#Quasisymmetric functions
Quasisymmetric functions were formally introduced by I. Gessel in [Ges84]. Earlier work on \(P\)-partitions anticipated this development. For an introduction to quasisymmetric functions, see [LMW13].
A function \(f\) is quasisymmetric if for every composition \(\alpha\) of length \(\ell,\) the coefficient of \(x_1^{\alpha_1} \dotsm x_\ell^{\alpha_\ell}\) is the same as the coefficient of \(x_{i_1}^{\alpha_1} \dotsm x_{i_\ell}^{\alpha_\ell},\) for any \(0 \lt i_1 \lt i_2 \lt \dotsb \lt i_{\ell}.\) The set of quasisymmetric functions forms a graded ring, \(\spaceQSym.\)
Quasisymmetric functions of degree \(n\) are usually indexed by either integer compositions of \(n,\) or subsets of \([n-1].\) Given a composition \(\alpha \vDash n\) with \(\ell\) parts, define \[S_\alpha \coloneqq \{\alpha_1, \alpha_1+\alpha_2,\dotsc, \alpha_1+\alpha_2+\dotsb+\alpha_{\ell-1}\}.\] The bijection \(\alpha \mapsto S_\alpha\) maps compositions of \(n\) to subsets of \([n-1].\) The partial order \(\alpha \leq \beta\) denotes refinement. That is, \(\beta\) can be obtained from \(\alpha\) by adding consecutive parts of \(\alpha.\) When \(\alpha \leq \beta,\) we illustrate this relationship with bars between parts of \(\alpha,\) such that parts between bars add to parts of \(\beta.\)
Example (Refinement and bars).
Taken from [AS19]. \[112|341|21|34|2 \quad\text{corresponds to}\quad \alpha = 11234121342, \quad \beta = 48372.\]
The most prevalent quasisymmetric functions are perhaps the Gessel quasisymmetric functions. P. Nadeau and V. Tewari study the interaction between quasisymmetric polynomials and Postnikov’s divided symmetrization operator [NT21]. Their formulas are motivated by Schubert calculus and use the Aval–Bergeron–Bergeron decomposition modulo positive degree quasisymmetric polynomials. N. Bergeron, L. Gagnon, P. Nadeau, H. Spink, and V. Tewari introduce equivariant equivariant quasisymmetry for polynomials in two alphabets [BGNS+25]. This leads to double fundamental and double forest polynomials, intended as quasisymmetric analogues of double Schur and double Schubert polynomials. R. M. Adin, I. M. Gessel, V. Reiner, and Y. Roichman introduce cyclic quasisymmetric functions [AGRR21]. These are quasisymmetric functions with compatibility under cyclic rotation of the variables, and the paper develops cyclic analogues of descent-set enumerators and related bases. X. Gao, L. Guo, and X.-S. Peng introduce multi-quasisymmetric functions with semigroup exponents [GGP24]. Their construction extends the usual quasisymmetric Hopf-algebra framework and interacts with Rota–Baxter algebra structures.
#Duality with noncommutative symmetric functions
The algebra \(\spaceQSym\) is graded-dual to the algebra NSym of noncommutative symmetric functions [GKLL+95]. Under the standard pairing, \[\langle \completeH_\alpha,\qmonom_\beta\rangle=\delta_{\alpha,\beta},\] where \(\completeH_\alpha\) denotes the noncommutative complete homogeneous basis and \(\qmonom_\beta\) denotes the monomial quasisymmetric basis. Thus product rules in \(\spaceQSym\) are dual to coproduct rules in NSym, and coproduct rules in \(\spaceQSym\) are dual to products in NSym.
This duality should be compared with the ordinary Hall inner product on symmetric functions. In the symmetric world, Schur functions are self-dual. In the QSym–NSym world, the two sides are different algebras: composition-indexed quasisymmetric functions live on the commutative side, while noncommutative complete, ribbon, immaculate, and noncommutative Schur functions live on the dual side.
The most useful dictionary is: \( \text{QSym basis} \) \( \text{Dual NSym basis} \) \( \text{What it records}\) \( \qmonom_\alpha \) \( \completeH_\alpha \) \( \text{ordered blocks of a composition}\) \( \gessel_\alpha \) \( R_\alpha \) \( \text{descent sets and ribbon shapes}\) \( \text{dual immaculate} \) \( \text{immaculate} \) \( \text{composition-tableau Schur analogues}\) \( \qPsi_\alpha,\qPhi_\alpha \) \( \Psi_\alpha,\Phi_\alpha \) \( \text{power-sum analogues}\) Here \(R_\alpha\) denotes the noncommutative ribbon basis. The first row is the defining pairing, while the second row is often the practical one: many combinatorial generating functions naturally expand in the fundamental functions \(\gessel_\alpha,\) and their dual statements use ribbons in NSym.
Example
For compositions of \(2,\) the duality gives \[\langle \completeH_{2},\qmonom_{2}\rangle=1,\qquad \langle \completeH_{2},\qmonom_{11}\rangle=0,\qquad \langle \completeH_{11},\qmonom_{11}\rangle=1.\] The refinement order appears when changing from monomial quasisymmetric functions to the fundamental basis; dually, the same incidence matrix changes the noncommutative complete basis to the ribbon basis.
The representation-theoretic shadow of this duality is the 0-Hecke algebra. The Grothendieck group of finite-dimensional \(0\)-Hecke modules identifies with \(\spaceQSym,\) while the Grothendieck group of projective modules identifies with NSym [KT97]. Under the quasisymmetric characteristic, simple \(0\)-Hecke modules map to the fundamental quasisymmetric functions. Induction of modules corresponds to multiplication, and restriction corresponds to coproduct. This is the direct analogue of how the Frobenius characteristic packages induction and restriction for symmetric-group representations.
#Monomial quasisymmetric functions
Given a composition \(\alpha\) with \(\ell\) parts, we define the monomial quasisymmetric functions as \[\qmonom_\alpha(\xvec) \coloneqq \sum_{i_1 \lt{} i_2 \lt{} \dotsb \lt{} i_\ell} x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2} \dotsm x_{i_\ell}^{\alpha_\ell}.\] The functions \(\qmonom_\alpha\) constitute a basis for the space of homogeneous quasisymmetric functions of degree \(n\) as \(\alpha\) ranges over all compositions of \(n.\)
The monomial quasisymmetric functions refine the monomial symmetric functions, \[\monomial_{\lambda}(\xvec) = \sum_{\alpha \sim \lambda}\qmonom_{\alpha}(\xvec)\] where we sum over all compositions \(\alpha\) that are a permutation of \(\lambda.\)
O. Pechenik and M. Satriano introduce double monomial quasisymmetric functions as torus-equivariant Schubert representatives for the James reduced product model of \(\spaceQSym\) [PS23]. They give a cellular basis and a combinatorial Littlewood–Richardson rule for the structure constants.
D. Grinberg and E. A. Vassilieva construct an enriched monomial basis of \(\spaceQSym\) from weighted posets [GV21]. This gives another bridge between \(P\)-partition enumerators and quasisymmetric bases.
#Powersum quasisymmetric functions (Psi)
There are two quasisymmetric refinements of the power-sum symmetric functions introduced in [BDHM+20], denoted \(\qPsi_\alpha\) and \(\qPhi_\alpha.\)
A. Lazzeroni introduces another powersum basis for quasisymmetric functions, defined by matrix fillings [Laz23]. It has a shuffle product, a deconcatenation coproduct, and a change-of-basis rule to the fundamental quasisymmetric functions using tuples of ribbons. The construction also lifts to quasisymmetric functions in non-commuting variables. J. M. Campbell uses quasisymmetric power sums to define quasi-immanants [Cam25]. These refine immanants by replacing cycle types with cycle compositions, and a special case attached to quasisymmetric Schur functions gives an analogue of second immanants.
Given a pair of compositions of \(n,\) \(\alpha \leq \beta,\) related by \[\alpha_{11} \alpha_{12} \dotsc \alpha_{1,i_1}| \alpha_{21} \alpha_{22} \dotsc \alpha_{2,i_2} | \dotsb | \alpha_{k1} \alpha_{k2} \dotsc \alpha_{k,i_k}\] let \[\pi(\alpha,\beta) \coloneqq \prod_{j=1}^k (\alpha_{j1})(\alpha_{j1}+\alpha_{j2})\dotsb (\alpha_{j1}+\alpha_{j2}+\dotsb +\alpha_{j,i_j}).\] The quasisymmetric power sum \(\qPsi_\alpha\) is defined as \[\qPsi_\alpha(\xvec) \coloneqq z_\alpha \sum_{\beta \geq \alpha} \frac{1}{\pi(\alpha,\beta)} \qmonom_\beta(\xvec).\] For example, \(\Psi_{231} = \frac{1}{10}\qmonom_6 + \frac{1}{4}\qmonom_{24}+ \frac{3}{5}\qmonom_{51}+\qmonom_{231}.\) It was shown in [BDHM+20] that quasisymmetric power sums refine the usual power sums as \[\powerSum_{\lambda}(\xvec) = \sum_{\alpha \sim \lambda}\qPsi_{\alpha}(\xvec).\]
The \(\qPsi_\alpha\) have a nice relationship with Gessel quasisymmetric functions.
S.-I. Choi, Y.-H. Kim, and Y.-T. Oh [CKO24] study quasisymmetric power-sum expansions through poset modules for the \(0\)-Hecke algebra. Their method applies to several Schur-like quasisymmetric bases, including the dual immaculate and extended Schur bases, and gives border-strip-tableau descriptions for the coefficients in these \(\qPsi\)-expansions.
#Powersum quasisymmetric functions (Phi)
#Powersum quasisymmetric functions (p)
In [AWW21], the authors introduce a third quasisymmetric refinement of the power-sum basis, denoted \(\qRho_\alpha.\) They refer to this basis as the combinatorial quasisymmetric power-sum basis .
R. I. Liu and M. Tang study shuffle bases of quasisymmetric functions and use infinitesimal characters to characterize when such bases give Hopf-algebra isomorphisms with the shuffle algebra of compositions [LT23]. Their framework gives general constructions of quasisymmetric power sums, recovering several earlier bases in a uniform way. M. Tang studies substring-compatible permutation statistics, constructing a substring coalgebra analogous to the shuffle algebra of a shuffle-compatible statistic [Tan25]. Under a weak shuffle compatibility hypothesis, the shuffle algebra and substring coalgebra combine to form a Hopf algebra; the paper conjectures that the descent, peak, and valley sets are the only nontrivial statistics satisfying both compatibilities.
Example (Substring coalgebra for descents).
The descent set is substring-compatible. If \(\pi=\pi_1\dotsm\pi_n\) and \(\sigma=\pi_i\dotsm\pi_j\) is a consecutive substring, then \(\DES(\sigma)\) is obtained from \(\DES(\pi)\cap\{i,\dotsc,j-1\}\) by subtracting \(i-1\) from each position.
For example, in the substring coalgebra for the descent statistic, \[\begin{aligned} \Delta([213]_{\DES}) &= [\,]_{\DES}\otimes [213]_{\DES} + [2]_{\DES}\otimes [13]_{\DES} \\ &\quad + [21]_{\DES}\otimes [3]_{\DES} + [213]_{\DES}\otimes [\,]_{\DES}. \end{aligned}\] Under the Hopf isomorphism \([ \pi]_{\DES}\mapsto \gessel_{\operatorname{comp}(\DES(\pi))},\) this becomes the usual coproduct \[\Delta(\gessel_{(1,2)}) = 1\otimes\gessel_{(1,2)} + \gessel_{(1)}\otimes\gessel_{(2)} + \gessel_{(1,1)}\otimes\gessel_{(1)} + \gessel_{(1,2)}\otimes 1.\]
O. Bouillot, J.-C. Novelli, and J.-Y. Thibon construct new bases of \(\spaceQSym\) and \(\mathrm{WQSym}\) from a two-parameter deformation of the quasi-shuffle [BNT22]. The deformation is defined using the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, and the product rule in the new bases is given by this operation.
H. Mlodecki constructs an explicit bidendriform automorphism of the Hopf algebra of word quasisymmetric functions \(\mathrm{WQSym}\) [Mlo24]. The construction uses decompositions of packed words through red and blue biplane forests and gives a concrete self-duality map compatible with the bidendriform structure.
N. Bergeron, K. Chan, F. Soltani, and M. Zabrocki study quasisymmetric polynomials in anticommuting variables [BCSZ23]. The quotient of the exterior algebra by the ideal generated by these quasisymmetric polynomials has a ballot-sequence basis and Hilbert series \[\sum_{k=0}^{\lfloor n/2\rfloor} f^{(n-k,k)} q^k.\]
Let \(\alpha\) be a composition and let \(\lambda\) be the partition obtained from \(\alpha\) by rearranging the parts in decreasing order. The monomial expansion of \(\qRho_\alpha\) is given by \[\qRho_\alpha(\xvec) = \sum_{\beta} R_{\alpha \beta} \qmonom_\beta(\xvec)\] where \(R_{\alpha \beta}\) is the number of ordered set partitions (see A000670) \[\gamma_1 | \gamma_2 | \dotsb | \gamma_k\] with the following two properties. The word \[\lambda_{\gamma_{11}}, \lambda_{\gamma_{12}},\dotsc,\; \lambda_{\gamma_{21}}, \lambda_{\gamma_{22}},\dotsc,\; \dotsc, \lambda_{\gamma_{k1}}, \lambda_{\gamma_{k2}},\dotsc,\] is equal to \(\alpha,\) and \(\lambda_{\gamma_{i1}}+ \lambda_{\gamma_{i2}}+\dotsb = \beta_i\) for all \(i.\) For example, if \(\lambda = 322211,\) then the set-partition \(235|16|4\) contributes to \(R_{\alpha \beta}\) where \[\alpha = (2,2,1,3,1,2), \qquad \beta = (5,4,2).\] The original article [AWW21] uses the enumeration of certain matrices to define the \(R_{\alpha \beta},\) but the above definition is equivalent to theirs.
Example (Table of \(\qRho_\alpha,\) for size \(4\)).
The following table shows the monomial expansion of \(\qRho_\alpha.\)
\( \alpha \) \( \qRho_\alpha \) \( 4 \) \( \qmonom_4 \) \( 13 \) \( \qmonom_{13} \) \( 22 \) \( \qmonom_4+2 \qmonom_{22} \) \( 31 \) \( \qmonom_4+\qmonom_{31} \) \( 112 \) \( \qmonom_{22}+2 \qmonom_{112} \) \( 121 \) \( 2 \qmonom_{13}+2 \qmonom_{121} \) \( 211 \) \( \qmonom_4+\qmonom_{22}+2 \qmonom_{31}+2 \qmonom_{211} \) \( 1111 \) \( \qmonom_4+4 \qmonom_{13}+6 \qmonom_{22}+4 \qmonom_{31}+12 \qmonom_{112}+12 \qmonom_{121}+12 \qmonom_{211}+24 \qmonom_{1111} \)The authors also give formulas for computing product and coproduct.
#Involutions on quasisymmetric functions
The standard involution on symmetric functions that sends \(\schurS_\lambda\) to \(\schurS_{\lambda'}\) can be extended to quasisymmetric functions by \(\omega.\) Two other useful involutions, \(\rho\) and \(\psi,\) are also defined via the Gessel quasisymmetric functions [LMW13]: \[\omega \gessel_{n,S}(x) = \gessel_{n,[n-1]\setminus (n-S)}(x), \qquad \rho \gessel_{n,S}(x) = \gessel_{n,n-S}(x), \qquad \psi \gessel_{n,S}(x) = \gessel_{n, [n-1]\setminus S}(x).\] Here, \(n-S\) is the set \(\{n-s : s \in S\}.\) Equivalently, \(\rho\gessel_\alpha=\gessel_{\alpha^r},\) where \(\alpha^r\) is the reverse composition. The map \(\omega\) restricts to the usual involution on \(\spaceSym,\) while \(\psi\) is the complementary descent-set involution. These maps satisfy \(\psi=\rho\circ\omega=\omega\circ\rho.\)
In [BDHM+20], it is shown that \(\omega\left( \qPsi_\alpha \right) = (-1)^{|\alpha|-\length(\alpha)}\qPsi_{\alpha^r},\) where \(\alpha^r\) denotes the reverse of \(\alpha.\) The same property holds for \(\qRho_\alpha,\) see [Thm. 5.7, AWW21].
Bibliography
- [AGRR21]Ron M. Adin, Ira M. Gessel, Victor Reiner and Yuval Roichman. Cyclic quasi-symmetric functions. Israel Journal of Mathematics, 243(1):437–500, 2021.
.bib
@article{AdinGesselReinerRoichman2021, author = {Adin, Ron M. and Gessel, Ira M. and Reiner, Victor and Roichman, Yuval}, title = {Cyclic quasi-symmetric functions}, year = {2021}, journal = {Israel Journal of Mathematics}, volume = {243}, number = {1}, pages = {437--500}, publisher = {Springer Science and Business Media LLC}, doi = {10.1007/s11856-021-2167-4}, url = {http://dx.doi.org/10.1007/s11856-021-2167-4}, issn = {1565-8511}, eprint = {1811.05440} } - [AS19]Per Alexandersson and Robin Sulzgruber. P-partitions and p-positivity. International Mathematics Research Notices, 2021(14):10848–10907, July 2019.
.bib
@article{AlexanderssonSulzgruber2019, title = {{P}-Partitions and {p}-Positivity}, volume = {2021}, ISSN = {1687-0247}, url = {http://dx.doi.org/10.1093/imrn/rnz130}, DOI = {10.1093/imrn/rnz130}, number = {14}, journal = {International Mathematics Research Notices}, publisher = {Oxford University Press (OUP)}, author = {Alexandersson, Per and Sulzgruber, Robin}, year = {2019}, month = jul, pages = {10848–10907} } - [AWW21]Farid Aliniaeifard, Victor Wang and Stephanie Willigenburg. ${P}$-partition power sums. arXiv:2112.06457, 2021.
.bib
@article{AliniaeifardWangWilligenburg2021x, Author = {Farid Aliniaeifard and Victor Wang and Stephanie van Willigenburg}, Title = {${P}$-partition power sums}, Year = {2021}, Eprint = {2112.06457}, url = {https://arxiv.org/abs/2112.06457}, journal = {arXiv e-prints} } - [BDHM+20]Cristina Ballantine, Zajj Daugherty, Angela Hicks, Sarah Mason and Elizabeth Niese. On quasisymmetric power sums. Journal of Combinatorial Theory, Series A, 175:105273, October 2020.
.bib
@article{BallantineDaughertyHicksMason2020, doi = {10.1016/j.jcta.2020.105273}, url2 = {https://doi.org/10.1016/j.jcta.2020.105273}, year = {2020}, month = oct, publisher = {Elsevier {BV}}, volume = {175}, pages = {105273}, author = {Cristina Ballantine and Zajj Daugherty and Angela Hicks and Sarah Mason and Elizabeth Niese}, title = {On quasisymmetric power sums}, journal = {Journal of Combinatorial Theory, Series A} } - [BCSZ23]Nantel Bergeron, Kelvin Chan, Farhad Soltani and Mike Zabrocki. Quasisymmetric harmonics of the exterior algebra. Canadian Mathematical Bulletin, 66(3):997–1013, 2023.
.bib
@article{BergeronChanSoltaniZabrocki2023, author = {Nantel Bergeron and Kelvin Chan and Farhad Soltani and Mike Zabrocki}, title = {Quasisymmetric harmonics of the exterior algebra}, year = {2023}, journal = {Canadian Mathematical Bulletin}, volume = {66}, number = {3}, pages = {997--1013}, doi = {10.4153/s0008439523000024}, url = {https://doi.org/10.4153/s0008439523000024}, eprint = {2206.02065} } - [BGNS+25]Nantel Bergeron, Lucas Gagnon, Philippe Nadeau, Hunter Spink and Vasu Tewari. Equivariant quasisymmetry and noncrossing partitions. arXiv:2504.15234, 2025.
.bib
@article{BergeronGagnonNadeauSpinkTewari2025x, author = {Nantel Bergeron and Lucas Gagnon and Philippe Nadeau and Hunter Spink and Vasu Tewari}, title = {Equivariant quasisymmetry and noncrossing partitions}, year = {2025}, eprint = {2504.15234}, url = {https://arxiv.org/abs/2504.15234}, journal = {arXiv e-prints} } - [BNT22]Olivier Bouillot, Jean-Christophe Novelli and Jean-Yves Thibon. A two-parameter deformation of the quasi-shuffle and new bases of quasi-symmetric functions. arXiv:2209.13317, 2022.
.bib
@article{BouillotNovelliThibon2022x, author = {Olivier Bouillot and Jean-Christophe Novelli and Jean-Yves Thibon}, title = {A two-parameter deformation of the quasi-shuffle and new bases of quasi-symmetric functions}, year = {2022}, eprint = {2209.13317}, url = {https://arxiv.org/abs/2209.13317}, journal = {arXiv e-prints} } - [Cam25]John M. Campbell. Quasi-immanants. arXiv:2501.15667, 2025.
.bib
@article{Campbell2025x, author = {John M. Campbell}, title = {Quasi-immanants}, year = {2025}, eprint = {2501.15667}, url = {https://arxiv.org/abs/2501.15667}, journal = {arXiv e-prints} } - [CKO24]Seung-Il Choi, Young-Hun Kim and Young-Tak Oh. Poset modules of the 0-Hecke algebras and related quasisymmetric power sum expansions. European Journal of Combinatorics, 120:103965, 2024.
.bib
@article{ChoiKimOh2024, author = {Choi, Seung-Il and Kim, Young-Hun and Oh, Young-Tak}, title = {Poset modules of the 0-{H}ecke algebras and related quasisymmetric power sum expansions}, year = {2024}, journal = {European Journal of Combinatorics}, volume = {120}, pages = {103965}, publisher = {Elsevier BV}, doi = {10.1016/j.ejc.2024.103965}, url = {http://dx.doi.org/10.1016/j.ejc.2024.103965}, issn = {0195-6698}, eprint = {2308.10456} } - [GGP24]Xing Gao, Li Guo and Xiao-Song Peng. Multi-quasisymmetric functions with semigroup exponents, Hopf algebras and Rota-Baxter algebras. arXiv:2406.14800, 2024.
.bib
@article{GaoGuoPeng2024x, author = {Xing Gao and Li Guo and Xiao-Song Peng}, title = {Multi-quasisymmetric functions with semigroup exponents, {H}opf algebras and {R}ota-{B}axter algebras}, year = {2024}, eprint = {2406.14800}, url = {https://arxiv.org/abs/2406.14800}, journal = {arXiv e-prints} } - [GKLL+95]Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh and Jean-Yves Thibon. Noncommutative symmetrical functions. Advances in Mathematics, 112(2):218–348, May 1995.
.bib
@article{GelfandKrobLascouxLeclercRetakhThibon1995, doi = {10.1006/aima.1995.1032}, url2 = {https://doi.org/10.1006/aima.1995.1032}, year = {1995}, month = may, publisher = {Elsevier {BV}}, volume = {112}, number = {2}, pages = {218--348}, author = {Israel M. Gelfand and Daniel Krob and Alain Lascoux and Bernard Leclerc and Vladimir S. Retakh and Jean-Yves Thibon}, title = {Noncommutative Symmetrical Functions}, journal = {Advances in Mathematics} } - [Ges84]Ira M. Gessel. Multipartite ${P}$-partitions and inner products of skew Schur functions. Combinatorics and algebra (Boulder, Colo., 1983), 34:289–317, 1984.
.bib
@incollection{Gessel1984, AUTHOR = {Ira M. Gessel}, TITLE = {Multipartite ${P}$-partitions and inner products of skew {S}chur functions}, BOOKTITLE = {Combinatorics and algebra ({B}oulder, {C}olo., 1983)}, SERIES = {Contemp. Math.}, VOLUME = {34}, PAGES = {289--317}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {1984}, MRCLASS = {05A17 (20C30)}, MRNUMBER = {777705}, MRREVIEWER = {Jacques D{\'{e}}sarm{\'{e}}nien}, DOI = {10.1090/conm/034/777705}, URL2 = {https://doi.org/10.1090/conm/034/777705}, } - [GV21]Darij Grinberg and Ekaterina A. Vassilieva. Weighted posets and the enriched monomial basis of QSym. 33th International conference on formal power series and algebraic combinatorics, 85B, 2021. 12 pages
.bib
@inproceedings{GrinbergVassilieva2021, author = {Darij Grinberg and Ekaterina A. Vassilieva}, title = {Weighted posets and the enriched monomial basis of {QS}ym}, url = {https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/58.html}, year = {2021}, booktitle = {33th {I}nternational Conference on Formal Power Series and Algebraic Combinatorics}, venue = {Ramat Gan}, publisher = {S{\'{e}}minaire Lotharingien de Combinatoire}, volume = {85B}, number = {58}, note = {12 pages} } - [KT97]Daniel Krob and Jean-Yves Thibon. Noncommutative symmetric functions IV : Quantum linear groups and Hecke algebras at $q=0$. Journal of Algebraic Combinatorics, 6(4):339–376, 1997.
.bib
@article{KrobThibon1997, doi = {10.1023/a:1008673127310}, url2 = {https://doi.org/10.1023/a:1008673127310}, year = {1997}, publisher = {Springer Nature}, volume = {6}, number = {4}, pages = {339--376}, author = {Daniel Krob and Jean-Yves Thibon}, title = {Noncommutative symmetric functions {IV} : Quantum linear groups and {H}ecke algebras at $q=0$}, journal = {Journal of Algebraic Combinatorics} } - [Laz23]Anthony Lazzeroni. Powersum bases in quasisymmetric functions and quasisymmetric functions in non-commuting variables. The Electronic Journal of Combinatorics, 30(4), December 2023.
.bib
@article{Lazzeroni2023, title = {Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-Commuting Variables}, volume = {30}, ISSN = {1077-8926}, url = {http://dx.doi.org/10.37236/11724}, DOI = {10.37236/11724}, number = {4}, journal = {The Electronic Journal of Combinatorics}, publisher = {The Electronic Journal of Combinatorics}, author = {Lazzeroni, Anthony}, year = {2023}, month = dec, eprint = {2112.13333} } - [LT23]Ricky Ini Liu and Michael Tang. Shuffle Bases and Quasisymmetric Power Sums. arXiv:2310.09371, 2023.
.bib
@article{LiuTang2023x, author = {Ricky Ini Liu and Michael Tang}, title = {Shuffle {B}ases and {Q}uasisymmetric {P}ower {S}ums}, year = {2023}, eprint = {2310.09371}, url = {https://arxiv.org/abs/2310.09371}, journal = {arXiv e-prints} } - [LMW13]Kurt W. Luoto, Stefan Mykytiuk and Stephanie Willigenburg. An introduction to quasisymmetric Schur functions: Hopf algebras, quasisymmetric functions, and Young composition tableaux (SpringerBriefs in mathematics). Springer, 2013.
.bib
@book{LuotoEtAl2013IntroQSymSchur, Author = {Kurt W. Luoto and Stefan Mykytiuk and Stephanie {van Willigenburg}}, Title = {An Introduction to Quasisymmetric {S}chur Functions: {H}opf Algebras, Quasisymmetric Functions, and {Y}oung Composition Tableaux (SpringerBriefs in Mathematics)}, Publisher = {Springer}, Year = {2013}, ISBN = {1461472997}, URL = {https://www.math.ubc.ca/\%7esteph/papers/QuasiSchurBook.pdf} } - [LMW13]Kurt Luoto, Stefan Mykytiuk and Stephanie Willigenburg. An introduction to quasisymmetric Schur functions. Springer New York, 2013.
.bib
@book{LuotoMykytiukWilligenburg2013, doi = {10.1007/978-1-4614-7300-8}, url = {https://doi.org/10.1007/978-1-4614-7300-8}, year = {2013}, publisher = {Springer New York}, author = {Kurt Luoto and Stefan Mykytiuk and Stephanie van Willigenburg}, title = {An Introduction to Quasisymmetric {S}chur Functions} } - [Mlo24]Hugo Mlodecki. Decompositions of packed words and self duality of word quasisymmetric functions. Combinatorial Theory, 4(1), 2024.
.bib
@article{Mlodecki2024, author = {Hugo Mlodecki}, title = {Decompositions of packed words and self duality of word quasisymmetric functions}, year = {2024}, journal = {Combinatorial Theory}, volume = {4}, number = {1}, doi = {10.5070/c64163836}, url = {https://doi.org/10.5070/c64163836}, eprint = {2205.13949} } - [NT21]Philippe Nadeau and Vasu Tewari. Divided symmetrization and quasisymmetric functions. Selecta Mathematica, 27(4), 2021.
.bib
@article{NadeauTewari2021Divided, author = {Philippe Nadeau and Vasu Tewari}, title = {Divided symmetrization and quasisymmetric functions}, year = {2021}, journal = {Selecta Mathematica}, volume = {27}, number = {4}, doi = {10.1007/s00029-021-00695-6}, eprint = {1908.10934} } - [PS23]Oliver Pechenik and Matthew Satriano. James reduced product schemes and double quasisymmetric functions. arXiv:2304.11508, 2023.
.bib
@article{PechenikSatriano2023x, author = {Oliver Pechenik and Matthew Satriano}, title = {James reduced product schemes and double quasisymmetric functions}, year = {2023}, eprint = {2304.11508}, url = {https://arxiv.org/abs/2304.11508}, journal = {arXiv e-prints} } - [Tan25]Michael Tang. Substring compatibility of permutation statistics. arXiv:2510.25524, 2025.
.bib
@article{Tang2025x, author = {Michael Tang}, title = {Substring compatibility of permutation statistics}, year = {2025}, eprint = {2510.25524}, url = {https://arxiv.org/abs/2510.25524}, journal = {arXiv e-prints} }