", "category":"Other", "page":"eulerianSymmetric", "name":"Eulerian quasisymmetric functions", "space":"Sym", "symbol":"$\\eulerianQ_{\\lambda,j}(\\xvec)$" },{ "href":"lah.htm#lah", "year":"2019", "id":"lah", "category":"Other", "page":"lah", "name":"Lah symmetric functions", "space":"Sym", "symbol":"$\\lah_{n,k}(\\xvec)$" },{ "href":"weightedBondSymmetric.htm#mobiusSymmetricFunction", "year":"2026", "id":"mobiusSymmetricFunction", "category":"Other", "page":"weightedBondSymmetric", "name":"Möbius symmetric functions of graphs", "space":"Sym", "symbol":"\\(M_G(\\xvec)\\)" },{ "href":"parkingFunctionSymmetric.htm#parking", "year":"1994", "id":"parking", "category":"Other", "page":"parkingFunctionSymmetric", "name":"Parking function symmetric functions", "space":"Sym", "symbol":"$\\parking_n$" },{ "href":"petrie.htm#petrie", "year":"1992", "id":"petrie", "category":"Other", "page":"petrie", "name":"Petrie symmetric functions", "space":"Sym", "symbol":"" },{ "href":"weightedBondSymmetric.htm#weightedBondChromaticSymmetric", "year":"2026", "id":"weightedBondChromaticSymmetric", "category":"Other", "page":"weightedBondSymmetric", "name":"Weighted-bond chromatic symmetric functions", "space":"Sym", "symbol":"\\(\\Psi_G(\\xvec)\\)" },{ "href":"stanleySymmetric.htm#stanleySymAffine", "year":"2006", "id":"stanleySymAffine", "category":"Schubert", "page":"stanleySymmetric", "name":"Affine Stanley symmetric functions", "space":"Sym", "symbol":"$\\stanleySymAffine_w(\\xvec)$" },{ "href":"grothendieck.htm#grothendieckCanonical", "year":"2017", "id":"grothendieckCanonical", "category":"Schubert", "page":"grothendieck", "name":"Canonical stable Grothendieck polynomials", "space":"Sym", "symbol":"$\\grothendieckStable^{(a,b)}_\\lambda(\\xvec)$" },{ "href":"stanleySymmetric.htm#stanleySymDouble", "year":"2018", "id":"stanleySymDouble", "category":"Schubert", "page":"stanleySymmetric", "name":"Double Stanley symmetric functions", "space":"Sym", "symbol":"$\\stanleySym_\\omega(\\xvec;\\yvec)$" },{ "href":"grothendieck.htm#grothendieckGrassmann", "year":"1994", "id":"grothendieckGrassmann", "category":"Schubert", "page":"grothendieck", "name":"Grassmann Grothendieck polynomials", "space":"Sym", "symbol":"$\\grothendieckStable_\\lambda(\\xvec)$" },{ "href":"stanleySymmetric.htm#stanleySymInvolution", "year":"2017", "id":"stanleySymInvolution", "category":"Schubert", "page":"stanleySymmetric", "name":"Involution Stanley symmetric functions", "space":"Sym", "symbol":"$\\stanleySymInvolution_{y}(\\xvec)$" },{ "href":"stanleySymmetric.htm#stanleySymAffineSkew", "year":"2006", "id":"stanleySymAffineSkew", "category":"Schubert", "page":"stanleySymmetric", "name":"Skew affine Stanley symmetric functions", "space":"Sym", "symbol":"$\\stanleySymAffine_{w/v}(\\xvec)$" },{ "href":"grothendieck.htm#grothendieckStable", "year":"1994", "id":"grothendieckStable", "category":"Schubert", "page":"grothendieck", "name":"Stable Grothendieck polynomials", "space":"Sym", "symbol":"$\\grothendieckStable_\\omega(\\xvec)$" },{ "href":"stanleySymmetric.htm#stanleySym", "year":"1984", "id":"stanleySym", "category":"Schubert", "page":"stanleySymmetric", "name":"Stanley symmetric functions", "space":"Sym", "symbol":"$\\stanleySym_\\omega(\\xvec)$" },{ "href":"stanleySymmetric.htm#stanleySymBC", "year":"1996", "id":"stanleySymBC", "category":"Schubert", "page":"stanleySymmetric", "name":"Type B/C Stanley symmetric functions", "space":"Sym", "symbol":"$\\stanleySym^B_\\omega(\\xvec)$" },{ "href":"stanleySymmetric.htm#schurAffine", "year":"2006", "id":"schurAffine", "category":"Schur", "page":"stanleySymmetric", "name":"Affine Schur functions", "space":"Sym", "symbol":"$\\stanleySymAffine_{\\lambda}(\\xvec)$" },{ "href":"superSymmetricSchur.htm#bigSchur", "year":"2017", "id":"bigSchur", "category":"Schur", "page":"superSymmetricSchur", "name":"Big Schur functions", "space":"Sym", "symbol":"" },{ "href":"kschur.htm#catalanSymmetric", "year":"2010", "id":"catalanSymmetric", "category":"Schur", "page":"kschur", "name":"Catalan symmetric functions", "space":"Sym", "symbol":"$\\catalanH_{\\Phi,\\gamma}(\\xvec;t)$" },{ "href":"grothendieck.htm#grothendieckDualStable", "year":"2007", "id":"grothendieckDualStable", "category":"Schur", "page":"grothendieck", "name":"Dual stable Grothendieck polynomials", "space":"Sym", "symbol":"$\\grothendieckDual_\\lambda(\\xvec)$" },{ "href":"schurShifted.htm#schurFactorial", "year":"1989", "id":"schurFactorial", "category":"Schur", "page":"schurShifted", "name":"Factorial Schur polynomials", "space":"Sym", "symbol":"$\\schurFactorial_\\mu(\\xvec)$" },{ "href":"schurMisc.htm#generalizedSchurP", "year":"2019", "id":"generalizedSchurP", "category":"Schur", "page":"schurMisc", "name":"Generalized Schur $P$-functions", "space":"Sym", "symbol":"$P^F_\\lambda(\\xvec)$" },{ "href":"schurMisc.htm#generalizedSchurQ", "year":"2019", "id":"generalizedSchurQ", "category":"Schur", "page":"schurMisc", "name":"Generalized Schur $Q$-functions", "space":"Sym", "symbol":"$Q^F_\\lambda(\\xvec)$" },{ "href":"grothendieck.htm#grothendieckRefinedDualStable", "year":"2016", "id":"grothendieckRefinedDualStable", "category":"Schur", "page":"grothendieck", "name":"Generalized and refined dual stable Grothendieck polynomials", "space":"Sym", "symbol":"$\\grothendieckRefinedDualStable_\\lambda(\\xvec)$" },{ "href":"hallLittlewood.htm#hallLittlewoodS", "year":"2024", "id":"hallLittlewoodS", "category":"Schur", "page":"hallLittlewood", "name":"Hall--Littlewood $S$-functions", "space":"Sym", "symbol":"$S_{\\lambda/\\mu}(X;t)$" },{ "href":"hallLittlewood.htm#hallLittlewoodP", "year":"1961", "id":"hallLittlewoodP", "category":"Schur", "page":"hallLittlewood", "name":"Hall--Littlewood P", "space":"Sym", "symbol":"$\\hallLittlewoodP_\\lambda(\\xvec;t)$" },{ "href":"hallLittlewood.htm#hallLittlewoodQ", "year":"1961", "id":"hallLittlewoodQ", "category":"Schur", "page":"hallLittlewood", "name":"Hall--Littlewood Q", "space":"Sym", "symbol":"$\\hallLittlewoodQ_\\lambda(\\xvec;t)$" },{ "href":"diagonalHarmonics.htm#hikita", "year":"2014", "id":"hikita", "category":"Schur", "page":"diagonalHarmonics", "name":"Hikita polynomials", "space":"Sym", "symbol":"$H_{m,n}(\\xvec;q,t)$" },{ "href":"jack.htm#jackJ", "year":"1970", "id":"jackJ", "category":"Schur", "page":"jack", "name":"Jack J polynomials", "space":"Sym", "symbol":"$\\jackJ_\\lambda(\\xvec)$" },{ "href":"jack.htm#jackP", "year":"1970", "id":"jackP", "category":"Schur", "page":"jack", "name":"Jack P polynomials", "space":"Sym", "symbol":"$\\jackP_\\lambda(\\xvec)$" },{ "href":"schurKP.htm#schurKP", "year":"2013", "id":"schurKP", "category":"Schur", "page":"schurKP", "name":"K-theoretic Schur P-functions", "space":"Sym", "symbol":"$\\schurKP_{\\lambda}(\\xvec)$" },{ "href":"schurKP.htm#schurKQ", "year":"2013", "id":"schurKQ", "category":"Schur", "page":"schurKP", "name":"K-theoretic Schur Q-functions", "space":"Sym", "symbol":"$\\schurKQ_{\\lambda}(\\xvec)$" },{ "href":"kschur.htm#KatalanSymmetric", "year":"2020", "id":"KatalanSymmetric", "category":"Schur", "page":"kschur", "name":"Katalan symmetric functions", "space":"Sym", "symbol":"$\\catalanH_{\\Phi,M,\\gamma}(\\xvec)$" },{ "href":"llt.htm#LLT", "year":"1997", "id":"LLT", "category":"Schur", "page":"llt", "name":"LLT polynomials", "space":"Sym", "symbol":"$\\LLT_\\nuvec(\\xvec;q)$" },{ "href":"macdonaldP.htm#macdonaldJ", "year":"1987", "id":"macdonaldJ", "category":"Schur", "page":"macdonaldP", "name":"Macdonald J polynomials", "space":"Sym", "symbol":"$\\macdonaldJ_\\lambda(\\xvec;q,t)$" },{ "href":"macdonaldP.htm#macdonaldP", "year":"1987", "id":"macdonaldP", "category":"Schur", "page":"macdonaldP", "name":"Macdonald P polynomials", "space":"Sym", "symbol":"$\\macdonaldP_\\lambda(\\xvec;q,t)$" },{ "href":"macdonaldP.htm#macdonaldPqq2", "year":"2020", "id":"macdonaldPqq2", "category":"Schur", "page":"macdonaldP", "name":"Macdonald P polynomials at $(q,q^2)$", "space":"Sym", "symbol":"$\\macdonaldP_\\lambda(\\xvec;q,q^2)$" },{ "href":"hallLittlewood.htm#imathHallLittlewoodH", "year":"2025", "id":"imathHallLittlewoodH", "category":"Schur", "page":"hallLittlewood", "name":"Modified $\\imath$ Hall--Littlewood functions", "space":"Sym", "symbol":"$V^\\imath_\\lambda(\\xvec;t,\\theta)$" },{ "href":"hallLittlewood.htm#hallLittlewoodH", "year":"1961", "id":"hallLittlewoodH", "category":"Schur", "page":"hallLittlewood", "name":"Modified Hall--Littlewood polynomials", "space":"Sym", "symbol":"$\\hallLittlewoodH_\\lambda(\\xvec;t)$" },{ "href":"macdonaldH.htm#macdonaldH", "year":"1987", "id":"macdonaldH", "category":"Schur", "page":"macdonaldH", "name":"Modified Macdonald polynomials", "space":"Sym", "symbol":"$\\macdonaldH_\\lambda(\\xvec;q,t)$" },{ "href":"petrie.htm#modularSchur", "year":"1992", "id":"modularSchur", "category":"Schur", "page":"petrie", "name":"Modular Schur functions", "space":"Sym", "symbol":"" },{ "href":"pGrothendieck.htm#pGrothendieck", "year":"2023", "id":"pGrothendieck", "category":"Schur", "page":"pGrothendieck", "name":"P-Grothendieck polynomials", "space":"Sym", "symbol":"" },{ "href":"schurRibbon.htm#schurRibbon", "year":"1901", "id":"schurRibbon", "category":"Schur", "page":"schurRibbon", "name":"Ribbon Schur polynomials", "space":"Sym", "symbol":"" },{ "href":"schurMisc.htm#schurP", "year":"1911", "id":"schurP", "category":"Schur", "page":"schurMisc", "name":"Schur's $P$ functions", "space":"Sym", "symbol":"$\\schurP_\\lambda(\\xvec)$" },{ "href":"schurMisc.htm#schurQ", "year":"1911", "id":"schurQ", "category":"Schur", "page":"schurMisc", "name":"Schur's $Q$ functions", "space":"Sym", "symbol":"$\\schurQ_\\lambda(\\xvec)$" },{ "href":"schur.htm#schurSkew", "year":"1901", "id":"schurSkew", "category":"Schur", "page":"schur", "name":"Skew Schur polynomials", "space":"Sym", "symbol":"" },{ "href":"stanleySymmetric.htm#schurAffineSkew", "year":"2006", "id":"schurAffineSkew", "category":"Schur", "page":"stanleySymmetric", "name":"Skew affine Schur functions", "space":"Sym", "symbol":"$\\stanleySymAffine_{\\lambda/\\mu}(\\xvec)$" },{ "href":"hallLittlewood.htm#hallLittlewoodT", "year":"1961", "id":"hallLittlewoodT", "category":"Schur", "page":"hallLittlewood", "name":"Transformed Hall--Littlewood", "space":"Sym", "symbol":"$\\hallLittlewoodT_\\lambda(\\xvec;t)$" },{ "href":"hallLittlewood.htm#typeCUniversalCharacter", "year":"2022", "id":"typeCUniversalCharacter", "category":"Schur", "page":"hallLittlewood", "name":"Type $C$ universal characters", "space":"Sym", "symbol":"$s^C_\\lambda$" },{ "href":"unicellular-llt.htm#unicellularLLT", "year":"2005", "id":"unicellularLLT", "category":"Schur", "page":"unicellular-llt", "name":"Unicellular LLT polynomials", "space":"Sym", "symbol":"$\\LLT_\\avec(\\xvec;q)$" },{ "href":"zonal.htm#zonal", "year":"1963", "id":"zonal", "category":"Schur", "page":"zonal", "name":"Zonal symmetric functions", "space":"Sym", "symbol":"$\\zonal_\\lambda(\\xvec)$" },{ "href":"cycleIndexPolynomial.htm#lyndonSymmetric", "year":"1993", "id":"lyndonSymmetric", "category":"Sym", "page":"cycleIndexPolynomial", "name":"Lyndon symmetric functions", "space":"Sym", "symbol":"$L_\\lambda(\\xvec)$" },{ "href":"kschur.htm#kSchur", "year":"2003", "id":"kSchur", "category":"Schur", "page":"kschur", "name":"$k$-Schur polynomials", "space":"Sym*", "symbol":"$\\kSchur^{(k)}_\\lambda(\\xvec)$" }], "graph_page":"polynomial-relations.htm", "edges":[{ "label":"Contains", "source_index":1, "id":"e1", "transitive":true, "type":"contains", "source":"KatalanSymmetric", "refs":[{ "key":"BlasiakMorseSeelinger2020", "href":"#BlasiakMorseSeelinger2020", "label":"BMS20", "tooltip":"Blasiak, Morse and Seelinger, $K$-theoretic Catalan functions (2020)" }], "poset":true, "target":"catalanSymmetric", "status":"theorem" },{ "label":"KTheoreticAnalogueOf", "source_index":2, "id":"e2", "transitive":false, "type":"k_theoretic_analogue_of", "source":"KatalanSymmetric", "refs":[{ "key":"BlasiakMorseSeelinger2020", "href":"#BlasiakMorseSeelinger2020", "label":"BMS20", "tooltip":"Blasiak, Morse and Seelinger, $K$-theoretic Catalan functions (2020)" }], "poset":false, "target":"catalanSymmetric", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e3", "transitive":true, "type":"contains", "source":"LLT", "refs":[{ "key":"Lascoux97ribbontableaux", "href":"#Lascoux97ribbontableaux", "label":"LLT97", "tooltip":"Lascoux, Leclerc and Thibon, Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties (1997)" }], "poset":true, "target":"hallLittlewoodH", "status":"theorem" },{ "label":"Contains", "source_index":2, "id":"e4", "transitive":true, "type":"contains", "source":"LLT", "refs":[{ "key":"Lascoux97ribbontableaux", "href":"#Lascoux97ribbontableaux", "label":"LLT97", "tooltip":"Lascoux, Leclerc and Thibon, Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties (1997)" }], "poset":true, "target":"hallLittlewoodT", "status":"theorem" },{ "label":"Contains", "source_index":4, "id":"e5", "transitive":true, "type":"contains", "source":"LLT", "refs":[{ "key":"Lascoux97ribbontableaux", "href":"#Lascoux97ribbontableaux", "label":"LLT97", "tooltip":"Lascoux, Leclerc and Thibon, Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties (1997)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"Contains", "source_index":3, "id":"e6", "transitive":true, "type":"contains", "source":"LLT", "refs":[{ "key":"Lascoux97ribbontableaux", "href":"#Lascoux97ribbontableaux", "label":"LLT97", "tooltip":"Lascoux, Leclerc and Thibon, Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties (1997)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "label":"Contains", "source_index":5, "id":"e7", "transitive":true, "type":"contains", "source":"LLT", "refs":[{ "key":"HaglundHaimanLoehr2005", "href":"#HaglundHaimanLoehr2005", "label":"HHL05", "tooltip":"Haglund, Haiman and Loehr, A combinatorial formula for Macdonald polynomials (2005)" }], "poset":true, "target":"unicellularLLT", "status":"theorem" },{ "attrs":{ "combinatorial_rule":"none_known" }, "label":"PositiveIn", "source_index":6, "id":"e8", "transitive":true, "type":"positive_in", "source":"LLT", "refs":[{ "key":"Lascoux97ribbontableaux", "href":"#Lascoux97ribbontableaux", "label":"LLT97", "tooltip":"Lascoux, Leclerc and Thibon, Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties (1997)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"finite-variable nonsymmetric edge" }, "label":"Generalizes", "source_index":1, "id":"e9", "transitive":true, "type":"generalizes", "source":"almostSymmetricSchur", "refs":[{ "key":"Weising2024AlmostSymmetricx", "href":"#Weising2024AlmostSymmetricx", "label":"Wei24", "tooltip":"Weising, Almost Symmetric Schur Functions (2024)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "scope":"fully symmetric edge" }, "label":"Generalizes", "source_index":2, "id":"e10", "transitive":true, "type":"generalizes", "source":"almostSymmetricSchur", "refs":[{ "key":"Weising2024AlmostSymmetricx", "href":"#Weising2024AlmostSymmetricx", "label":"Wei24", "tooltip":"Weising, Almost Symmetric Schur Functions (2024)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.14 and Cor. 3.11" }, "label":"PositiveIn", "source_index":2, "id":"e11", "transitive":true, "type":"positive_in", "source":"atom", "refs":[{ "key":"Searles2019PolynomialBases", "href":"#Searles2019PolynomialBases", "label":"Sea19", "tooltip":"Searles, Polynomial bases: Positivity and Schur multiplication (2019)" },{ "key":"HicksNiese2024x", "href":"#HicksNiese2024x", "label":"HN24", "tooltip":"Hicks and Niese, Quasisymmetric divided difference operators and polynomial bases (2024)" }], "poset":true, "target":"fundamentalParticle", "status":"theorem" },{ "attrs":{ "proof":"semistandard augmented fillings", "scope":"nonsymmetric monomials" }, "label":"PositiveIn", "source_index":1, "id":"e12", "transitive":true, "type":"positive_in", "source":"atom", "refs":[{ "key":"Mason2009", "href":"#Mason2009", "label":"Mas09", "tooltip":"Mason, An explicit construction of type A Demazure atoms (2009)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e13", "transitive":true, "type":"positive_in", "source":"augmentedMonomial", "refs":[{ "key":"Merca2015", "href":"#Merca2015", "label":"Mer15", "tooltip":"Merca, Augmented monomials in terms of power sums (2015)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "map":"omega and reverse index" }, "label":"TransformsTo", "source_index":1, "id":"e14", "transitive":false, "type":"transforms_to", "source":"backwardExtSchur", "refs":[{ "key":"Daugherty2024Extended", "href":"#Daugherty2024Extended", "label":"Dau24", "tooltip":"Daugherty, Extended Schur functions and bases related by involutions (2024)" }], "poset":false, "target":"extSchur", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e15", "transitive":true, "type":"positive_in", "source":"bigSchur", "refs":[{ "key":"Shigechi2017x", "href":"#Shigechi2017x", "label":"Shi17", "tooltip":"Shigechi, Shifted tableaux and products of Schur’s symmetric functions (2017)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e16", "transitive":true, "type":"positive_in", "source":"booleanProduct", "refs":[{ "key":"BilleyRhoadesTewari2019", "href":"#BilleyRhoadesTewari2019", "label":"BRT19", "tooltip":"Billey, Rhoades and Tewari, Boolean product polynomials, Schur positivity, and Chern plethysm (2019)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"full root ideal" }, "label":"Contains", "source_index":1, "id":"e17", "transitive":true, "type":"contains", "source":"catalanSymmetric", "refs":[{ "key":"Panyushev2010", "href":"#Panyushev2010", "label":"Pan10", "tooltip":"Panyushev, Generalised Kostka–Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles (2010)" }], "poset":true, "target":"hallLittlewoodT", "status":"theorem" },{ "attrs":{ "semiring":"NN[t]" }, "label":"PositiveIn", "source_index":3, "id":"e18", "transitive":true, "type":"positive_in", "source":"catalanSymmetric", "refs":[{ "key":"BlasiakMorsePun2020x", "href":"#BlasiakMorsePun2020x", "label":"BMP20", "tooltip":"Blasiak, Morse and Pun, Demazure crystals and the Schur positivity of Catalan functions (2020)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"SpecializesTo", "source_index":2, "id":"e19", "transitive":true, "type":"specializes_to", "source":"catalanSymmetric", "refs":[{ "key":"BlasiakMorsePunSummers2018", "href":"#BlasiakMorsePunSummers2018", "label":"BMPS18", "tooltip":"Blasiak, Morse, Pun and Summers, $k$-Schur expansions of Catalan functions (2018)" }], "poset":true, "target":"kSchur", "status":"theorem" },{ "attrs":{ "scope":"complete multipartite graphs at q=1" }, "label":"Contains", "source_index":6, "id":"e20", "transitive":true, "type":"contains", "source":"chromaticQuasisymmetric", "refs":[{ "key":"CrewSpirkl2020x", "href":"#CrewSpirkl2020x", "label":"CS21", "tooltip":"Crew and Spirkl, A Complete Multipartite Basis for the Chromatic Symmetric Function (2021)" }], "poset":true, "target":"completeMultipartite", "status":"theorem" },{ "attrs":{ "scope":"(3+1)-free incomparability graphs at q=1", "semiring":"NN" }, "label":"PositiveIn", "source_index":5, "id":"e21", "transitive":true, "type":"positive_in", "source":"chromaticQuasisymmetric", "refs":[{ "key":"Hikita2024x", "href":"#Hikita2024x", "label":"Hik24", "tooltip":"Hikita, A proof of the Stanley–Stembridge conjecture (2024)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "attrs":{ "semiring":"NN[q]" }, "label":"PositiveIn", "source_index":1, "id":"e22", "transitive":true, "type":"positive_in", "source":"chromaticQuasisymmetric", "refs":[{ "key":"ShareshianWachs2012", "href":"#ShareshianWachs2012", "label":"SW12", "tooltip":"Shareshian and Wachs, Chromatic quasisymmetric functions and Hessenberg varieties (2012)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "scope":"after applying omega, when symmetric" }, "label":"PositiveIn", "source_index":3, "id":"e23", "transitive":true, "type":"positive_in", "source":"chromaticQuasisymmetric", "refs":[{ "key":"AlexanderssonSulzgruber2019", "href":"#AlexanderssonSulzgruber2019", "label":"AS19", "tooltip":"Alexandersson and Sulzgruber, P-partitions and p-positivity (2019)" },{ "key":"Ellzey2017", "href":"#Ellzey2017", "label":"Ell17", "tooltip":"Ellzey, A directed graph generalization of chromatic quasisymmetric functions (2017)" },{ "key":"Athanasiadis2015", "href":"#Athanasiadis2015", "label":"Ath15", "tooltip":"Athanasiadis, Power sum expansion of chromatic quasisymmetric functions (2015)" }], "poset":true, "target":"powerSum", "status":"theorem" },{ "attrs":{ "scope":"after applying omega" }, "label":"PositiveIn", "source_index":2, "id":"e24", "transitive":true, "type":"positive_in", "source":"chromaticQuasisymmetric", "refs":[{ "key":"AlexanderssonSulzgruber2019", "href":"#AlexanderssonSulzgruber2019", "label":"AS19", "tooltip":"Alexandersson and Sulzgruber, P-partitions and p-positivity (2019)" }], "poset":true, "target":"qPsi", "status":"theorem" },{ "attrs":{ "scope":"(3+1)-free incomparability graphs at q=1 and unit interval graphs with q parameter" }, "label":"PositiveIn", "source_index":4, "id":"e25", "transitive":true, "type":"positive_in", "source":"chromaticQuasisymmetric", "refs":[{ "key":"Gasharov1996", "href":"#Gasharov1996", "label":"Gas96", "tooltip":"Gasharov, Incomparability graphs of $(3+1)$-free posets are $s$-positive (1996)" },{ "key":"ShareshianWachs2012", "href":"#ShareshianWachs2012", "label":"SW12", "tooltip":"Shareshian and Wachs, Chromatic quasisymmetric functions and Hessenberg varieties (2012)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "sign_rule":"maximal stable partitions", "proof":"Cor.~3", "scope":"chromatic symmetric functions at q=1" }, "label":"SignedIn", "source_index":7, "id":"e26", "transitive":false, "type":"signed_in", "source":"chromaticQuasisymmetric", "refs":[{ "key":"CrewSpirkl2020x", "href":"#CrewSpirkl2020x", "label":"CS21", "tooltip":"Crew and Spirkl, A Complete Multipartite Basis for the Chromatic Symmetric Function (2021)" }], "poset":false, "target":"completeMultipartite", "status":"theorem" },{ "attrs":{ "proof":"Thm. 4.13" }, "label":"PositiveIn", "source_index":3, "id":"e27", "transitive":true, "type":"positive_in", "source":"completeFlaggedHomogeneous", "refs":[{ "key":"Ehrhard2023x", "href":"#Ehrhard2023x", "label":"Ehr23", "tooltip":"Ehrhard, Complete flagged homogeneous polynomials (2023)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "proof":"Thm. 4.12" }, "label":"PositiveIn", "source_index":2, "id":"e28", "transitive":true, "type":"positive_in", "source":"completeFlaggedHomogeneous", "refs":[{ "key":"Ehrhard2023x", "href":"#Ehrhard2023x", "label":"Ehr23", "tooltip":"Ehrhard, Complete flagged homogeneous polynomials (2023)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "proof":"Prop. 2.3" }, "label":"StableLimit", "source_index":1, "id":"e29", "transitive":true, "type":"stable_limit", "source":"completeFlaggedHomogeneous", "refs":[{ "key":"Ehrhard2023x", "href":"#Ehrhard2023x", "label":"Ehr23", "tooltip":"Ehrhard, Complete flagged homogeneous polynomials (2023)" }], "poset":false, "target":"completeH", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e30", "transitive":true, "type":"positive_in", "source":"completeH", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "semiring":"QQ_nonnegative" }, "label":"PositiveIn", "source_index":2, "id":"e31", "transitive":true, "type":"positive_in", "source":"completeH", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":true, "target":"powerSum", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e32", "transitive":true, "type":"positive_in", "source":"completeH", "refs":[{ "key":"Sagan2001", "href":"#Sagan2001", "label":"Sag01", "tooltip":"Sagan, The symmetric group (2001)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e33", "transitive":true, "type":"positive_in", "source":"cycleIndexPolynomial", "refs":[{ "key":"LoehrWarrington2019", "href":"#LoehrWarrington2019", "label":"LW19", "tooltip":"Loehr and Warrington, Quasisymmetric and Schur expansions of cycle index polynomials (2019)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e34", "transitive":true, "type":"positive_in", "source":"diSchur", "refs":[{ "key":"BergBergeronSaliolaSerranoZabrocki2014", "href":"#BergBergeronSaliolaSerranoZabrocki2014", "label":"BBSS+14", "tooltip":"Berg, Bergeron, Saliola, Serrano and Zabrocki, A lift of the Schur and Hall–Littlewood bases to non-commutative symmetric functions (2014)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "proof":"combinatorial formula and 0-Hecke filtration" }, "label":"PositiveIn", "source_index":2, "id":"e35", "transitive":true, "type":"positive_in", "source":"diSchur", "refs":[{ "key":"AllenHallamMason2018", "href":"#AllenHallamMason2018", "label":"AHM18", "tooltip":"Allen, Hallam and Mason, Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions (2018)" },{ "key":"LeeOh2025x", "href":"#LeeOh2025x", "label":"LO25", "tooltip":"Lee and Oh, Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions (2025)" }], "poset":true, "target":"yqSchur", "status":"theorem" },{ "attrs":{ "map":"psi" }, "label":"TransformsTo", "source_index":3, "id":"e36", "transitive":false, "type":"transforms_to", "source":"diSchur", "refs":[{ "key":"NieseSundaramWilligenburgVegaWang2023", "href":"#NieseSundaramWilligenburgVegaWang2023", "label":"NSWV+23", "tooltip":"Niese, Sundaram, Willigenburg, Vega and Wang, Row-strict dual immaculate functions (2023)" }], "poset":false, "target":"rsdiSchur", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e37", "transitive":true, "type":"positive_in", "source":"elementaryE", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e38", "transitive":true, "type":"positive_in", "source":"elementaryE", "refs":[{ "key":"Sagan2001", "href":"#Sagan2001", "label":"Sag01", "tooltip":"Sagan, The symmetric group (2001)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e39", "transitive":true, "type":"positive_in", "source":"eulerianSymmetric", "refs":[{ "key":"ShareshianWachs2010", "href":"#ShareshianWachs2010", "label":"SW10", "tooltip":"Shareshian and Wachs, Eulerian quasisymmetric functions (2010)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "scope":"lambda=(n)" }, "label":"PositiveIn", "source_index":3, "id":"e40", "transitive":true, "type":"positive_in", "source":"eulerianSymmetric", "refs":[{ "key":"SaganShareshianWachs2011", "href":"#SaganShareshianWachs2011", "label":"SSW11", "tooltip":"Sagan, Shareshian and Wachs, Eulerian quasisymmetric functions and cyclic sieving (2011)" }], "poset":true, "target":"powerSum", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e41", "transitive":true, "type":"positive_in", "source":"eulerianSymmetric", "refs":[{ "key":"HendersonWachs2012", "href":"#HendersonWachs2012", "label":"HW12", "tooltip":"Henderson and Wachs, Unimodality of Eulerian quasisymmetric functions (2012)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"partition compositions" }, "label":"Contains", "source_index":1, "id":"e42", "transitive":true, "type":"contains", "source":"extSchur", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e43", "transitive":true, "type":"positive_in", "source":"extSchur", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "scope":"stable Kohnert limit" }, "label":"PositiveIn", "source_index":2, "id":"e44", "transitive":true, "type":"positive_in", "source":"extSchur", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "attrs":{ "map":"omega and reverse index" }, "label":"TransformsTo", "source_index":6, "id":"e45", "transitive":false, "type":"transforms_to", "source":"extSchur", "refs":[{ "key":"Daugherty2024Extended", "href":"#Daugherty2024Extended", "label":"Dau24", "tooltip":"Daugherty, Extended Schur functions and bases related by involutions (2024)" }], "poset":false, "target":"backwardExtSchur", "status":"theorem" },{ "attrs":{ "map":"rho and reverse index" }, "label":"TransformsTo", "source_index":5, "id":"e46", "transitive":false, "type":"transforms_to", "source":"extSchur", "refs":[{ "key":"Daugherty2024Extended", "href":"#Daugherty2024Extended", "label":"Dau24", "tooltip":"Daugherty, Extended Schur functions and bases related by involutions (2024)" }], "poset":false, "target":"flippedExtSchur", "status":"theorem" },{ "attrs":{ "map":"psi" }, "label":"TransformsTo", "source_index":4, "id":"e47", "transitive":false, "type":"transforms_to", "source":"extSchur", "refs":[{ "key":"Daugherty2024Extended", "href":"#Daugherty2024Extended", "label":"Dau24", "tooltip":"Daugherty, Extended Schur functions and bases related by involutions (2024)" },{ "key":"NieseSundaramWilligenburgVegaWang2023", "href":"#NieseSundaramWilligenburgVegaWang2023", "label":"NSWV+23", "tooltip":"Niese, Sundaram, Willigenburg, Vega and Wang, Row-strict dual immaculate functions (2023)" }], "poset":false, "target":"rsExtSchur", "status":"theorem" },{ "label":"Generalizes", "source_index":1, "id":"e48", "transitive":true, "type":"generalizes", "source":"flaggedLLT", "refs":[{ "key":"BlasiakHaimanMorsePunSeelinger2025x", "href":"#BlasiakHaimanMorsePunSeelinger2025x", "label":"BHMP+25", "tooltip":"Blasiak, Haiman, Morse, Pun and Seelinger, Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials (2025)" }], "poset":true, "target":"LLT", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e49", "transitive":true, "type":"positive_in", "source":"flaggedLLT", "refs":[{ "key":"BlasiakHaimanMorsePunSeelinger2025x", "href":"#BlasiakHaimanMorsePunSeelinger2025x", "label":"BHMP+25", "tooltip":"Blasiak, Haiman, Morse, Pun and Seelinger, Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials (2025)" }], "poset":true, "target":"atom", "status":"conjecture" },{ "attrs":{ "sign_rule":"K-theoretic signs" }, "label":"SignedIn", "source_index":1, "id":"e50", "transitive":false, "type":"signed_in", "source":"flaggedRefinedSkewStableGrothendieck", "refs":[{ "key":"Kundu2024x", "href":"#Kundu2024x", "label":"Kun24", "tooltip":"Kundu, Key expansion of the flagged refined skew stable Grothendieck polynomial (2024)" }], "poset":false, "target":"key", "status":"theorem" },{ "attrs":{ "map":"rho and reverse index" }, "label":"TransformsTo", "source_index":1, "id":"e51", "transitive":false, "type":"transforms_to", "source":"flippedExtSchur", "refs":[{ "key":"Daugherty2024Extended", "href":"#Daugherty2024Extended", "label":"Dau24", "tooltip":"Daugherty, Extended Schur functions and bases related by involutions (2024)" }], "poset":false, "target":"extSchur", "status":"theorem" },{ "attrs":{ "proof":"Thm. 1.3" }, "label":"PositiveIn", "source_index":1, "id":"e52", "transitive":true, "type":"positive_in", "source":"forestPolynomials", "refs":[{ "key":"NadeauTewari2023x", "href":"#NadeauTewari2023x", "label":"NT23", "tooltip":"Nadeau and Tewari, Forest polynomials and the class of the permutahedral variety (2023)" }], "poset":true, "target":"slideF", "status":"theorem" },{ "attrs":{ "pairing":"Hall inner product" }, "label":"DualTo", "source_index":2, "id":"e53", "transitive":false, "type":"dual_to", "source":"forgotten", "refs":[{ "key":"StanleyEC2", "href":"#StanleyEC2", "label":"Sta01", "tooltip":"Stanley, Enumerative Combinatorics: Volume 2 (2001)" }], "poset":false, "target":"elementaryE", "status":"theorem" },{ "attrs":{ "map":"omega" }, "label":"TransformsTo", "source_index":1, "id":"e54", "transitive":false, "type":"transforms_to", "source":"forgotten", "refs":[{ "key":"StanleyEC2", "href":"#StanleyEC2", "label":"Sta01", "tooltip":"Stanley, Enumerative Combinatorics: Volume 2 (2001)" }], "poset":false, "target":"monomial", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.9 and Cor. 3.11", "scope":"nonsymmetric monomials" }, "label":"PositiveIn", "source_index":1, "id":"e55", "transitive":true, "type":"positive_in", "source":"fundamentalParticle", "refs":[{ "key":"HicksNiese2024x", "href":"#HicksNiese2024x", "label":"HN24", "tooltip":"Hicks and Niese, Quasisymmetric divided difference operators and polynomial bases (2024)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "scope":"identity basement" }, "label":"Contains", "source_index":1, "id":"e56", "transitive":true, "type":"contains", "source":"generalizedDemazureAtoms", "refs":[{ "key":"HaglundMasonRemmel2012", "href":"#HaglundMasonRemmel2012", "label":"HMR12", "tooltip":"Haglund, Mason and Remmel, Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm (2012)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "scope":"reverse-identity basement" }, "label":"Contains", "source_index":2, "id":"e57", "transitive":true, "type":"contains", "source":"generalizedDemazureAtoms", "refs":[{ "key":"HaglundMasonRemmel2012", "href":"#HaglundMasonRemmel2012", "label":"HMR12", "tooltip":"Haglund, Mason and Remmel, Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm (2012)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e58", "transitive":true, "type":"positive_in", "source":"generalizedDemazureAtoms", "refs":[{ "key":"HaglundMasonRemmel2012", "href":"#HaglundMasonRemmel2012", "label":"HMR12", "tooltip":"Haglund, Mason and Remmel, Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm (2012)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "scope":"$f_d(u)=u^d$ specialization" }, "label":"Generalizes", "source_index":1, "id":"e59", "transitive":true, "type":"generalizes", "source":"generalizedSchurP", "refs":[{ "key":"Okada2019x", "href":"#Okada2019x", "label":"Oka19", "tooltip":"Okada, A Generalization of Schur’s ${P}$- and ${Q}$-Functions (2019)" }], "poset":true, "target":"schurP", "status":"theorem" },{ "attrs":{ "scope":"$f_d(u)=2u^d$ specialization" }, "label":"Generalizes", "source_index":1, "id":"e60", "transitive":true, "type":"generalizes", "source":"generalizedSchurQ", "refs":[{ "key":"Okada2019x", "href":"#Okada2019x", "label":"Oka19", "tooltip":"Okada, A Generalization of Schur’s ${P}$- and ${Q}$-Functions (2019)" }], "poset":true, "target":"schurQ", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e61", "transitive":true, "type":"positive_in", "source":"gessel", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "attrs":{ "scope":"Grassmannian permutations" }, "label":"Contains", "source_index":1, "id":"e62", "transitive":true, "type":"contains", "source":"grothendieck", "refs":[{ "key":"Lascoux1990Grothendieck", "href":"#Lascoux1990Grothendieck", "label":"Las90", "tooltip":"Lascoux, Anneau de Grothendieck de la variété de drapeaux (1990)" }], "poset":true, "target":"grothendieckGrassmann", "status":"theorem" },{ "attrs":{ "parameter":"beta" }, "label":"KTheoreticAnalogueOf", "source_index":3, "id":"e63", "transitive":false, "type":"k_theoretic_analogue_of", "source":"grothendieck", "refs":[{ "key":"Lascoux1990Grothendieck", "href":"#Lascoux1990Grothendieck", "label":"Las90", "tooltip":"Lascoux, Anneau de Grothendieck de la variété de drapeaux (1990)" }], "poset":false, "target":"schubert", "status":"theorem" },{ "attrs":{ "proof":"Thm. 1.9" }, "label":"PositiveIn", "source_index":4, "id":"e64", "transitive":true, "type":"positive_in", "source":"grothendieck", "refs":[{ "key":"ShimozonoYu2021", "href":"#ShimozonoYu2021", "label":"SY23", "tooltip":"Shimozono and Yu, Grothendieck-to-Lascoux expansions (2023)" }], "poset":true, "target":"lascoux", "status":"theorem" },{ "label":"StableLimit", "source_index":2, "id":"e65", "transitive":true, "type":"stable_limit", "source":"grothendieck", "refs":[{ "key":"FominKirillov1994grothendieck", "href":"#FominKirillov1994grothendieck", "label":"FK94", "tooltip":"Fomin and Kirillov, Grothendieck polynomials and the Yang–Baxter equation (1994)" }], "poset":false, "target":"grothendieckStable", "status":"theorem" },{ "attrs":{ "map":"(a,b)=(0,-1)" }, "label":"SpecializesTo", "source_index":2, "id":"e66", "transitive":true, "type":"specializes_to", "source":"grothendieckCanonical", "refs":[{ "key":"Yeliussizov2017", "href":"#Yeliussizov2017", "label":"Yel17", "tooltip":"Yeliussizov, Duality and deformations of stable Grothendieck polynomials (2017)" }], "poset":true, "target":"grothendieckStable", "status":"theorem" },{ "attrs":{ "map":"(a,b)=(0,0)" }, "label":"SpecializesTo", "source_index":1, "id":"e67", "transitive":true, "type":"specializes_to", "source":"grothendieckCanonical", "refs":[{ "key":"Yeliussizov2017", "href":"#Yeliussizov2017", "label":"Yel17", "tooltip":"Yeliussizov, Duality and deformations of stable Grothendieck polynomials (2017)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"single variables" }, "label":"SpecializesTo", "source_index":1, "id":"e68", "transitive":true, "type":"specializes_to", "source":"grothendieckDouble", "refs":[{ "key":"FominKirillov1994grothendieck", "href":"#FominKirillov1994grothendieck", "label":"FK94", "tooltip":"Fomin and Kirillov, Grothendieck polynomials and the Yang–Baxter equation (1994)" }], "poset":true, "target":"grothendieck", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e69", "transitive":true, "type":"positive_in", "source":"grothendieckDualStable", "refs":[{ "key":"LamPylyavskyy2007combinatorialhopf", "href":"#LamPylyavskyy2007combinatorialhopf", "label":"LP07", "tooltip":"Lam and Pylyavskyy, Combinatorial Hopf Algebras and K-homology of Grassmanians (2007)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"t_i=1" }, "label":"RefinedBy", "source_index":1, "id":"e70", "transitive":true, "type":"refined_by", "source":"grothendieckDualStable", "refs":[{ "key":"GalashinGrinbergLiu2016", "href":"#GalashinGrinbergLiu2016", "label":"GGL16", "tooltip":"Galashin, Grinberg and Liu, Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions (2016)" }], "poset":false, "target":"grothendieckRefinedDualStable", "status":"theorem" },{ "label":"Generalizes", "source_index":1, "id":"e71", "transitive":true, "type":"generalizes", "source":"grothendieckFactorial", "refs":[{ "key":"MatsumuraSugimoto2019", "href":"#MatsumuraSugimoto2019", "label":"MS19", "tooltip":"Matsumura and Sugimoto, Factorial flagged Grothendieck polynomials (2019)" }], "poset":true, "target":"grothendieckGrassmann", "status":"theorem" },{ "attrs":{ "map":"forget flags" }, "label":"SpecializesTo", "source_index":2, "id":"e72", "transitive":true, "type":"specializes_to", "source":"grothendieckFactorial", "refs":[{ "key":"MatsumuraSugimoto2019", "href":"#MatsumuraSugimoto2019", "label":"MS19", "tooltip":"Matsumura and Sugimoto, Factorial flagged Grothendieck polynomials (2019)" }], "poset":true, "target":"grothendieckDouble", "status":"theorem" },{ "attrs":{ "map":"beta=0" }, "label":"SpecializesTo", "source_index":3, "id":"e73", "transitive":true, "type":"specializes_to", "source":"grothendieckFactorial", "refs":[{ "key":"MatsumuraSugimoto2019", "href":"#MatsumuraSugimoto2019", "label":"MS19", "tooltip":"Matsumura and Sugimoto, Factorial flagged Grothendieck polynomials (2019)" }], "poset":true, "target":"schurFactorial", "status":"theorem" },{ "attrs":{ "parameter":"beta" }, "label":"KTheoreticAnalogueOf", "source_index":2, "id":"e74", "transitive":false, "type":"k_theoretic_analogue_of", "source":"grothendieckGrassmann", "refs":[{ "key":"FominKirillov1994grothendieck", "href":"#FominKirillov1994grothendieck", "label":"FK94", "tooltip":"Fomin and Kirillov, Grothendieck polynomials and the Yang–Baxter equation (1994)" }], "poset":false, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"Thm. 1.2", "semiring":"NN[beta]" }, "label":"PositiveIn", "source_index":3, "id":"e75", "transitive":true, "type":"positive_in", "source":"grothendieckGrassmann", "refs":[{ "key":"Monical2017", "href":"#Monical2017", "label":"Mon17", "tooltip":"Monical, Set-valued skyline fillings (2017)" }], "poset":true, "target":"lascouxAtom", "status":"theorem" },{ "attrs":{ "map":"highest degree term" }, "label":"SpecializesTo", "source_index":1, "id":"e76", "transitive":true, "type":"specializes_to", "source":"grothendieckGrassmann", "refs":[{ "key":"FominKirillov1994grothendieck", "href":"#FominKirillov1994grothendieck", "label":"FK94", "tooltip":"Fomin and Kirillov, Grothendieck polynomials and the Yang–Baxter equation (1994)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"Grassmannian permutations" }, "label":"Contains", "source_index":1, "id":"e77", "transitive":true, "type":"contains", "source":"grothendieckStable", "refs":[{ "key":"FominKirillov1994grothendieck", "href":"#FominKirillov1994grothendieck", "label":"FK94", "tooltip":"Fomin and Kirillov, Grothendieck polynomials and the Yang–Baxter equation (1994)" }], "poset":true, "target":"grothendieckGrassmann", "status":"theorem" },{ "attrs":{ "parameter":"beta" }, "label":"KTheoreticAnalogueOf", "source_index":2, "id":"e78", "transitive":false, "type":"k_theoretic_analogue_of", "source":"grothendieckStable", "refs":[{ "key":"FominKirillov1994grothendieck", "href":"#FominKirillov1994grothendieck", "label":"FK94", "tooltip":"Fomin and Kirillov, Grothendieck polynomials and the Yang–Baxter equation (1994)" }], "poset":false, "target":"stanleySym", "status":"theorem" },{ "attrs":{ "proof":"Hecke insertion", "semiring":"NN" }, "label":"PositiveIn", "source_index":3, "id":"e79", "transitive":true, "type":"positive_in", "source":"grothendieckStable", "refs":[{ "key":"BuchKreschShimozonoTamvakisYong2007", "href":"#BuchKreschShimozonoTamvakisYong2007", "label":"BKST+07", "tooltip":"Buch, Kresch, Shimozono, Tamvakis and Yong, Stable Grothendieck polynomials and K-theoretic factor sequences (2007)" }], "poset":true, "target":"grothendieckGrassmann", "status":"theorem" },{ "label":"SignedIn", "source_index":4, "id":"e80", "transitive":false, "type":"signed_in", "source":"grothendieckStable", "refs":[{ "key":"Lascoux2001", "href":"#Lascoux2001", "label":"Las01", "tooltip":"Lascoux, Transitions on Grothendieck polynomials (2001)" }], "poset":false, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e81", "transitive":true, "type":"positive_in", "source":"hallLittlewoodH", "refs":[{ "key":"Littlewood1961", "href":"#Littlewood1961", "label":"Lit61", "tooltip":"Littlewood, On certain symmetric functions (1961)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"Eq. (4)", "sign_rule":"$(-1)^{|\\mu|-|\\lambda|}$" }, "label":"SignedIn", "source_index":2, "id":"e82", "transitive":false, "type":"signed_in", "source":"hallLittlewoodH", "refs":[{ "key":"BandlowMorse2010", "href":"#BandlowMorse2010", "label":"BM10", "tooltip":"Bandlow and Morse, The expansion of Hall–Littlewood functions in the dual Grothendieck polynomial basis (2010)" }], "poset":false, "target":"grothendieckDualStable", "status":"theorem" },{ "attrs":{ "map":"rescaling and plethystic transform" }, "label":"TransformsTo", "source_index":3, "id":"e83", "transitive":false, "type":"transforms_to", "source":"hallLittlewoodH", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":false, "target":"hallLittlewoodQ", "status":"theorem" },{ "attrs":{ "proof":"Cor.~30", "semiring":"NN[t]" }, "label":"PositiveIn", "source_index":5, "id":"e84", "transitive":true, "type":"positive_in", "source":"hallLittlewoodP", "refs":[{ "key":"Alexandersson2015gbMacdonald", "href":"#Alexandersson2015gbMacdonald", "label":"Ale19", "tooltip":"Alexandersson, Non-symmetric Macdonald polynomials and Demazure–Lusztig operators (2019)" }], "poset":true, "target":"permutedBasementTAtom", "status":"theorem" },{ "attrs":{ "proof":"sum over rearrangements" }, "label":"PositiveIn", "source_index":4, "id":"e85", "transitive":true, "type":"positive_in", "source":"hallLittlewoodP", "refs":[{ "key":"HaglundLuotoMasonWilligenburg2011", "href":"#HaglundLuotoMasonWilligenburg2011", "label":"HLMW11", "tooltip":"Haglund, Luoto, Mason and Willigenburg, Quasisymmetric Schur functions (2011)" },{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"tAtom", "status":"theorem" },{ "attrs":{ "map":"t=1" }, "label":"SpecializesTo", "source_index":1, "id":"e86", "transitive":true, "type":"specializes_to", "source":"hallLittlewoodP", "refs":[{ "key":"Littlewood1961", "href":"#Littlewood1961", "label":"Lit61", "tooltip":"Littlewood, On certain symmetric functions (1961)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "map":"t=-1" }, "label":"SpecializesTo", "source_index":3, "id":"e87", "transitive":true, "type":"specializes_to", "source":"hallLittlewoodP", "refs":[{ "key":"Stembridge1989", "href":"#Stembridge1989", "label":"Ste89", "tooltip":"Stembridge, Shifted tableaux and the projective representations of symmetric groups (1989)" }], "poset":true, "target":"schurP", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":2, "id":"e88", "transitive":true, "type":"specializes_to", "source":"hallLittlewoodP", "refs":[{ "key":"Littlewood1961", "href":"#Littlewood1961", "label":"Lit61", "tooltip":"Littlewood, On certain symmetric functions (1961)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":1, "id":"e89", "transitive":true, "type":"specializes_to", "source":"hallLittlewoodQ", "refs":[{ "key":"Littlewood1961", "href":"#Littlewood1961", "label":"Lit61", "tooltip":"Littlewood, On certain symmetric functions (1961)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":1, "id":"e90", "transitive":true, "type":"specializes_to", "source":"hallLittlewoodS", "refs":[{ "key":"GrinbergVassilieva2024x", "href":"#GrinbergVassilieva2024x", "label":"GV24", "tooltip":"Grinberg and Vassilieva, Quasisymmetric expansion of Hall-Littlewood symmetric functions (2024)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "attrs":{ "semiring":"NN[q]" }, "label":"PositiveIn", "source_index":1, "id":"e91", "transitive":true, "type":"positive_in", "source":"hallLittlewoodT", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"vertical-strip LLT polynomials" }, "label":"PositiveIn", "source_index":2, "id":"e92", "transitive":true, "type":"positive_in", "source":"hikita", "refs":[{ "key":"BergeronGarsiaLevenXin2015", "href":"#BergeronGarsiaLevenXin2015", "label":"BGLX15", "tooltip":"Bergeron, Garsia, Leven and Xin, Compositional $(km, kn)$-shuffle conjectures (2015)" }], "poset":true, "target":"LLT", "status":"theorem" },{ "attrs":{ "semiring":"NN[q,t]" }, "label":"PositiveIn", "source_index":1, "id":"e93", "transitive":true, "type":"positive_in", "source":"hikita", "refs":[{ "key":"Hikita2014", "href":"#Hikita2014", "label":"Hik14", "tooltip":"Hikita, Affine springer fibers of type A and combinatorics of diagonal coinvariants (2014)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "scope":"y alphabet empty" }, "label":"Contains", "source_index":1, "id":"e94", "transitive":true, "type":"contains", "source":"hookSchur", "refs":[{ "key":"YangRemmel1998", "href":"#YangRemmel1998", "label":"YR98", "tooltip":"Yang and Remmel, Hook-Schur functions analogues of Littlewood’s identities and their bijective proofs (1998)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "attrs":{ "map":"y=x" }, "label":"SpecializesTo", "source_index":2, "id":"e95", "transitive":true, "type":"specializes_to", "source":"hookSchur", "refs":[{ "key":"Shigechi2017x", "href":"#Shigechi2017x", "label":"Shi17", "tooltip":"Shigechi, Shifted tableaux and products of Schur’s symmetric functions (2017)" }], "poset":true, "target":"bigSchur", "status":"theorem" },{ "attrs":{ "scope":"$\\theta=0$ specialization" }, "label":"Generalizes", "source_index":1, "id":"e96", "transitive":true, "type":"generalizes", "source":"imathHallLittlewoodH", "refs":[{ "key":"LuRuanWang2025", "href":"#LuRuanWang2025", "label":"LRW25", "tooltip":"Lu, Ruan and Wang, $\\imath$Hall algebra of Jordan quiver and $\\imath$Hall–Littlewood functions (2025)" }], "poset":true, "target":"hallLittlewoodH", "status":"theorem" },{ "attrs":{ "map":"$\\theta=0$" }, "label":"SpecializesTo", "source_index":2, "id":"e97", "transitive":true, "type":"specializes_to", "source":"imathHallLittlewoodH", "refs":[{ "key":"LuRuanWang2025", "href":"#LuRuanWang2025", "label":"LRW25", "tooltip":"Lu, Ruan and Wang, $\\imath$Hall algebra of Jordan quiver and $\\imath$Hall–Littlewood functions (2025)" }], "poset":true, "target":"hallLittlewoodH", "status":"theorem" },{ "attrs":{ "map":"$t=0,\\theta=1$" }, "label":"SpecializesTo", "source_index":3, "id":"e98", "transitive":true, "type":"specializes_to", "source":"imathHallLittlewoodH", "refs":[{ "key":"LuRuanWang2025", "href":"#LuRuanWang2025", "label":"LRW25", "tooltip":"Lu, Ruan and Wang, $\\imath$Hall algebra of Jordan quiver and $\\imath$Hall–Littlewood functions (2025)" }], "poset":true, "target":"typeCUniversalCharacter", "status":"theorem" },{ "attrs":{ "semiring":"NN[a]" }, "label":"PositiveIn", "source_index":5, "id":"e99", "transitive":true, "type":"positive_in", "source":"jackJ", "refs":[{ "key":"KnopSahi1997", "href":"#KnopSahi1997", "label":"KS97", "tooltip":"Knop and Sahi, A recursion and a combinatorial formula for Jack polynomials (1997)" }], "poset":true, "target":"augmentedMonomial", "status":"theorem" },{ "attrs":{ "map":"alpha=0 plus conjugation" }, "label":"SpecializesTo", "source_index":1, "id":"e100", "transitive":true, "type":"specializes_to", "source":"jackJ", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "attrs":{ "map":"alpha=infinity" }, "label":"SpecializesTo", "source_index":4, "id":"e101", "transitive":true, "type":"specializes_to", "source":"jackJ", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "map":"alpha=1" }, "label":"SpecializesTo", "source_index":2, "id":"e102", "transitive":true, "type":"specializes_to", "source":"jackJ", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"alpha=2" }, "label":"SpecializesTo", "source_index":3, "id":"e103", "transitive":true, "type":"specializes_to", "source":"jackJ", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"zonal", "status":"theorem" },{ "attrs":{ "map":"alpha=infinity" }, "label":"SpecializesTo", "source_index":1, "id":"e104", "transitive":true, "type":"specializes_to", "source":"jackP", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "map":"alpha=1" }, "label":"SpecializesTo", "source_index":2, "id":"e105", "transitive":true, "type":"specializes_to", "source":"jackP", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"alpha=2" }, "label":"SpecializesTo", "source_index":3, "id":"e106", "transitive":true, "type":"specializes_to", "source":"jackP", "refs":[{ "key":"Jack1970", "href":"#Jack1970", "label":"Jac70", "tooltip":"Jack, A class of symmetric polynomials with a parameter (1970)" }], "poset":true, "target":"zonal", "status":"theorem" },{ "attrs":{ "map":"multiply by $H_\\lambda(a)$" }, "label":"TransformsTo", "source_index":4, "id":"e107", "transitive":false, "type":"transforms_to", "source":"jackP", "refs":[{ "key":"Stanley1989", "href":"#Stanley1989", "label":"Sta89", "tooltip":"Stanley, Some combinatorial properties of Jack symmetric functions (1989)" }], "poset":false, "target":"jackJ", "status":"theorem" },{ "attrs":{ "map":"top-degree component" }, "label":"SpecializesTo", "source_index":1, "id":"e108", "transitive":true, "type":"specializes_to", "source":"jackShifted", "refs":[{ "key":"OkounkovOlshanski1997", "href":"#OkounkovOlshanski1997", "label":"OO97", "tooltip":"Okounkov and Olshanski, Shifted Jack polynomials, binomial formula, and applications (1997)" }], "poset":true, "target":"jackP", "status":"theorem" },{ "label":"SpecializesTo", "source_index":2, "id":"e109", "transitive":true, "type":"specializes_to", "source":"jackShifted", "refs":[{ "key":"Sahi1994", "href":"#Sahi1994", "label":"Sah94", "tooltip":"Sahi, The spectrum of certain invariant differential operators associated to a Hermitian symmetric space (1994)" }], "poset":true, "target":"schurShifted", "status":"theorem" },{ "attrs":{ "proof":"positive k-branching" }, "label":"PositiveIn", "source_index":1, "id":"e110", "transitive":true, "type":"positive_in", "source":"kSchur", "refs":[{ "key":"BlasiakMorseSummers2019", "href":"#BlasiakMorseSummers2019", "label":"BMPS19", "tooltip":"Blasiak, Morse, Pun and Summers, Catalan functions and $k$-Schur positivity (2019)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"flagged Schur polynomials" }, "label":"Contains", "source_index":1, "id":"e111", "transitive":true, "type":"contains", "source":"key", "refs":[{ "key":"ReinerShimozono1995", "href":"#ReinerShimozono1995", "label":"RS95", "tooltip":"Reiner and Shimozono, Key polynomials and a flagged Littlewood–Richardson rule (1995)" }], "poset":true, "target":"schurFlagged", "status":"theorem" },{ "attrs":{ "scope":"dominant weights" }, "label":"Contains", "source_index":2, "id":"e112", "transitive":true, "type":"contains", "source":"key", "refs":[{ "key":"Lascoux1990Keys", "href":"#Lascoux1990Keys", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Keys & standard bases (1990)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e113", "transitive":true, "type":"positive_in", "source":"key", "refs":[{ "key":"Lascoux1990Keys", "href":"#Lascoux1990Keys", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Keys & standard bases (1990)" }], "poset":true, "target":"atom", "status":"theorem" },{ "label":"PositiveIn", "source_index":4, "id":"e114", "transitive":true, "type":"positive_in", "source":"key", "refs":[{ "key":"Lascoux1990Keys", "href":"#Lascoux1990Keys", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Keys & standard bases (1990)" }], "poset":true, "target":"generalizedDemazureAtoms", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.4" }, "label":"PositiveIn", "source_index":5, "id":"e115", "transitive":true, "type":"positive_in", "source":"key", "refs":[{ "key":"AssafSearles2018", "href":"#AssafSearles2018", "label":"AS18", "tooltip":"Assaf and Searles, Kohnert tableaux and a lifting of quasi-Schur functions (2018)" }], "poset":true, "target":"quasiKey", "status":"theorem" },{ "attrs":{ "proof":"Thm. 2.13" }, "label":"PositiveIn", "source_index":6, "id":"e116", "transitive":true, "type":"positive_in", "source":"key", "refs":[{ "key":"AssafSearles2018", "href":"#AssafSearles2018", "label":"AS18", "tooltip":"Assaf and Searles, Kohnert tableaux and a lifting of quasi-Schur functions (2018)" }], "poset":true, "target":"slideF", "status":"theorem" },{ "attrs":{ "proof":"Thm. 7.6" }, "label":"SignedIn", "source_index":7, "id":"e117", "transitive":false, "type":"signed_in", "source":"key", "refs":[{ "key":"Ehrhard2023x", "href":"#Ehrhard2023x", "label":"Ehr23", "tooltip":"Ehrhard, Complete flagged homogeneous polynomials (2023)" }], "poset":false, "target":"completeFlaggedHomogeneous", "status":"theorem" },{ "attrs":{ "scope":"ordinary shapes" }, "label":"Contains", "source_index":1, "id":"e118", "transitive":true, "type":"contains", "source":"keySkew", "refs":[{ "key":"AssafvanWilligenburg2019", "href":"#AssafvanWilligenburg2019", "label":"AW19", "tooltip":"Assaf and Willigenburg, Skew key polynomials and a generalized Littlewood–Richardson rule (2019)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "proof":"key-positive expansion refined by key-to-atom expansion" }, "label":"PositiveIn", "source_index":3, "id":"e119", "transitive":true, "type":"positive_in", "source":"keySkew", "refs":[{ "key":"AssafvanWilligenburg2019", "href":"#AssafvanWilligenburg2019", "label":"AW19", "tooltip":"Assaf and Willigenburg, Skew key polynomials and a generalized Littlewood–Richardson rule (2019)" },{ "key":"Lascoux1990Keys", "href":"#Lascoux1990Keys", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Keys & standard bases (1990)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "proof":"generalized Littlewood--Richardson rule" }, "label":"PositiveIn", "source_index":2, "id":"e120", "transitive":true, "type":"positive_in", "source":"keySkew", "refs":[{ "key":"AssafvanWilligenburg2019", "href":"#AssafvanWilligenburg2019", "label":"AW19", "tooltip":"Assaf and Willigenburg, Skew key polynomials and a generalized Littlewood–Richardson rule (2019)" }], "poset":true, "target":"key", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e121", "transitive":true, "type":"contains", "source":"kohnert", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"key", "status":"theorem" },{ "label":"Contains", "source_index":2, "id":"e122", "transitive":true, "type":"contains", "source":"kohnert", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"lock", "status":"theorem" },{ "label":"Contains", "source_index":3, "id":"e123", "transitive":true, "type":"contains", "source":"kohnert", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "label":"Contains", "source_index":4, "id":"e124", "transitive":true, "type":"contains", "source":"kohnert", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":5, "id":"e125", "transitive":true, "type":"positive_in", "source":"kohnert", "refs":[{ "key":"Assaf2021", "href":"#Assaf2021", "label":"Ass21", "tooltip":"Assaf, Demazure crystals for Kohnert polynomials (2021)" }], "poset":true, "target":"key", "status":"theorem" },{ "label":"PositiveIn", "source_index":6, "id":"e126", "transitive":true, "type":"positive_in", "source":"kohnert", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"slideM", "status":"theorem" },{ "attrs":{ "proof":"Def. 3.10" }, "label":"StableLimit", "source_index":7, "id":"e127", "transitive":true, "type":"stable_limit", "source":"kohnert", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":false, "target":"kohnertQuasi", "status":"theorem" },{ "attrs":{ "proof":"Cor. 6.14", "scope":"lock diagrams" }, "label":"Contains", "source_index":1, "id":"e128", "transitive":true, "type":"contains", "source":"kohnertQuasi", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"extSchur", "status":"theorem" },{ "attrs":{ "proof":"Thm. 4.17" }, "label":"PositiveIn", "source_index":2, "id":"e129", "transitive":true, "type":"positive_in", "source":"kohnertQuasi", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.11" }, "label":"PositiveIn", "source_index":3, "id":"e130", "transitive":true, "type":"positive_in", "source":"kohnertQuasi", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "label":"KTheoreticAnalogueOf", "source_index":1, "id":"e131", "transitive":false, "type":"k_theoretic_analogue_of", "source":"kromaticSymmetric", "refs":[{ "key":"CrewPechenikSpirkl2023x", "href":"#CrewPechenikSpirkl2023x", "label":"CPS26", "tooltip":"Crew, Pechenik and Spirkl, The Kromatic symmetric function: A K-theoretic analog of ${X}_{G}$ (2026)" }], "poset":false, "target":"chromaticQuasisymmetric", "status":"theorem" },{ "attrs":{ "scope":"claw-free incomparability graphs of posets" }, "label":"PositiveIn", "source_index":2, "id":"e132", "transitive":true, "type":"positive_in", "source":"kromaticSymmetric", "refs":[{ "key":"CrewPechenikSpirkl2023x", "href":"#CrewPechenikSpirkl2023x", "label":"CPS26", "tooltip":"Crew, Pechenik and Spirkl, The Kromatic symmetric function: A K-theoretic analog of ${X}_{G}$ (2026)" }], "poset":true, "target":"grothendieckStable", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e133", "transitive":true, "type":"positive_in", "source":"lah", "refs":[{ "key":"PetreolleSokal2019x", "href":"#PetreolleSokal2019x", "label":"PS21", "tooltip":"Pétréolle and Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions (2021)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "label":"Generalizes", "source_index":1, "id":"e134", "transitive":true, "type":"generalizes", "source":"lascoux", "refs":[{ "key":"Lascoux2001", "href":"#Lascoux2001", "label":"Las01", "tooltip":"Lascoux, Transitions on Grothendieck polynomials (2001)" }], "poset":true, "target":"grothendieckGrassmann", "status":"theorem" },{ "attrs":{ "parameter":"beta" }, "label":"KTheoreticAnalogueOf", "source_index":2, "id":"e135", "transitive":false, "type":"k_theoretic_analogue_of", "source":"lascoux", "refs":[{ "key":"Lascoux2001", "href":"#Lascoux2001", "label":"Las01", "tooltip":"Lascoux, Transitions on Grothendieck polynomials (2001)" }], "poset":false, "target":"key", "status":"theorem" },{ "attrs":{ "proof":"via quasiLascoux basis" }, "label":"PositiveIn", "source_index":3, "id":"e136", "transitive":true, "type":"positive_in", "source":"lascoux", "refs":[{ "key":"MonicalPechenikSearles2021", "href":"#MonicalPechenikSearles2021", "label":"MPS21", "tooltip":"Monical, Pechenik and Searles, Polynomials from combinatorial $K$-theory (2021)" }], "poset":true, "target":"lascouxAtom", "status":"theorem" },{ "attrs":{ "map":"beta=0" }, "label":"SpecializesTo", "source_index":4, "id":"e137", "transitive":true, "type":"specializes_to", "source":"lascoux", "refs":[{ "key":"Lascoux2001", "href":"#Lascoux2001", "label":"Las01", "tooltip":"Lascoux, Transitions on Grothendieck polynomials (2001)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "parameter":"beta" }, "label":"KTheoreticAnalogueOf", "source_index":1, "id":"e138", "transitive":false, "type":"k_theoretic_analogue_of", "source":"lascouxAtom", "refs":[{ "key":"Lascoux2004", "href":"#Lascoux2004", "label":"Las04", "tooltip":"Lascoux, Schubert & Grothendieck: Un bilan bidécennal (2004)" }], "poset":false, "target":"atom", "status":"theorem" },{ "attrs":{ "map":"beta=0" }, "label":"SpecializesTo", "source_index":2, "id":"e139", "transitive":true, "type":"specializes_to", "source":"lascouxAtom", "refs":[{ "key":"Lascoux2004", "href":"#Lascoux2004", "label":"Las04", "tooltip":"Lascoux, Schubert & Grothendieck: Un bilan bidécennal (2004)" }], "poset":true, "target":"atom", "status":"theorem" },{ "label":"Generalizes", "source_index":1, "id":"e140", "transitive":true, "type":"generalizes", "source":"littlewoodSchur", "refs":[{ "key":"Riedtmann2018x", "href":"#Riedtmann2018x", "label":"Rie18", "tooltip":"Riedtmann, Overlap Identities for Littlewood–Schur Functions (2018)" }], "poset":true, "target":"hookSchur", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e141", "transitive":true, "type":"contains", "source":"lock", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"Thm. 6.11" }, "label":"PositiveIn", "source_index":3, "id":"e142", "transitive":true, "type":"positive_in", "source":"lock", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"slideF", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.7" }, "label":"PositiveIn", "source_index":4, "id":"e143", "transitive":true, "type":"positive_in", "source":"lock", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":true, "target":"slideM", "status":"theorem" },{ "label":"StableLimit", "source_index":2, "id":"e144", "transitive":true, "type":"stable_limit", "source":"lock", "refs":[{ "key":"AssafSearles2019", "href":"#AssafSearles2019", "label":"AS22", "tooltip":"Assaf and Searles, Kohnert polynomials (2022)" }], "poset":false, "target":"extSchur", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e145", "transitive":true, "type":"positive_in", "source":"lyndonSymmetric", "refs":[{ "key":"GesselReutenauer1993", "href":"#GesselReutenauer1993", "label":"GR93", "tooltip":"Gessel and Reutenauer, Counting permutations with given cycle structure and descent set (1993)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e146", "transitive":true, "type":"positive_in", "source":"lyndonSymmetric", "refs":[{ "key":"GesselReutenauer1993", "href":"#GesselReutenauer1993", "label":"GR93", "tooltip":"Gessel and Reutenauer, Counting permutations with given cycle structure and descent set (1993)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"q,t -> infinity" }, "label":"SpecializesTo", "source_index":2, "id":"e147", "transitive":true, "type":"specializes_to", "source":"macdonaldE", "refs":[{ "key":"Mason2009", "href":"#Mason2009", "label":"Mas09", "tooltip":"Mason, An explicit construction of type A Demazure atoms (2009)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "map":"q=t=0" }, "label":"SpecializesTo", "source_index":4, "id":"e148", "transitive":true, "type":"specializes_to", "source":"macdonaldE", "refs":[{ "key":"RamYip2011", "href":"#RamYip2011", "label":"RY11", "tooltip":"Ram and Yip, A combinatorial formula for Macdonald polynomials (2011)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":1, "id":"e149", "transitive":true, "type":"specializes_to", "source":"macdonaldE", "refs":[{ "key":"Opdam1995", "href":"#Opdam1995", "label":"Opd95", "tooltip":"Opdam, Harmonic analysis for certain representations of graded Hecke algebras (1995)" }], "poset":true, "target":"macdonaldEspec", "status":"theorem" },{ "attrs":{ "scope":"symmetric case" }, "label":"SpecializesTo", "source_index":3, "id":"e150", "transitive":true, "type":"specializes_to", "source":"macdonaldE", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"macdonaldP", "status":"theorem" },{ "attrs":{ "scope":"symmetric limit" }, "label":"StableLimit", "source_index":5, "id":"e151", "transitive":true, "type":"stable_limit", "source":"macdonaldE", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":false, "target":"macdonaldP", "status":"theorem" },{ "attrs":{ "map":"q=t=0" }, "label":"SpecializesTo", "source_index":1, "id":"e152", "transitive":true, "type":"specializes_to", "source":"macdonaldEQuasi", "refs":[{ "key":"CorteelMandelshtamMasonWilliams2019x", "href":"#CorteelMandelshtamMasonWilliams2019x", "label":"CHMM+19", "tooltip":"Corteel, Haglund, Mandelshtam, Mason and Williams, Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials (2019)" }], "poset":true, "target":"qSchur", "status":"theorem" },{ "attrs":{ "scope":"identity basement" }, "label":"Contains", "source_index":1, "id":"e153", "transitive":true, "type":"contains", "source":"macdonaldEperm", "refs":[{ "key":"Ferreira2011", "href":"#Ferreira2011", "label":"Fer11", "tooltip":"Ferreira, Row-strict quasisymmetric Schur functions, characterizations of Demazure atoms, and permuted basement nonsymmetric Macdonald polynomials (2011)" }], "poset":true, "target":"macdonaldE", "status":"theorem" },{ "attrs":{ "semiring":"NN[q]", "proof":"Cor. 4.7", "scope":"t=0" }, "label":"PositiveIn", "source_index":4, "id":"e154", "transitive":true, "type":"positive_in", "source":"macdonaldEperm", "refs":[{ "key":"MandelshtamNiergarthSingh2026x", "href":"#MandelshtamNiergarthSingh2026x", "label":"MNS26", "tooltip":"Mandelshtam, Niergarth and Singh, Positive expansions of permuted basement and quasisymmetric Macdonald polynomials at $t=0$ (2026)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "map":"q=t=0" }, "label":"SpecializesTo", "source_index":3, "id":"e155", "transitive":true, "type":"specializes_to", "source":"macdonaldEperm", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"permutedBasementAtom", "status":"theorem" },{ "attrs":{ "map":"q=0" }, "label":"SpecializesTo", "source_index":2, "id":"e156", "transitive":true, "type":"specializes_to", "source":"macdonaldEperm", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"permutedBasementTAtom", "status":"theorem" },{ "attrs":{ "scope":"symmetric/partition case" }, "label":"Contains", "source_index":1, "id":"e157", "transitive":true, "type":"contains", "source":"macdonaldEspec", "refs":[{ "key":"Opdam1995", "href":"#Opdam1995", "label":"Opd95", "tooltip":"Opdam, Harmonic analysis for certain representations of graded Hecke algebras (1995)" }], "poset":true, "target":"qWhittaker", "status":"theorem" },{ "attrs":{ "proof":"Cor. 4.7", "semiring":"NN[q]" }, "label":"PositiveIn", "source_index":4, "id":"e158", "transitive":true, "type":"positive_in", "source":"macdonaldEspec", "refs":[{ "key":"MandelshtamNiergarthSingh2026x", "href":"#MandelshtamNiergarthSingh2026x", "label":"MNS26", "tooltip":"Mandelshtam, Niergarth and Singh, Positive expansions of permuted basement and quasisymmetric Macdonald polynomials at $t=0$ (2026)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "semiring":"NN[q]" }, "label":"PositiveIn", "source_index":3, "id":"e159", "transitive":true, "type":"positive_in", "source":"macdonaldEspec", "refs":[{ "key":"AlexanderssonSawhney2017", "href":"#AlexanderssonSawhney2017", "label":"AS17", "tooltip":"Alexandersson and Sawhney, A major-index preserving map on fillings (2017)" },{ "key":"Assaf2018Kostka", "href":"#Assaf2018Kostka", "label":"Ass18", "tooltip":"Assaf, Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials (2018)" },{ "key":"AssafGonzalez2018", "href":"#AssafGonzalez2018", "label":"AG18", "tooltip":"Assaf and González, Crystal graphs, key tabloids, and nonsymmetric Macdonald polynomials (2018)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "map":"q=1" }, "label":"SpecializesTo", "source_index":2, "id":"e160", "transitive":true, "type":"specializes_to", "source":"macdonaldEspec", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"nonSymmetricElementary", "status":"theorem" },{ "attrs":{ "semiring":"NN[t]" }, "label":"PositiveIn", "source_index":1, "id":"e161", "transitive":true, "type":"positive_in", "source":"macdonaldH", "refs":[{ "key":"HaglundHaimanLoehr2005", "href":"#HaglundHaimanLoehr2005", "label":"HHL05", "tooltip":"Haglund, Haiman and Loehr, A combinatorial formula for Macdonald polynomials (2005)" }], "poset":true, "target":"LLT", "status":"theorem" },{ "attrs":{ "semiring":"NN[q,t]", "proof":"Cor. 9.4", "scope":"k-bounded partitions" }, "label":"PositiveIn", "source_index":2, "id":"e162", "transitive":true, "type":"positive_in", "source":"macdonaldH", "refs":[{ "key":"Kato2025x", "href":"#Kato2025x", "label":"Kat25", "tooltip":"Kato, Categorification of $k$-Schur functions and refined Macdonald positivity (2025)" }], "poset":true, "target":"kSchur", "status":"theorem" },{ "attrs":{ "semiring":"NN[q,t]" }, "label":"PositiveIn", "source_index":3, "id":"e163", "transitive":true, "type":"positive_in", "source":"macdonaldH", "refs":[{ "key":"Haiman2001", "href":"#Haiman2001", "label":"Hai01", "tooltip":"Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture (2001)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"q=0" }, "label":"SpecializesTo", "source_index":4, "id":"e164", "transitive":true, "type":"specializes_to", "source":"macdonaldH", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"hallLittlewoodH", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e165", "transitive":true, "type":"positive_in", "source":"macdonaldJ", "refs":[{ "key":"CorteelMandelshtamMasonWilliams2022", "href":"#CorteelMandelshtamMasonWilliams2022", "label":"CHMM+22", "tooltip":"Corteel, Haglund, Mandelshtam, Mason and Williams, Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials (2022)" }], "poset":true, "target":"macdonaldJQuasi", "status":"theorem" },{ "attrs":{ "map":"Jack limit" }, "label":"SpecializesTo", "source_index":1, "id":"e166", "transitive":true, "type":"specializes_to", "source":"macdonaldJ", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"jackJ", "status":"theorem" },{ "attrs":{ "map":"plethystic transform" }, "label":"TransformsTo", "source_index":2, "id":"e167", "transitive":false, "type":"transforms_to", "source":"macdonaldJ", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":false, "target":"macdonaldH", "status":"theorem" },{ "attrs":{ "scope":"column partitions" }, "label":"Contains", "source_index":1, "id":"e168", "transitive":true, "type":"contains", "source":"macdonaldP", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "attrs":{ "scope":"$t=q^2$ specialization" }, "label":"Contains", "source_index":2, "id":"e169", "transitive":true, "type":"contains", "source":"macdonaldP", "refs":[{ "key":"He2020", "href":"#He2020", "label":"He20", "tooltip":"He, A Characteristic Map for the Symmetric Space of Symplectic Forms Over a Finite Field (2020)" }], "poset":true, "target":"macdonaldPqq2", "status":"theorem" },{ "attrs":{ "map":"q=0" }, "label":"SpecializesTo", "source_index":3, "id":"e170", "transitive":true, "type":"specializes_to", "source":"macdonaldP", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"hallLittlewoodP", "status":"theorem" },{ "attrs":{ "map":"Jack limit" }, "label":"SpecializesTo", "source_index":4, "id":"e171", "transitive":true, "type":"specializes_to", "source":"macdonaldP", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"jackP", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":5, "id":"e172", "transitive":true, "type":"specializes_to", "source":"macdonaldP", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"qWhittaker", "status":"theorem" },{ "attrs":{ "map":"q=t" }, "label":"SpecializesTo", "source_index":6, "id":"e173", "transitive":true, "type":"specializes_to", "source":"macdonaldP", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"integral-form rescaling" }, "label":"TransformsTo", "source_index":7, "id":"e174", "transitive":false, "type":"transforms_to", "source":"macdonaldP", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":false, "target":"macdonaldJ", "status":"theorem" },{ "attrs":{ "scope":"skew coefficients after evaluation at odd prime powers" }, "label":"PositiveIn", "source_index":1, "id":"e175", "transitive":true, "type":"positive_in", "source":"macdonaldPqq2", "refs":[{ "key":"He2020", "href":"#He2020", "label":"He20", "tooltip":"He, A Characteristic Map for the Symmetric Space of Symplectic Forms Over a Finite Field (2020)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"one-row partitions" }, "label":"Contains", "source_index":1, "id":"e176", "transitive":true, "type":"contains", "source":"modularSchur", "refs":[{ "key":"DotyWalker1992", "href":"#DotyWalker1992", "label":"DW92", "tooltip":"Doty and Walker, Modular symmetric functions and irreducible modular representations of general linear groups (1992)" }], "poset":true, "target":"petrie", "status":"theorem" },{ "attrs":{ "parameter":"k greater than largest part" }, "label":"SpecializesTo", "source_index":2, "id":"e177", "transitive":true, "type":"specializes_to", "source":"modularSchur", "refs":[{ "key":"DotyWalker1992", "href":"#DotyWalker1992", "label":"DW92", "tooltip":"Doty and Walker, Modular symmetric functions and irreducible modular representations of general linear groups (1992)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"refinement" }, "label":"RefinedBy", "source_index":1, "id":"e178", "transitive":true, "type":"refined_by", "source":"monomial", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":false, "target":"qmonom", "status":"theorem" },{ "label":"Generalizes", "source_index":1, "id":"e179", "transitive":true, "type":"generalizes", "source":"multiSchur", "refs":[{ "key":"Iwao2023", "href":"#Iwao2023", "label":"Iwa23", "tooltip":"Iwao, Free fermions and Schur expansions of multi-Schur functions (2023)" }], "poset":true, "target":"hookSchur", "status":"theorem" },{ "label":"Generalizes", "source_index":2, "id":"e180", "transitive":true, "type":"generalizes", "source":"multiSchur", "refs":[{ "key":"Iwao2023", "href":"#Iwao2023", "label":"Iwa23", "tooltip":"Iwao, Free fermions and Schur expansions of multi-Schur functions (2023)" }], "poset":true, "target":"schurFlagged", "status":"theorem" },{ "attrs":{ "pairing":"canonical NSym--QSym pairing" }, "label":"NSymQSymDual", "source_index":3, "id":"e181", "transitive":false, "type":"nsym_qsym_dual", "source":"nonComHomogeneous", "refs":[{ "key":"GelfandKrobLascouxLeclercRetakhThibon1995", "href":"#GelfandKrobLascouxLeclercRetakhThibon1995", "label":"GKLL+95", "tooltip":"Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon, Noncommutative symmetrical functions (1995)" }], "poset":false, "target":"qmonom", "status":"theorem" },{ "attrs":{ "proof":"dual to the $\\qPhi$ monomial expansion" }, "label":"PositiveIn", "source_index":2, "id":"e182", "transitive":true, "type":"positive_in", "source":"nonComHomogeneous", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":true, "target":"nonComPowerPhi", "status":"theorem" },{ "attrs":{ "proof":"dual to the $\\qPsi$ monomial expansion" }, "label":"PositiveIn", "source_index":1, "id":"e183", "transitive":true, "type":"positive_in", "source":"nonComHomogeneous", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":true, "target":"nonComPowerPsi", "status":"theorem" },{ "attrs":{ "pairing":"canonical NSym--QSym pairing" }, "label":"NSymQSymDual", "source_index":1, "id":"e184", "transitive":false, "type":"nsym_qsym_dual", "source":"nonComPowerPhi", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":false, "target":"qPhi", "status":"theorem" },{ "attrs":{ "pairing":"canonical NSym--QSym pairing" }, "label":"NSymQSymDual", "source_index":1, "id":"e185", "transitive":false, "type":"nsym_qsym_dual", "source":"nonComPowerPsi", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":false, "target":"qPsi", "status":"theorem" },{ "attrs":{ "scope":"partition shapes" }, "label":"Generalizes", "source_index":1, "id":"e186", "transitive":true, "type":"generalizes", "source":"nonSymmetricElementary", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "attrs":{ "proof":"Kostka numbers" }, "label":"PositiveIn", "source_index":2, "id":"e187", "transitive":true, "type":"positive_in", "source":"nonSymmetricElementary", "refs":[{ "key":"Assaf2018Kostka", "href":"#Assaf2018Kostka", "label":"Ass18", "tooltip":"Assaf, Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials (2018)" },{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"key", "status":"theorem" },{ "label":"KTheoreticAnalogueOf", "source_index":1, "id":"e188", "transitive":false, "type":"k_theoretic_analogue_of", "source":"pGrothendieck", "refs":[{ "key":"Hawkes2023", "href":"#Hawkes2023", "label":"Haw23", "tooltip":"Hawkes, P-schur positive P-Grothendieck polynomials (2023)" }], "poset":false, "target":"schurP", "status":"theorem" },{ "attrs":{ "scope":"weak P-Grothendieck polynomials" }, "label":"PositiveIn", "source_index":2, "id":"e189", "transitive":true, "type":"positive_in", "source":"pGrothendieck", "refs":[{ "key":"Hawkes2023", "href":"#Hawkes2023", "label":"Haw23", "tooltip":"Hawkes, P-schur positive P-Grothendieck polynomials (2023)" }], "poset":true, "target":"schurP", "status":"theorem" },{ "label":"Contains", "source_index":3, "id":"e190", "transitive":true, "type":"contains", "source":"pPartition", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"completeH", "status":"theorem" },{ "label":"Contains", "source_index":2, "id":"e191", "transitive":true, "type":"contains", "source":"pPartition", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "label":"Contains", "source_index":4, "id":"e192", "transitive":true, "type":"contains", "source":"pPartition", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e193", "transitive":true, "type":"contains", "source":"pPartition", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "label":"PositiveIn", "source_index":5, "id":"e194", "transitive":true, "type":"positive_in", "source":"pPartition", "refs":[{ "key":"StanleyEC2", "href":"#StanleyEC2", "label":"Sta01", "tooltip":"Stanley, Enumerative Combinatorics: Volume 2 (2001)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "scope":"order-preserving labeling" }, "label":"PositiveIn", "source_index":6, "id":"e195", "transitive":true, "type":"positive_in", "source":"pPartition", "refs":[{ "key":"AlexanderssonSulzgruber2019", "href":"#AlexanderssonSulzgruber2019", "label":"AS19", "tooltip":"Alexandersson and Sulzgruber, P-partitions and p-positivity (2019)" }], "poset":true, "target":"qPsi", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e196", "transitive":true, "type":"positive_in", "source":"pPartitionsFlagged", "refs":[{ "key":"AssafBergeron2019", "href":"#AssafBergeron2019", "label":"AB19", "tooltip":"Assaf and Bergeron, Flagged $(\\mathcal{ P },ρ)$-partitions (2019)" }], "poset":true, "target":"slideF", "status":"theorem" },{ "attrs":{ "scope":"unbounded flags" }, "label":"SpecializesTo", "source_index":1, "id":"e197", "transitive":true, "type":"specializes_to", "source":"pPartitionsFlagged", "refs":[{ "key":"AssafBergeron2019", "href":"#AssafBergeron2019", "label":"AB19", "tooltip":"Assaf and Bergeron, Flagged $(\\mathcal{ P },ρ)$-partitions (2019)" }], "poset":true, "target":"pPartition", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e198", "transitive":true, "type":"positive_in", "source":"parking", "refs":[{ "key":"StanleyEC2", "href":"#StanleyEC2", "label":"Sta01", "tooltip":"Stanley, Enumerative Combinatorics: Volume 2 (2001)" }], "poset":true, "target":"completeH", "status":"theorem" },{ "attrs":{ "semiring":"QQ_nonnegative" }, "label":"PositiveIn", "source_index":1, "id":"e199", "transitive":true, "type":"positive_in", "source":"parking", "refs":[{ "key":"StanleyEC2", "href":"#StanleyEC2", "label":"Sta01", "tooltip":"Stanley, Enumerative Combinatorics: Volume 2 (2001)" }], "poset":true, "target":"powerSum", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e200", "transitive":true, "type":"positive_in", "source":"parking", "refs":[{ "key":"StanleyEC2", "href":"#StanleyEC2", "label":"Sta01", "tooltip":"Stanley, Enumerative Combinatorics: Volume 2 (2001)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e201", "transitive":true, "type":"positive_in", "source":"peakQSym", "refs":[{ "key":"Stembridge1997", "href":"#Stembridge1997", "label":"Ste97", "tooltip":"Stembridge, Enriched $P$-partitions (1997)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "scope":"standard shifted tableaux" }, "label":"Contains", "source_index":1, "id":"e202", "transitive":true, "type":"contains", "source":"peakYqSchur", "refs":[{ "key":"Searles2025", "href":"#Searles2025", "label":"Sea25", "tooltip":"Searles, Diagram supermodules for 0-Hecke–Clifford algebras (2025)" }], "poset":true, "target":"schurQ", "status":"theorem" },{ "attrs":{ "proof":"Thm. 5.9" }, "label":"PositiveIn", "source_index":2, "id":"e203", "transitive":true, "type":"positive_in", "source":"peakYqSchur", "refs":[{ "key":"Searles2025", "href":"#Searles2025", "label":"Sea25", "tooltip":"Searles, Diagram supermodules for 0-Hecke–Clifford algebras (2025)" }], "poset":true, "target":"peakQSym", "status":"theorem" },{ "attrs":{ "scope":"identity basement, up to convention" }, "label":"Contains", "source_index":1, "id":"e204", "transitive":true, "type":"contains", "source":"permutedBasementAtom", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "scope":"longest basement, up to convention" }, "label":"Contains", "source_index":2, "id":"e205", "transitive":true, "type":"contains", "source":"permutedBasementAtom", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "scope":"identity basement, up to convention" }, "label":"Contains", "source_index":1, "id":"e206", "transitive":true, "type":"contains", "source":"permutedBasementTAtom", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"tAtom", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":2, "id":"e207", "transitive":true, "type":"specializes_to", "source":"permutedBasementTAtom", "refs":[{ "key":"AlexanderssonSawhney2019", "href":"#AlexanderssonSawhney2019", "label":"AS19", "tooltip":"Alexandersson and Sawhney, Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$ (2019)" }], "poset":true, "target":"permutedBasementAtom", "status":"theorem" },{ "attrs":{ "scope":"k=2" }, "label":"Contains", "source_index":1, "id":"e208", "transitive":true, "type":"contains", "source":"petrie", "refs":[{ "key":"DotyWalker1992", "href":"#DotyWalker1992", "label":"DW92", "tooltip":"Doty and Walker, Modular symmetric functions and irreducible modular representations of general linear groups (1992)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e209", "transitive":true, "type":"positive_in", "source":"petrie", "refs":[{ "key":"DotyWalker1992", "href":"#DotyWalker1992", "label":"DW92", "tooltip":"Doty and Walker, Modular symmetric functions and irreducible modular representations of general linear groups (1992)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e210", "transitive":true, "type":"positive_in", "source":"powerSum", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":true, "target":"qPhi", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e211", "transitive":true, "type":"positive_in", "source":"powerSum", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":true, "target":"qPsi", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e212", "transitive":true, "type":"positive_in", "source":"qPhi", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e213", "transitive":true, "type":"positive_in", "source":"qPsi", "refs":[{ "key":"BallantineDaughertyHicksMason2020", "href":"#BallantineDaughertyHicksMason2020", "label":"BDHM+20", "tooltip":"Ballantine, Daugherty, Hicks, Mason and Niese, On quasisymmetric power sums (2020)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e214", "transitive":true, "type":"positive_in", "source":"qRho", "refs":[{ "key":"AliniaeifardWangWilligenburg2021x", "href":"#AliniaeifardWangWilligenburg2021x", "label":"AWW21", "tooltip":"Aliniaeifard, Wang and Willigenburg, ${P}$-partition power sums (2021)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e215", "transitive":true, "type":"positive_in", "source":"qSchur", "refs":[{ "key":"HaglundLuotoMasonWilligenburg2011", "href":"#HaglundLuotoMasonWilligenburg2011", "label":"HLMW11", "tooltip":"Haglund, Luoto, Mason and Willigenburg, Quasisymmetric Schur functions (2011)" }], "poset":true, "target":"atom", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e216", "transitive":true, "type":"positive_in", "source":"qSchur", "refs":[{ "key":"HaglundLuotoMasonWilligenburg2011", "href":"#HaglundLuotoMasonWilligenburg2011", "label":"HLMW11", "tooltip":"Haglund, Luoto, Mason and Willigenburg, Quasisymmetric Schur functions (2011)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e217", "transitive":true, "type":"positive_in", "source":"qSchur", "refs":[{ "key":"HaglundLuotoMasonWilligenburg2011", "href":"#HaglundLuotoMasonWilligenburg2011", "label":"HLMW11", "tooltip":"Haglund, Luoto, Mason and Willigenburg, Quasisymmetric Schur functions (2011)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "attrs":{ "map":"omega" }, "label":"TransformsTo", "source_index":5, "id":"e218", "transitive":false, "type":"transforms_to", "source":"qSchur", "refs":[{ "key":"MasonRemmel2013", "href":"#MasonRemmel2013", "label":"MR13", "tooltip":"Mason and Remmel, Row-strict quasisymmetric Schur functions (2013)" }], "poset":false, "target":"rsqSchur", "status":"theorem" },{ "attrs":{ "map":"rho and reverse index" }, "label":"TransformsTo", "source_index":4, "id":"e219", "transitive":false, "type":"transforms_to", "source":"qSchur", "refs":[{ "key":"LuotoMykytiukWilligenburg2013", "href":"#LuotoMykytiukWilligenburg2013", "label":"LMW13", "tooltip":"Luoto, Mykytiuk and Willigenburg, An introduction to quasisymmetric Schur functions (2013)" }], "poset":false, "target":"yqSchur", "status":"theorem" },{ "attrs":{ "scope":"peak-algebra lift" }, "label":"Generalizes", "source_index":1, "id":"e220", "transitive":true, "type":"generalizes", "source":"qSchurQ", "refs":[{ "key":"JingLi2015", "href":"#JingLi2015", "label":"JL15", "tooltip":"Jing and Li, A lift of Schur’s Q-functions to the peak algebra (2015)" }], "poset":true, "target":"schurQ", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.9" }, "label":"PositiveIn", "source_index":2, "id":"e221", "transitive":true, "type":"positive_in", "source":"qSchurQ", "refs":[{ "key":"Oguz2019", "href":"#Oguz2019", "label":"Og19", "tooltip":"Oğuz, A note on Jing and Li’s type B quasisymmetric Schur functions (2019)" }], "poset":true, "target":"peakQSym", "status":"theorem" },{ "attrs":{ "proof":"Thms. 4.5 and 4.16" }, "label":"PositiveIn", "source_index":3, "id":"e222", "transitive":true, "type":"positive_in", "source":"qSchurQ", "refs":[{ "key":"ChoiNamOh2025", "href":"#ChoiNamOh2025", "label":"CNO25", "tooltip":"Choi, Nam and Oh, Quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions (2025)" },{ "key":"SearlesSlatteryHolmes2026", "href":"#SearlesSlatteryHolmes2026", "label":"SS26", "tooltip":"Searles and Slattery-Holmes, Expanding quasisymmetric Schur Q-functions into peak Young quasisymmetric Schur functions (2026)" }], "poset":true, "target":"peakYqSchur", "status":"theorem" },{ "attrs":{ "semiring":"NN[q]" }, "label":"PositiveIn", "source_index":2, "id":"e223", "transitive":true, "type":"positive_in", "source":"qWhittaker", "refs":[{ "key":"Assaf2018Kostka", "href":"#Assaf2018Kostka", "label":"Ass18", "tooltip":"Assaf, Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials (2018)" },{ "key":"AssafGonzalez2018", "href":"#AssafGonzalez2018", "label":"AG18", "tooltip":"Assaf and González, Crystal graphs, key tabloids, and nonsymmetric Macdonald polynomials (2018)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"q=0" }, "label":"SpecializesTo", "source_index":1, "id":"e224", "transitive":true, "type":"specializes_to", "source":"qWhittaker", "refs":[{ "key":"Macdonald1987", "href":"#Macdonald1987", "label":"Mac88", "tooltip":"Macdonald, A new class of symmetric functions (1988)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"definition", "scope":"nonsymmetric monomials" }, "label":"PositiveIn", "source_index":1, "id":"e225", "transitive":true, "type":"positive_in", "source":"qmonom", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.1" }, "label":"PositiveIn", "source_index":1, "id":"e226", "transitive":true, "type":"positive_in", "source":"quasiKey", "refs":[{ "key":"Searles2019PolynomialBases", "href":"#Searles2019PolynomialBases", "label":"Sea19", "tooltip":"Searles, Polynomial bases: Positivity and Schur multiplication (2019)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.7" }, "label":"PositiveIn", "source_index":2, "id":"e227", "transitive":true, "type":"positive_in", "source":"quasiKey", "refs":[{ "key":"AssafSearles2018", "href":"#AssafSearles2018", "label":"AS18", "tooltip":"Assaf and Searles, Kohnert tableaux and a lifting of quasi-Schur functions (2018)" }], "poset":true, "target":"slideF", "status":"theorem" },{ "attrs":{ "proof":"Thm. 4.16" }, "label":"StableLimit", "source_index":3, "id":"e228", "transitive":true, "type":"stable_limit", "source":"quasiKey", "refs":[{ "key":"AssafSearles2018", "href":"#AssafSearles2018", "label":"AS18", "tooltip":"Assaf and Searles, Kohnert tableaux and a lifting of quasi-Schur functions (2018)" }], "poset":false, "target":"qSchur", "status":"theorem" },{ "attrs":{ "map":"psi" }, "label":"TransformsTo", "source_index":1, "id":"e229", "transitive":false, "type":"transforms_to", "source":"rsExtSchur", "refs":[{ "key":"Daugherty2024Extended", "href":"#Daugherty2024Extended", "label":"Dau24", "tooltip":"Daugherty, Extended Schur functions and bases related by involutions (2024)" },{ "key":"NieseSundaramWilligenburgVegaWang2023", "href":"#NieseSundaramWilligenburgVegaWang2023", "label":"NSWV+23", "tooltip":"Niese, Sundaram, Willigenburg, Vega and Wang, Row-strict dual immaculate functions (2023)" }], "poset":false, "target":"extSchur", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e230", "transitive":true, "type":"positive_in", "source":"rsdiSchur", "refs":[{ "key":"NieseSundaramWilligenburgVegaWang2023", "href":"#NieseSundaramWilligenburgVegaWang2023", "label":"NSWV+23", "tooltip":"Niese, Sundaram, Willigenburg, Vega and Wang, Row-strict dual immaculate functions (2023)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "map":"psi" }, "label":"TransformsTo", "source_index":2, "id":"e231", "transitive":false, "type":"transforms_to", "source":"rsdiSchur", "refs":[{ "key":"NieseSundaramWilligenburgVegaWang2023", "href":"#NieseSundaramWilligenburgVegaWang2023", "label":"NSWV+23", "tooltip":"Niese, Sundaram, Willigenburg, Vega and Wang, Row-strict dual immaculate functions (2023)" }], "poset":false, "target":"diSchur", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e232", "transitive":true, "type":"positive_in", "source":"rsqSchur", "refs":[{ "key":"MasonRemmel2013", "href":"#MasonRemmel2013", "label":"MR13", "tooltip":"Mason and Remmel, Row-strict quasisymmetric Schur functions (2013)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "map":"omega" }, "label":"TransformsTo", "source_index":2, "id":"e233", "transitive":false, "type":"transforms_to", "source":"rsqSchur", "refs":[{ "key":"MasonRemmel2013", "href":"#MasonRemmel2013", "label":"MR13", "tooltip":"Mason and Remmel, Row-strict quasisymmetric Schur functions (2013)" }], "poset":false, "target":"qSchur", "status":"theorem" },{ "attrs":{ "proof":"Prop. 1" }, "label":"PositiveIn", "source_index":1, "id":"e234", "transitive":true, "type":"positive_in", "source":"rsyqSchur", "refs":[{ "key":"MasonNiese2015", "href":"#MasonNiese2015", "label":"MN15", "tooltip":"Mason and Niese, Skew row-strict quasisymmetric Schur functions (2015)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "map":"omega" }, "label":"TransformsTo", "source_index":2, "id":"e235", "transitive":false, "type":"transforms_to", "source":"rsyqSchur", "refs":[{ "key":"MasonNiese2015", "href":"#MasonNiese2015", "label":"MN15", "tooltip":"Mason and Niese, Skew row-strict quasisymmetric Schur functions (2015)" }], "poset":false, "target":"yqSchur", "status":"theorem" },{ "attrs":{ "scope":"vexillary permutations" }, "label":"Contains", "source_index":1, "id":"e236", "transitive":true, "type":"contains", "source":"schubert", "refs":[{ "key":"LascouxSchutzenberger1982", "href":"#LascouxSchutzenberger1982", "label":"LS82", "tooltip":"Lascoux and Schützenberger, Polynômes de Schubert (1982)" },{ "key":"ReinerShimozono1995", "href":"#ReinerShimozono1995", "label":"RS95", "tooltip":"Reiner and Shimozono, Key polynomials and a flagged Littlewood–Richardson rule (1995)" }], "poset":true, "target":"schurFlagged", "status":"theorem" },{ "attrs":{ "scope":"Grassmannian permutations" }, "label":"Contains", "source_index":2, "id":"e237", "transitive":true, "type":"contains", "source":"schubert", "refs":[{ "key":"LascouxSchutzenberger1982", "href":"#LascouxSchutzenberger1982", "label":"LS82", "tooltip":"Lascoux and Schützenberger, Polynômes de Schubert (1982)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"Thm. 1.4" }, "label":"PositiveIn", "source_index":7, "id":"e238", "transitive":true, "type":"positive_in", "source":"schubert", "refs":[{ "key":"NadeauTewari2023x", "href":"#NadeauTewari2023x", "label":"NT23", "tooltip":"Nadeau and Tewari, Forest polynomials and the class of the permutahedral variety (2023)" }], "poset":true, "target":"forestPolynomials", "status":"theorem" },{ "label":"PositiveIn", "source_index":4, "id":"e239", "transitive":true, "type":"positive_in", "source":"schubert", "refs":[{ "key":"LascouxSchutzenberger1990", "href":"#LascouxSchutzenberger1990", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Tableaux and noncommutative Schubert polynomials (1990)" }], "poset":true, "target":"key", "status":"theorem" },{ "attrs":{ "proof":"Thm. A" }, "label":"PositiveIn", "source_index":5, "id":"e240", "transitive":true, "type":"positive_in", "source":"schubert", "refs":[{ "key":"Searles2019PolynomialBases", "href":"#Searles2019PolynomialBases", "label":"Sea19", "tooltip":"Searles, Polynomial bases: Positivity and Schur multiplication (2019)" }], "poset":true, "target":"quasiKey", "status":"theorem" },{ "label":"PositiveIn", "source_index":6, "id":"e241", "transitive":true, "type":"positive_in", "source":"schubert", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":true, "target":"slideF", "status":"theorem" },{ "label":"StableLimit", "source_index":3, "id":"e242", "transitive":true, "type":"stable_limit", "source":"schubert", "refs":[{ "key":"Stanley1984", "href":"#Stanley1984", "label":"Sta84", "tooltip":"Stanley, On the number of reduced decompositions of elements of Coxeter groups (1984)" }], "poset":false, "target":"stanleySym", "status":"theorem" },{ "attrs":{ "scope":"$0$-Grassmannian affine permutations" }, "label":"Contains", "source_index":1, "id":"e243", "transitive":true, "type":"contains", "source":"schubertAffine", "refs":[{ "key":"Lee2015", "href":"#Lee2015", "label":"Lee15", "tooltip":"Lee, Combinatorial description of the cohomology of the affine flag variety (2015)" }], "poset":true, "target":"schurAffine", "status":"theorem" },{ "attrs":{ "scope":"finite permutations", "map":"$p_j=0$" }, "label":"SpecializesTo", "source_index":3, "id":"e244", "transitive":true, "type":"specializes_to", "source":"schubertAffine", "refs":[{ "key":"Lee2015", "href":"#Lee2015", "label":"Lee15", "tooltip":"Lee, Combinatorial description of the cohomology of the affine flag variety (2015)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "attrs":{ "map":"$x_i=0$" }, "label":"SpecializesTo", "source_index":2, "id":"e245", "transitive":true, "type":"specializes_to", "source":"schubertAffine", "refs":[{ "key":"Lee2015", "href":"#Lee2015", "label":"Lee15", "tooltip":"Lee, Combinatorial description of the cohomology of the affine flag variety (2015)" }], "poset":true, "target":"stanleySymAffine", "status":"theorem" },{ "attrs":{ "scope":"Grassmannian permutations and coefficient specialization" }, "label":"Contains", "source_index":1, "id":"e246", "transitive":true, "type":"contains", "source":"schubertDouble", "refs":[{ "key":"ChenLiLouck2002", "href":"#ChenLiLouck2002", "label":"CLL02", "tooltip":"Chen, Li and Louck, The Flagged Double Schur Function (2002)" }], "poset":true, "target":"schurFactorial", "status":"theorem" },{ "attrs":{ "map":"y=0" }, "label":"SpecializesTo", "source_index":2, "id":"e247", "transitive":true, "type":"specializes_to", "source":"schubertDouble", "refs":[{ "key":"LascouxSchutzenberger1982", "href":"#LascouxSchutzenberger1982", "label":"LS82", "tooltip":"Lascoux and Schützenberger, Polynômes de Schubert (1982)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "attrs":{ "scope":"$c$-variables specialized to Schur $Q$-polynomials, type $C$ double case" }, "label":"SpecializesTo", "source_index":3, "id":"e248", "transitive":true, "type":"specializes_to", "source":"schubertEnriched", "refs":[{ "key":"AndersonFulton2021x", "href":"#AndersonFulton2021x", "label":"AF21", "tooltip":"Anderson and Fulton, Schubert Polynomials in Types A and C (2021)" }], "poset":true, "target":"schubertBCD", "status":"theorem" },{ "attrs":{ "scope":"$c$-variables specialized to power series" }, "label":"SpecializesTo", "source_index":2, "id":"e249", "transitive":true, "type":"specializes_to", "source":"schubertEnriched", "refs":[{ "key":"AndersonFulton2021x", "href":"#AndersonFulton2021x", "label":"AF21", "tooltip":"Anderson and Fulton, Schubert Polynomials in Types A and C (2021)" }], "poset":true, "target":"schubertBackStableDouble", "status":"theorem" },{ "attrs":{ "map":"$c_i=0$ and Anderson--Fulton $\\eta_j=-y_j$" }, "label":"SpecializesTo", "source_index":1, "id":"e250", "transitive":true, "type":"specializes_to", "source":"schubertEnriched", "refs":[{ "key":"AndersonFulton2021x", "href":"#AndersonFulton2021x", "label":"AF21", "tooltip":"Anderson and Fulton, Schubert Polynomials in Types A and C (2021)" }], "poset":true, "target":"schubertDouble", "status":"theorem" },{ "label":"StableLimit", "source_index":1, "id":"e251", "transitive":true, "type":"stable_limit", "source":"schubertInvolution", "refs":[{ "key":"HamakerMarbergPawlowski2018", "href":"#HamakerMarbergPawlowski2018", "label":"HMP18", "tooltip":"Hamaker, Marberg and Pawlowski, Transition formulas for involution Schubert polynomials (2018)" }], "poset":false, "target":"stanleySymInvolution", "status":"theorem" },{ "attrs":{ "map":"q=0" }, "label":"SpecializesTo", "source_index":1, "id":"e252", "transitive":true, "type":"specializes_to", "source":"schubertQuantum", "refs":[{ "key":"FominGelfandPostnikov1997", "href":"#FominGelfandPostnikov1997", "label":"FGP97", "tooltip":"Fomin, Gelfand and Postnikov, Quantum Schubert polynomials (1997)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "attrs":{ "scope":"trivial skewing permutation" }, "label":"Contains", "source_index":1, "id":"e253", "transitive":true, "type":"contains", "source":"schubertSkew", "refs":[{ "key":"LenartSottile2003", "href":"#LenartSottile2003", "label":"LS03", "tooltip":"Lenart and Sottile, Skew Schubert polynomials (2003)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e254", "transitive":true, "type":"positive_in", "source":"schubertSkew", "refs":[{ "key":"LenartSottile2003", "href":"#LenartSottile2003", "label":"LS03", "tooltip":"Lenart and Sottile, Skew Schubert polynomials (2003)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "label":"SpecializesTo", "source_index":1, "id":"e255", "transitive":true, "type":"specializes_to", "source":"schubertUniversal", "refs":[{ "key":"Fulton1999", "href":"#Fulton1999", "label":"Ful99", "tooltip":"Fulton, Universal Schubert polynomials (1999)" }], "poset":true, "target":"schubert", "status":"theorem" },{ "attrs":{ "pairing":"Hall inner product" }, "label":"DualTo", "source_index":1, "id":"e256", "transitive":false, "type":"dual_to", "source":"schurAffine", "refs":[{ "key":"MorseSchilling2015", "href":"#MorseSchilling2015", "label":"MS15", "tooltip":"Morse and Schilling, Crystal approach to affine Schubert calculus (2015)" },{ "key":"LapointeMorse2008", "href":"#LapointeMorse2008", "label":"LM08", "tooltip":"Lapointe and Morse, Quantum cohomology and the $k$-Schur basis (2008)" }], "poset":false, "target":"kSchur", "status":"theorem" },{ "attrs":{ "scope":"empty inner shape" }, "label":"Contains", "source_index":1, "id":"e257", "transitive":true, "type":"contains", "source":"schurAffineSkew", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"schurAffine", "status":"theorem" },{ "label":"Contains", "source_index":2, "id":"e258", "transitive":true, "type":"contains", "source":"schurAffineSkew", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"schurCylindric", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e259", "transitive":true, "type":"positive_in", "source":"schurAffineSkew", "refs":[{ "key":"LapointeMorse2008", "href":"#LapointeMorse2008", "label":"LM08", "tooltip":"Lapointe and Morse, Quantum cohomology and the $k$-Schur basis (2008)" }], "poset":true, "target":"schurAffine", "status":"theorem" },{ "label":"PositiveIn", "source_index":4, "id":"e260", "transitive":true, "type":"positive_in", "source":"schurAffineSkew", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"degree zero" }, "label":"Contains", "source_index":1, "id":"e261", "transitive":true, "type":"contains", "source":"schurCylindric", "refs":[{ "key":"Postnikov2005", "href":"#Postnikov2005", "label":"Pos05", "tooltip":"Postnikov, Affine approach to quantum Schubert calculus (2005)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "attrs":{ "map":"top-degree component" }, "label":"SpecializesTo", "source_index":2, "id":"e262", "transitive":true, "type":"specializes_to", "source":"schurFactorial", "refs":[{ "key":"BiedenharnLouck1989", "href":"#BiedenharnLouck1989", "label":"BL89", "tooltip":"Biedenharn and Louck, A new class of symmetric polynomials defined in terms of tableaux (1989)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"SpecializesTo", "source_index":1, "id":"e263", "transitive":true, "type":"specializes_to", "source":"schurFactorial", "refs":[{ "key":"BiedenharnLouck1989", "href":"#BiedenharnLouck1989", "label":"BL89", "tooltip":"Biedenharn and Louck, A new class of symmetric polynomials defined in terms of tableaux (1989)" }], "poset":true, "target":"schurShifted", "status":"theorem" },{ "attrs":{ "scope":"unrestricted flag" }, "label":"Contains", "source_index":1, "id":"e264", "transitive":true, "type":"contains", "source":"schurFlagged", "refs":[{ "key":"LascouxSchutzenberger1982", "href":"#LascouxSchutzenberger1982", "label":"LS82", "tooltip":"Lascoux and Schützenberger, Polynômes de Schubert (1982)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"unrestricted row intervals" }, "label":"Contains", "source_index":2, "id":"e265", "transitive":true, "type":"contains", "source":"schurFlagged", "refs":[{ "key":"GesselViennot1989", "href":"#GesselViennot1989", "label":"GV89", "tooltip":"Gessel and Viennot, Determinants, paths, and plane partitions (1989)" },{ "key":"Wachs1985", "href":"#Wachs1985", "label":"Wac85", "tooltip":"Wachs, Flagged Schur functions, Schubert polynomials, and symmetrizing operators (1985)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e266", "transitive":true, "type":"positive_in", "source":"schurFlagged", "refs":[{ "key":"ReinerShimozono1995", "href":"#ReinerShimozono1995", "label":"RS95", "tooltip":"Reiner and Shimozono, Key polynomials and a flagged Littlewood–Richardson rule (1995)" }], "poset":true, "target":"key", "status":"theorem" },{ "label":"KTheoreticAnalogueOf", "source_index":1, "id":"e267", "transitive":false, "type":"k_theoretic_analogue_of", "source":"schurKP", "refs":[{ "key":"IkedaNaruse2013", "href":"#IkedaNaruse2013", "label":"IN13", "tooltip":"Ikeda and Naruse, K-theoretic analogues of factorial Schur P- and Q-functions (2013)" }], "poset":false, "target":"schurP", "status":"theorem" },{ "label":"KTheoreticAnalogueOf", "source_index":1, "id":"e268", "transitive":false, "type":"k_theoretic_analogue_of", "source":"schurKQ", "refs":[{ "key":"IkedaNaruse2013", "href":"#IkedaNaruse2013", "label":"IN13", "tooltip":"Ikeda and Naruse, K-theoretic analogues of factorial Schur P- and Q-functions (2013)" }], "poset":false, "target":"schurQ", "status":"theorem" },{ "attrs":{ "sign_rule":"Theorem 1.1", "note":"NN[beta]-positive in the cases of Corollary 1.2" }, "label":"SignedIn", "source_index":2, "id":"e269", "transitive":false, "type":"signed_in", "source":"schurKQ", "refs":[{ "key":"ChiuMarberg2024", "href":"#ChiuMarberg2024", "label":"CM24", "tooltip":"Chiu and Marberg, Expanding K-theoretic Schur Q-functions (2024)" }], "poset":false, "target":"schurKP", "status":"theorem" },{ "attrs":{ "map":"r=1" }, "label":"SpecializesTo", "source_index":1, "id":"e270", "transitive":true, "type":"specializes_to", "source":"schurLoop", "refs":[{ "key":"LamPylyavskyy2012", "href":"#LamPylyavskyy2012", "label":"LP12", "tooltip":"Lam and Pylyavskyy, Total positivity in loop groups, I: Whirls and curls (2012)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"omega and conjugate index" }, "label":"TransformsTo", "source_index":1, "id":"e271", "transitive":false, "type":"transforms_to", "source":"schurOrthogonal", "refs":[{ "key":"Hamel1997", "href":"#Hamel1997", "label":"Ham97", "tooltip":"Hamel, Determinantal forms for symplectic and orthogonal Schur functions (1997)" }], "poset":false, "target":"schurSymplectic", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e272", "transitive":true, "type":"positive_in", "source":"schurP", "refs":[{ "key":"Stembridge1989", "href":"#Stembridge1989", "label":"Ste89", "tooltip":"Stembridge, Shifted tableaux and the projective representations of symmetric groups (1989)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"multiply by $2^{\\length(\\lambda)}$" }, "label":"TransformsTo", "source_index":2, "id":"e273", "transitive":false, "type":"transforms_to", "source":"schurP", "refs":[{ "key":"Schur1911", "href":"#Schur1911", "label":"Sch11", "tooltip":"Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen (1911)" }], "poset":false, "target":"schurQ", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e274", "transitive":true, "type":"positive_in", "source":"schurQ", "refs":[{ "key":"Stembridge1989", "href":"#Stembridge1989", "label":"Ste89", "tooltip":"Stembridge, Shifted tableaux and the projective representations of symmetric groups (1989)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "proof":"skew Schur fundamental expansion" }, "label":"PositiveIn", "source_index":2, "id":"e275", "transitive":true, "type":"positive_in", "source":"schurRibbon", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e276", "transitive":true, "type":"positive_in", "source":"schurRibbon", "refs":[{ "key":"Sagan2001", "href":"#Sagan2001", "label":"Sag01", "tooltip":"Sagan, The symmetric group (2001)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "sign_rule":"coarsening parity" }, "label":"SignedIn", "source_index":3, "id":"e277", "transitive":false, "type":"signed_in", "source":"schurRibbon", "refs":[{ "key":"Moustakas2023x", "href":"#Moustakas2023x", "label":"Mou23", "tooltip":"Moustakas, Descent representations and colored quasisymmetric functions (2023)" }], "poset":false, "target":"completeH", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e278", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"Lascoux1990Keys", "href":"#Lascoux1990Keys", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Keys & standard bases (1990)" }], "poset":true, "target":"atom", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e279", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"Lascoux1990Keys", "href":"#Lascoux1990Keys", "label":"LS90", "tooltip":"Lascoux and Schützenberger, Keys & standard bases (1990)" }], "poset":true, "target":"generalizedDemazureAtoms", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e280", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "semiring":"NN[t]" }, "label":"PositiveIn", "source_index":5, "id":"e281", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":true, "target":"hallLittlewoodP", "status":"theorem" },{ "label":"PositiveIn", "source_index":4, "id":"e282", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"Macdonald1995", "href":"#Macdonald1995", "label":"Mac95", "tooltip":"Macdonald, Symmetric functions and Hall polynomials (1995)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"PositiveIn", "source_index":6, "id":"e283", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"HaglundLuotoMasonWilligenburg2011", "href":"#HaglundLuotoMasonWilligenburg2011", "label":"HLMW11", "tooltip":"Haglund, Luoto, Mason and Willigenburg, Quasisymmetric Schur functions (2011)" }], "poset":true, "target":"qSchur", "status":"theorem" },{ "label":"PositiveIn", "source_index":7, "id":"e284", "transitive":true, "type":"positive_in", "source":"schurS", "refs":[{ "key":"MasonRemmel2013", "href":"#MasonRemmel2013", "label":"MR13", "tooltip":"Mason and Remmel, Row-strict quasisymmetric Schur functions (2013)" }], "poset":true, "target":"rsqSchur", "status":"theorem" },{ "attrs":{ "map":"top-degree component" }, "label":"SpecializesTo", "source_index":1, "id":"e285", "transitive":true, "type":"specializes_to", "source":"schurShifted", "refs":[{ "key":"OkounkovOlshanski1996", "href":"#OkounkovOlshanski1996", "label":"OO96", "tooltip":"Okounkov and Olshanski, Shifted Schur functions (1996)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"ribbon skew shapes" }, "label":"Contains", "source_index":2, "id":"e286", "transitive":true, "type":"contains", "source":"schurSkew", "refs":[{ "key":"Schur1901", "href":"#Schur1901", "label":"Sch01", "tooltip":"Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen (1901)" }], "poset":true, "target":"schurRibbon", "status":"theorem" },{ "attrs":{ "scope":"empty inner shape" }, "label":"Contains", "source_index":1, "id":"e287", "transitive":true, "type":"contains", "source":"schurSkew", "refs":[{ "key":"Schur1901", "href":"#Schur1901", "label":"Sch01", "tooltip":"Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen (1901)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e288", "transitive":true, "type":"positive_in", "source":"schurSkew", "refs":[{ "key":"Gessel1984", "href":"#Gessel1984", "label":"Ges84", "tooltip":"Gessel, Multipartite ${P}$-partitions and inner products of skew Schur functions (1984)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "label":"PositiveIn", "source_index":4, "id":"e289", "transitive":true, "type":"positive_in", "source":"schurSkew", "refs":[{ "key":"Sagan2001", "href":"#Sagan2001", "label":"Sag01", "tooltip":"Sagan, The symmetric group (2001)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"PositiveIn", "source_index":5, "id":"e290", "transitive":true, "type":"positive_in", "source":"schurSkew", "refs":[{ "key":"Sagan2001", "href":"#Sagan2001", "label":"Sag01", "tooltip":"Sagan, The symmetric group (2001)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "map":"omega and conjugate index" }, "label":"TransformsTo", "source_index":1, "id":"e291", "transitive":false, "type":"transforms_to", "source":"schurSymplectic", "refs":[{ "key":"Hamel1997", "href":"#Hamel1997", "label":"Ham97", "tooltip":"Hamel, Determinantal forms for symplectic and orthogonal Schur functions (1997)" }], "poset":false, "target":"schurOrthogonal", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e292", "transitive":true, "type":"contains", "source":"schurToric", "refs":[{ "key":"Postnikov2005", "href":"#Postnikov2005", "label":"Pos05", "tooltip":"Postnikov, Affine approach to quantum Schubert calculus (2005)" }], "poset":true, "target":"schurSkew", "status":"theorem" },{ "attrs":{ "scope":"finite-variable toric shapes" }, "label":"PositiveIn", "source_index":2, "id":"e293", "transitive":true, "type":"positive_in", "source":"schurToric", "refs":[{ "key":"Postnikov2005", "href":"#Postnikov2005", "label":"Pos05", "tooltip":"Postnikov, Affine approach to quantum Schubert calculus (2005)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e294", "transitive":true, "type":"contains", "source":"slideF", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "attrs":{ "proof":"Thm. 3.13 and Cor. 3.11" }, "label":"PositiveIn", "source_index":4, "id":"e295", "transitive":true, "type":"positive_in", "source":"slideF", "refs":[{ "key":"Searles2019PolynomialBases", "href":"#Searles2019PolynomialBases", "label":"Sea19", "tooltip":"Searles, Polynomial bases: Positivity and Schur multiplication (2019)" },{ "key":"HicksNiese2024x", "href":"#HicksNiese2024x", "label":"HN24", "tooltip":"Hicks and Niese, Quasisymmetric divided difference operators and polynomial bases (2024)" }], "poset":true, "target":"fundamentalParticle", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e296", "transitive":true, "type":"positive_in", "source":"slideF", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":true, "target":"slideM", "status":"theorem" },{ "label":"StableLimit", "source_index":2, "id":"e297", "transitive":true, "type":"stable_limit", "source":"slideF", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":false, "target":"gessel", "status":"theorem" },{ "label":"Contains", "source_index":1, "id":"e298", "transitive":true, "type":"contains", "source":"slideM", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "attrs":{ "proof":"definition", "scope":"nonsymmetric monomials" }, "label":"PositiveIn", "source_index":3, "id":"e299", "transitive":true, "type":"positive_in", "source":"slideM", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":true, "target":"monomial", "status":"theorem" },{ "label":"StableLimit", "source_index":2, "id":"e300", "transitive":true, "type":"stable_limit", "source":"slideM", "refs":[{ "key":"AssafSearles2016", "href":"#AssafSearles2016", "label":"AS17", "tooltip":"Assaf and Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams (2017)" }], "poset":false, "target":"qmonom", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e301", "transitive":true, "type":"positive_in", "source":"stanleySym", "refs":[{ "key":"MorseSchilling2015", "href":"#MorseSchilling2015", "label":"MS15", "tooltip":"Morse and Schilling, Crystal approach to affine Schubert calculus (2015)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"affine Grassmannian permutations" }, "label":"Contains", "source_index":1, "id":"e302", "transitive":true, "type":"contains", "source":"stanleySymAffine", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"schurAffine", "status":"theorem" },{ "label":"Contains", "source_index":2, "id":"e303", "transitive":true, "type":"contains", "source":"stanleySymAffine", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"schurCylindric", "status":"theorem" },{ "attrs":{ "scope":"finite permutations" }, "label":"Contains", "source_index":3, "id":"e304", "transitive":true, "type":"contains", "source":"stanleySymAffine", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"stanleySym", "status":"theorem" },{ "label":"PositiveIn", "source_index":5, "id":"e305", "transitive":true, "type":"positive_in", "source":"stanleySymAffine", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"schurAffine", "status":"conjecture" },{ "attrs":{ "scope":"rectangular Schur coefficients", "note":"partial result" }, "label":"PositiveIn", "source_index":4, "id":"e306", "transitive":true, "type":"positive_in", "source":"stanleySymAffine", "refs":[{ "key":"MorseSchilling2015", "href":"#MorseSchilling2015", "label":"MS15", "tooltip":"Morse and Schilling, Crystal approach to affine Schubert calculus (2015)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"$v=e$" }, "label":"Contains", "source_index":1, "id":"e307", "transitive":true, "type":"contains", "source":"stanleySymAffineSkew", "refs":[{ "key":"Lam2006", "href":"#Lam2006", "label":"Lam06", "tooltip":"Lam, Affine Stanley symmetric functions (2006)" }], "poset":true, "target":"stanleySymAffine", "status":"theorem" },{ "attrs":{ "scope":"type B" }, "label":"PositiveIn", "source_index":1, "id":"e308", "transitive":true, "type":"positive_in", "source":"stanleySymBC", "refs":[{ "key":"Lam1996", "href":"#Lam1996", "label":"Lam96", "tooltip":"Lam, $B_n$ Stanley symmetric functions (1996)" }], "poset":true, "target":"schurP", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e309", "transitive":true, "type":"positive_in", "source":"stanleySymBC", "refs":[{ "key":"HawkesParamonovSchilling2017", "href":"#HawkesParamonovSchilling2017", "label":"HPS17", "tooltip":"Hawkes, Paramonov and Schilling, Crystal analysis of type C Stanley symmetric functions (2017)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"omega in A_n" }, "label":"PositiveIn", "source_index":1, "id":"e310", "transitive":true, "type":"positive_in", "source":"stanleySymDouble", "refs":[{ "key":"Hawkes2020", "href":"#Hawkes2020", "label":"Haw20", "tooltip":"Hawkes, Crystal structures for double Stanley symmetric functions (2020)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":3, "id":"e311", "transitive":true, "type":"positive_in", "source":"stanleySymInvolution", "refs":[{ "key":"HamakerMarbergPawlowski2017", "href":"#HamakerMarbergPawlowski2017", "label":"HMP17", "tooltip":"Hamaker, Marberg and Pawlowski, Schur P-positivity and involution Stanley symmetric functions (2017)" }], "poset":true, "target":"schurP", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e312", "transitive":true, "type":"positive_in", "source":"stanleySymInvolution", "refs":[{ "key":"HamakerMarbergPawlowski2017", "href":"#HamakerMarbergPawlowski2017", "label":"HMP17", "tooltip":"Hamaker, Marberg and Pawlowski, Schur P-positivity and involution Stanley symmetric functions (2017)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e313", "transitive":true, "type":"positive_in", "source":"stanleySymInvolution", "refs":[{ "key":"HamakerMarbergPawlowski2017", "href":"#HamakerMarbergPawlowski2017", "label":"HMP17", "tooltip":"Hamaker, Marberg and Pawlowski, Schur P-positivity and involution Stanley symmetric functions (2017)" }], "poset":true, "target":"stanleySym", "status":"theorem" },{ "attrs":{ "map":"t=0" }, "label":"SpecializesTo", "source_index":1, "id":"e314", "transitive":true, "type":"specializes_to", "source":"tAtom", "refs":[{ "key":"HaglundLuotoMasonWilligenburg2011", "href":"#HaglundLuotoMasonWilligenburg2011", "label":"HLMW11", "tooltip":"Haglund, Luoto, Mason and Willigenburg, Quasisymmetric Schur functions (2011)" }], "poset":true, "target":"atom", "status":"theorem" },{ "attrs":{ "scope":"stable singularities with positive sum" }, "label":"PositiveIn", "source_index":1, "id":"e315", "transitive":true, "type":"positive_in", "source":"thomPolynomial", "refs":[{ "key":"PragaczWeber2007", "href":"#PragaczWeber2007", "label":"PW07", "tooltip":"Pragacz and Weber, Positivity of Schur function expansions of Thom polynomials (2007)" }], "poset":true, "target":"hookSchur", "status":"theorem" },{ "attrs":{ "proof":"Thm. 2", "sign_rule":"i-maximal partitions" }, "label":"SignedIn", "source_index":1, "id":"e316", "transitive":false, "type":"signed_in", "source":"tutteSymmetric", "refs":[{ "key":"CrewSpirkl2020x", "href":"#CrewSpirkl2020x", "label":"CS21", "tooltip":"Crew and Spirkl, A Complete Multipartite Basis for the Chromatic Symmetric Function (2021)" }], "poset":false, "target":"completeMultipartite", "status":"theorem" },{ "attrs":{ "scope":"unit interval graphs after q -> q+1" }, "label":"PositiveIn", "source_index":3, "id":"e317", "transitive":true, "type":"positive_in", "source":"unicellularLLT", "refs":[{ "key":"AlexanderssonSulzgruber2020x", "href":"#AlexanderssonSulzgruber2020x", "label":"AS20", "tooltip":"Alexandersson and Sulzgruber, A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions (2020)" }], "poset":true, "target":"elementaryE", "status":"theorem" },{ "attrs":{ "scope":"bandwidth at most 3", "note":"3-Schur positivity" }, "label":"PositiveIn", "source_index":2, "id":"e318", "transitive":true, "type":"positive_in", "source":"unicellularLLT", "refs":[{ "key":"Miller2019", "href":"#Miller2019", "label":"Mil19", "tooltip":"Miller, On the $k$-Schur positivity of $k$-bandwidth LLT polynomials (2019)" }], "poset":true, "target":"kSchur", "status":"theorem" },{ "attrs":{ "proof":"explicit combinatorial formula", "scope":"max_i a_i <= 2" }, "label":"PositiveIn", "source_index":1, "id":"e319", "transitive":true, "type":"positive_in", "source":"unicellularLLT", "refs":[{ "key":"Blasiak2016", "href":"#Blasiak2016", "label":"Bla16", "tooltip":"Blasiak, Haglund’s conjecture on 3-column Macdonald polynomials (2016)" }], "poset":true, "target":"schurS", "status":"theorem" },{ "attrs":{ "scope":"top homogeneous component" }, "label":"Contains", "source_index":1, "id":"e320", "transitive":true, "type":"contains", "source":"weightedBondChromaticSymmetric", "refs":[{ "key":"DLeonWachs2026x", "href":"#DLeonWachs2026x", "label":"DW26", "tooltip":"D’León and Wachs, Weighted bond posets and a new chromatic symmetric function (2026)" }], "poset":true, "target":"mobiusSymmetricFunction", "status":"theorem" },{ "label":"PositiveIn", "source_index":1, "id":"e321", "transitive":true, "type":"positive_in", "source":"yqSchur", "refs":[{ "key":"LuotoMykytiukWilligenburg2013", "href":"#LuotoMykytiukWilligenburg2013", "label":"LMW13", "tooltip":"Luoto, Mykytiuk and Willigenburg, An introduction to quasisymmetric Schur functions (2013)" }], "poset":true, "target":"gessel", "status":"theorem" },{ "label":"PositiveIn", "source_index":2, "id":"e322", "transitive":true, "type":"positive_in", "source":"yqSchur", "refs":[{ "key":"LuotoMykytiukWilligenburg2013", "href":"#LuotoMykytiukWilligenburg2013", "label":"LMW13", "tooltip":"Luoto, Mykytiuk and Willigenburg, An introduction to quasisymmetric Schur functions (2013)" }], "poset":true, "target":"qmonom", "status":"theorem" },{ "attrs":{ "map":"rho and reverse index" }, "label":"TransformsTo", "source_index":3, "id":"e323", "transitive":false, "type":"transforms_to", "source":"yqSchur", "refs":[{ "key":"LuotoMykytiukWilligenburg2013", "href":"#LuotoMykytiukWilligenburg2013", "label":"LMW13", "tooltip":"Luoto, Mykytiuk and Willigenburg, An introduction to quasisymmetric Schur functions (2013)" }], "poset":false, "target":"qSchur", "status":"theorem" },{ "attrs":{ "map":"omega" }, "label":"TransformsTo", "source_index":4, "id":"e324", "transitive":false, "type":"transforms_to", "source":"yqSchur", "refs":[{ "key":"MasonNiese2015", "href":"#MasonNiese2015", "label":"MN15", "tooltip":"Mason and Niese, Skew row-strict quasisymmetric Schur functions (2015)" }], "poset":false, "target":"rsyqSchur", "status":"theorem" }], "status_counts":{ "conjecture":2, "theorem":322 }, "relation_types":[{ "transitive":true, "type":"contains", "count":57, "label":"Contains", "poset":true },{ "transitive":true, "type":"degenerates_to", "count":0, "label":"DegeneratesTo", "poset":true },{ "transitive":true, "type":"generalizes", "count":13, "label":"Generalizes", "poset":true },{ "transitive":true, "type":"positive_in", "count":140, "label":"PositiveIn", "poset":true },{ "transitive":true, "type":"specializes_to", "count":57, "label":"SpecializesTo", "poset":true },{ "transitive":false, "type":"dual_to", "count":2, "label":"DualTo", "poset":false },{ "transitive":false, "type":"k_theoretic_analogue_of", "count":10, "label":"KTheoreticAnalogueOf", "poset":false },{ "transitive":false, "type":"nsym_qsym_dual", "count":3, "label":"NSymQSymDual", "poset":false },{ "transitive":true, "type":"refined_by", "count":2, "label":"RefinedBy", "poset":false },{ "transitive":false, "type":"signed_in", "count":8, "label":"SignedIn", "poset":false },{ "transitive":true, "type":"stable_limit", "count":10, "label":"StableLimit", "poset":false },{ "transitive":false, "type":"transforms_to", "count":22, "label":"TransformsTo", "poset":false }], "generated_from":"polydata", "references":["Alexandersson2015gbMacdonald","AlexanderssonSawhney2017","AlexanderssonSawhney2019","AlexanderssonSulzgruber2019","AlexanderssonSulzgruber2020x","AliniaeifardWangWilligenburg2021x","AllenHallamMason2018","AndersonFulton2021x","Assaf2018Kostka","Assaf2021","AssafBergeron2019","AssafGonzalez2018","AssafSearles2016","AssafSearles2018","AssafSearles2019","AssafvanWilligenburg2019","Athanasiadis2015","BallantineDaughertyHicksMason2020","BandlowMorse2010","BergBergeronSaliolaSerranoZabrocki2014","BergeronGarsiaLevenXin2015","BiedenharnLouck1989","BilleyRhoadesTewari2019","Blasiak2016","BlasiakHaimanMorsePunSeelinger2025x","BlasiakMorsePun2020x","BlasiakMorsePunSummers2018","BlasiakMorseSeelinger2020","BlasiakMorseSummers2019","BuchKreschShimozonoTamvakisYong2007","ChenLiLouck2002","ChiuMarberg2024","ChoiNamOh2025","CorteelMandelshtamMasonWilliams2019x","CorteelMandelshtamMasonWilliams2022","CrewPechenikSpirkl2023x","CrewSpirkl2020x","DLeonWachs2026x","Daugherty2024Extended","DotyWalker1992","Ehrhard2023x","Ellzey2017","Ferreira2011","FominGelfandPostnikov1997","FominKirillov1994grothendieck","Fulton1999","GalashinGrinbergLiu2016","Gasharov1996","GelfandKrobLascouxLeclercRetakhThibon1995","Gessel1984","GesselReutenauer1993","GesselViennot1989","GrinbergVassilieva2024x","HaglundHaimanLoehr2005","HaglundLuotoMasonWilligenburg2011","HaglundMasonRemmel2012","Haiman2001","HamakerMarbergPawlowski2017","HamakerMarbergPawlowski2018","Hamel1997","Hawkes2020","Hawkes2023","HawkesParamonovSchilling2017","He2020","HendersonWachs2012","HicksNiese2024x","Hikita2014","Hikita2024x","IkedaNaruse2013","Iwao2023","Jack1970","JingLi2015","Kato2025x","KnopSahi1997","Kundu2024x","Lam1996","Lam2006","LamPylyavskyy2007combinatorialhopf","LamPylyavskyy2012","LapointeMorse2008","Lascoux1990Grothendieck","Lascoux1990Keys","Lascoux2001","Lascoux2004","Lascoux97ribbontableaux","LascouxSchutzenberger1982","LascouxSchutzenberger1990","Lee2015","LeeOh2025x","LenartSottile2003","Littlewood1961","LoehrWarrington2019","LuRuanWang2025","LuotoMykytiukWilligenburg2013","Macdonald1987","Macdonald1995","MandelshtamNiergarthSingh2026x","Mason2009","MasonNiese2015","MasonRemmel2013","MatsumuraSugimoto2019","Merca2015","Miller2019","Monical2017","MonicalPechenikSearles2021","MorseSchilling2015","Moustakas2023x","NadeauTewari2023x","NieseSundaramWilligenburgVegaWang2023","Oguz2019","Okada2019x","OkounkovOlshanski1996","OkounkovOlshanski1997","Opdam1995","Panyushev2010","PetreolleSokal2019x","Postnikov2005","PragaczWeber2007","RamYip2011","ReinerShimozono1995","Riedtmann2018x","Sagan2001","SaganShareshianWachs2011","Sahi1994","Schur1901","Schur1911","Searles2019PolynomialBases","Searles2025","SearlesSlatteryHolmes2026","ShareshianWachs2010","ShareshianWachs2012","Shigechi2017x","ShimozonoYu2021","Stanley1984","Stanley1989","StanleyEC2","Stembridge1989","Stembridge1997","Wachs1985","Weising2024AlmostSymmetricx","YangRemmel1998","Yeliussizov2017"], "version":1 }