Index and list of polynomials

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#Index of polynomials

Below is the list of all families of symmetric functions and related families of polynomials currently covered.

Next to each link is the vector space where they live, year when they were introduced, and my personal judgement of how much information I have managed to write down about the family.

Name	Space	Category	Year	Rating	Relations
$k$-Schur polynomials	Sym*	Schur	2003	★★★★★	PositiveIn Schur polynomials [BMPS19]
Affine fixed-point free Stanley symmetric functions	Sym	Schubert	2019	★★★★★
Affine involution Stanley symmetric functions	Sym	Schubert	2018	★★★★★
Affine Schubert polynomials	All	Schubert	2006	★★★★★	Contains Affine Schur functions [Lee15] SpecializesTo Affine Stanley symmetric functions [Lee15] SpecializesTo Schubert polynomials [Lee15]
Affine Schur functions	Sym	Schur	2006	★★★★★	DualTo $k$-Schur polynomials [MS15] [LM08]
Affine Stanley symmetric functions	Sym	Schubert	2006	★★★★★	Contains Affine Schur functions [Lam06] Contains Cylindric Schur polynomials [Lam06] Contains Stanley symmetric functions [Lam06] PositiveIn Schur polynomials [MS15] PositiveIn Affine Schur functions [Lam06]
Almost symmetric Schur functions	Other	Schur	2024	★★★★★	Generalizes Key polynomials [Wei24] Generalizes Schur polynomials [Wei24]
Augmented monomial symmetric functions	Sym	Elementary	2015	★★★★★	PositiveIn Monomial symmetric functions [Mer15]
Back stable double Schubert polynomials	All	Schubert	2018	★★★★★
Backward extended Schur functions	QSym	Schur	2024	★★★★★	TransformsTo Extended Schur functions [Dau24]
Big Schur functions	Sym	Schur	2017	★★★★★	PositiveIn Monomial symmetric functions [Shi17]
Boolean product polynomials	Sym	Other	2018	★★★★★	PositiveIn Schur polynomials [BRT19]
Bottom Schur functions	Sym	Schur	2004	★★★★★
Canonical stable Grothendieck polynomials	Sym	Schubert	2017	★★★★★	SpecializesTo Schur polynomials [Yel17] SpecializesTo Stable Grothendieck polynomials [Yel17]
Catalan symmetric functions	Sym	Schur	2010	★★★★★	Contains Transformed Hall--Littlewood [Pan10] SpecializesTo $k$-Schur polynomials [BMPS18] PositiveIn Schur polynomials [BMP20]
Character symmetric functions	Sym	Schur	2016	★★★★★
Chromatic quasisymmetric functions	Sym		2012	★★★★★	PositiveIn Gessel quasisymmetric functions [SW12] PositiveIn Powersum quasisymmetric functions (Psi) [AS19] PositiveIn Power-sum symmetric functions [AS19] [Ell17] [Ath15] PositiveIn Schur polynomials [Gas96] [SW12] PositiveIn Elementary symmetric functions [Hik24] Contains Complete multipartite basis [CS21] SignedIn Complete multipartite basis [CS21]
Colored Eulerian quasisymmetric functions	Sym	Other	2012	★★★★★
Complete flagged homogeneous polynomials	All	Schur	2023	★★★★★	StableLimit Complete homogeneous symmetric functions [Ehr23] PositiveIn Key polynomials [Ehr23] PositiveIn Demazure atoms [Ehr23]
Complete homogeneous symmetric functions	Sym	Elementary	1700	★★★★★	PositiveIn Monomial symmetric functions [Mac95] PositiveIn Power-sum symmetric functions [Mac95] PositiveIn Schur polynomials [Sag01]
Complete multipartite basis	Sym	Other	2021	★★★★★
Cycle index polynomials	Sym	Other	1937	★★★★★	PositiveIn Schur polynomials [LW19]
Cylindric complete homogeneous polynomials	CSym	Elementary	2020	★★★★★
Cylindric elementary polynomials	CSym	Elementary	2020	★★★★★
Cylindric Schur polynomials	Sym		2005	★★★★★	Contains Skew Schur polynomials [Pos05]
Demazure $t$-atoms	All	Schur	2011	★★★★★	SpecializesTo Demazure atoms [HLMW11]
Demazure atoms	All	Schur	1990	★★★★★	PositiveIn Monomial symmetric functions [Mas09] PositiveIn Fundamental particle basis [Sea19] [HN24]
Double Grothendieck polynomials	All	Schubert	1994	★★★★★	SpecializesTo Grothendieck polynomials [FK94]
Double Schubert polynomials	All	Schubert	1982	★★★★★	Contains Factorial Schur polynomials [CLL02] SpecializesTo Schubert polynomials [LS82]
Double Stanley symmetric functions	Sym	Schubert	2018	★★★★★	PositiveIn Schur polynomials [Haw20]
Dual immaculate Schur functions	QSym	Schur	2014	★★★★★	PositiveIn Gessel quasisymmetric functions [BBSS+14] PositiveIn Young quasisymmetric Schur functions [AHM18] [LO25] TransformsTo Row-strict dual immaculate Schur functions [NSWV+23]
Dual stable Grothendieck polynomials	Sym	Schur	2007	★★★★★	RefinedBy Generalized and refined dual stable Grothendieck polynomials [GGL16] PositiveIn Schur polynomials [LP07]
Elementary symmetric functions	Sym	Elementary	1700	★★★★★	PositiveIn Monomial symmetric functions [Mac95] PositiveIn Schur polynomials [Sag01]
Elliptic Schur functions	Other	Schur	2007	★★★★★
Enriched Schubert polynomials	All	Schubert	2021	★★★★★	SpecializesTo Double Schubert polynomials [AF21] SpecializesTo Back stable double Schubert polynomials [AF21] SpecializesTo Type B/C/D Schubert polynomials [AF21]
Eulerian quasisymmetric functions	Sym	Other	2010	★★★★★	PositiveIn Gessel quasisymmetric functions [SW10] PositiveIn Schur polynomials [HW12] PositiveIn Power-sum symmetric functions [SSW11]
Extended chromatic symmetric functions	Sym		2019	★★★★★
Extended Schur functions	QSym	Schur	2019	★★★★★	Contains Schur polynomials [AS22] PositiveIn Monomial quasisymmetric functions [AS22] PositiveIn Gessel quasisymmetric functions [AS22] TransformsTo Row-strict extended Schur functions [Dau24] [NSWV+23] TransformsTo Flipped extended Schur functions [Dau24] TransformsTo Backward extended Schur functions [Dau24]
Factorial Schur polynomials	Sym	Schur	1989	★★★★★	SpecializesTo Shifted Schur polynomials [BL89] SpecializesTo Schur polynomials [BL89]
Factorial supersymmetric Schur polynomials	SuperSym	Schur	1996	★★★★★
Flagged $(P,w)$-partition generating functions	All	QuasiElementary	2019	★★★★★	SpecializesTo P-partitions [AB19] PositiveIn Fundamental slide polynomials [AB19]
Flagged factorial Grothendieck polynomials	All	Schubert	2019	★★★★★	Generalizes Grassmann Grothendieck polynomials [MS19] SpecializesTo Double Grothendieck polynomials [MS19] SpecializesTo Factorial Schur polynomials [MS19]
Flagged LLT polynomials	All	Schur	2025	★★★★★	Generalizes LLT polynomials [BHMP+25] PositiveIn Demazure atoms [BHMP+25]
Flagged refined skew stable Grothendieck polynomials	All	Schubert	2024	★★★★★	SignedIn Key polynomials [Kun24]
Flipped extended Schur functions	QSym	Schur	2024	★★★★★	TransformsTo Extended Schur functions [Dau24]
Forest polynomials	All	Schur	2023	★★★★★	PositiveIn Fundamental slide polynomials [NT23]
Forgotten symmetric functions	Sym	Elementary	1900	★★★★★	TransformsTo Monomial symmetric functions [Sta01] DualTo Elementary symmetric functions [Sta01]
Fundamental particle basis	All	Schur	2020	★★★★★	PositiveIn Monomial symmetric functions [HN24]
Fundamental slide polynomials	All	Schur	2016	★★★★★	Contains Gessel quasisymmetric functions [AS17] StableLimit Gessel quasisymmetric functions [AS17] PositiveIn Monomial slide polynomials [AS17] PositiveIn Fundamental particle basis [Sea19] [HN24]
Generalized and refined dual stable Grothendieck polynomials	Sym	Schur	2016	★★★★★
Generalized Demazure atoms	All	Schur	2012	★★★★★	Contains Demazure atoms [HMR12] Contains Schur polynomials [HMR12] PositiveIn Demazure atoms [HMR12]
Generalized Schur $P$-functions	Sym	Schur	2019	★★★★★	Generalizes Schur's $P$ functions [Oka19]
Generalized Schur $Q$-functions	Sym	Schur	2019	★★★★★	Generalizes Schur's $Q$ functions [Oka19]
Gessel quasisymmetric functions	QSym	QuasiElementary	1984	★★★★★	PositiveIn Monomial quasisymmetric functions [Ges84]
Grassmann Grothendieck polynomials	Sym	Schubert	1994	★★★★★	SpecializesTo Schur polynomials [FK94] KTheoreticAnalogueOf Schur polynomials [FK94] PositiveIn Lascoux atom [Mon17]
Grothendieck polynomials	All	Schubert	1990	★★★★★	Contains Grassmann Grothendieck polynomials [Las90] StableLimit Stable Grothendieck polynomials [FK94] KTheoreticAnalogueOf Schubert polynomials [Las90] PositiveIn Lascoux polynomials [SY23]
Hall--Littlewood $S$-functions	Sym	Schur	2024	★★★★★	SpecializesTo Skew Schur polynomials [GV24]
Hall--Littlewood P	Sym	Schur	1961	★★★★★	SpecializesTo Monomial symmetric functions [Lit61] SpecializesTo Schur polynomials [Lit61] SpecializesTo Schur's $P$ functions [Ste89] PositiveIn Demazure $t$-atoms [HLMW11] [AS19] PositiveIn Permuted-basement $t$-atoms [Ale19]
Hall--Littlewood Q	Sym	Schur	1961	★★★★★	SpecializesTo Schur polynomials [Lit61]
Hikita polynomials	Sym	Schur	2014	★★★★★	PositiveIn Gessel quasisymmetric functions [Hik14] PositiveIn LLT polynomials [BGLX15]
Hivert polynomials	QSym	Schur	2000	★★★★★
Hook Schur polynomials	SuperSym	Schur	1983	★★★★★	Contains Skew Schur polynomials [YR98] SpecializesTo Big Schur functions [Shi17]
Immaculate Schur functions	NSym	Schur	2014	★★★★★
Induced character basis	Sym	Schur	2018	★★★★★
Induced trivial basis	Sym	Schur	2018	★★★★★
Interpolation Macdonald P polynomials	ShiftedSym	Schur	1998	★★★★★
Involution Schubert polynomials	All	Schubert	2018	★★★★★	StableLimit Involution Stanley symmetric functions [HMP18]
Involution Stanley symmetric functions	Sym	Schubert	2017	★★★★★	PositiveIn Stanley symmetric functions [HMP17] PositiveIn Schur polynomials [HMP17] PositiveIn Schur's $P$ functions [HMP17]
Jack interpolation polynomials	All	Schur	1994	★★★★★
Jack J polynomials	Sym	Schur	1970	★★★★★	SpecializesTo Elementary symmetric functions [Jac70] SpecializesTo Schur polynomials [Jac70] SpecializesTo Zonal symmetric functions [Jac70] SpecializesTo Monomial symmetric functions [Jac70] PositiveIn Augmented monomial symmetric functions [KS97]
Jack P polynomials	Sym	Schur	1970	★★★★★	SpecializesTo Monomial symmetric functions [Jac70] SpecializesTo Schur polynomials [Jac70] SpecializesTo Zonal symmetric functions [Jac70] TransformsTo Jack J polynomials [Sta89]
K-theoretic Schur P-functions	Sym	Schur	2013	★★★★★	KTheoreticAnalogueOf Schur's $P$ functions [IN13]
K-theoretic Schur Q-functions	Sym	Schur	2013	★★★★★	KTheoreticAnalogueOf Schur's $Q$ functions [IN13] SignedIn K-theoretic Schur P-functions [CM24]
Katalan symmetric functions	Sym	Schur	2020	★★★★★	Contains Catalan symmetric functions [BMS20] KTheoreticAnalogueOf Catalan symmetric functions [BMS20]
Key polynomials	All	Schur	1974	★★★★★	Contains Schur polynomials (flagged) [RS95] Contains Schur polynomials [LS90] PositiveIn Demazure atoms [LS90] PositiveIn Generalized Demazure atoms [LS90] PositiveIn Quasi-key polynomials [AS18] PositiveIn Fundamental slide polynomials [AS18] SignedIn Complete flagged homogeneous polynomials [Ehr23]
Kohnert polynomials	All	Schur	2019	★★★★★	Contains Key polynomials [AS22] Contains Lock polynomials [AS22] Contains Schubert polynomials [AS22] Contains Schur polynomials [AS22] PositiveIn Key polynomials [Ass21] PositiveIn Monomial slide polynomials [AS22] StableLimit Kohnert quasisymmetric functions [AS22]
Kohnert quasisymmetric functions	QSym	Schur	2019	★★★★★	Contains Extended Schur functions [AS22] PositiveIn Gessel quasisymmetric functions [AS22] PositiveIn Monomial quasisymmetric functions [AS22]
Koornwinder polynomials	Other	Macdonald	1992	★★★★★
Kromatic symmetric functions	Sym		2023	★★★★★	KTheoreticAnalogueOf Chromatic quasisymmetric functions [CPS26] PositiveIn Stable Grothendieck polynomials [CPS26]
Lah symmetric functions	Sym	Other	2019	★★★★★	PositiveIn Elementary symmetric functions [PS21]
Lascoux atom	All	Schur	2004	★★★★★	KTheoreticAnalogueOf Demazure atoms [Las04] SpecializesTo Demazure atoms [Las04]
Lascoux polynomials	All	Schur	2001	★★★★★	Generalizes Grassmann Grothendieck polynomials [Las01] KTheoreticAnalogueOf Key polynomials [Las01] PositiveIn Lascoux atom [MPS21] SpecializesTo Key polynomials [Las01]
Lecture hall Schur functions	All	Schur	2018	★★★★★
Littlewood--Schur functions	SuperSym	Schur	2018	★★★★★	Generalizes Hook Schur polynomials [Rie18]
LLT polynomials	Sym	Schur	1997	★★★★★	Contains Modified Hall--Littlewood polynomials [LLT97] Contains Transformed Hall--Littlewood [LLT97] Contains Skew Schur polynomials [LLT97] Contains Schur polynomials [LLT97] Contains Unicellular LLT polynomials [HHL05] PositiveIn Schur polynomials [LLT97]
Lock polynomials	All	Schur	2019	★★★★★	Contains Schur polynomials [AS22] StableLimit Extended Schur functions [AS22] PositiveIn Fundamental slide polynomials [AS22] PositiveIn Monomial slide polynomials [AS22]
Loop Schur polynomials	LSym	Schur	2012	★★★★★	SpecializesTo Schur polynomials [LP12]
Lyndon symmetric functions	Sym	Sym	1993	★★★★★	PositiveIn Gessel quasisymmetric functions [GR93] PositiveIn Schur polynomials [GR93]
Macdonald E polynomials	All	Schur	1995	★★★★★	SpecializesTo Non-symmetric q-Whittaker polynomials [Opd95] SpecializesTo Demazure atoms [Mas09] SpecializesTo Macdonald P polynomials [Mac88] SpecializesTo Key polynomials [RY11] StableLimit Macdonald P polynomials [Mac88]
Macdonald J polynomials	Sym	Schur	1987	★★★★★	SpecializesTo Jack J polynomials [Mac88] TransformsTo Modified Macdonald polynomials [Mac88] PositiveIn Quasisymmetric Macdonald J polynomials [CHMM+22]
Macdonald P polynomials	Sym	Schur	1987	★★★★★	Contains Elementary symmetric functions [Mac88] Contains Macdonald P polynomials at $(q,q^2)$ [He20] SpecializesTo Hall--Littlewood P [Mac88] SpecializesTo Jack P polynomials [Mac88] SpecializesTo q-Whittaker functions [Mac88] SpecializesTo Schur polynomials [Mac88] TransformsTo Macdonald J polynomials [Mac88]
Macdonald P polynomials at $(q,q^2)$	Sym	Schur	2020	★★★★★	PositiveIn Schur polynomials [He20]
Modified $\imath$ Hall--Littlewood functions	Sym	Schur	2025	★★★★★	Generalizes Modified Hall--Littlewood polynomials [LRW25] SpecializesTo Modified Hall--Littlewood polynomials [LRW25] SpecializesTo Type $C$ universal characters [LRW25]
Modified Hall--Littlewood polynomials	Sym	Schur	1961	★★★★★	PositiveIn Schur polynomials [Lit61] SignedIn Dual stable Grothendieck polynomials [BM10] TransformsTo Hall--Littlewood Q [Mac95]
Modified Macdonald polynomials	Sym	Schur	1987	★★★★★	PositiveIn LLT polynomials [HHL05] PositiveIn $k$-Schur polynomials [Kat25] PositiveIn Schur polynomials [Hai01] SpecializesTo Modified Hall--Littlewood polynomials [Mac88]
Modular Schur functions	Sym	Schur	1992	★★★★★	Contains Petrie symmetric functions [DW92] SpecializesTo Schur polynomials [DW92]
Monomial quasisymmetric functions	QSym	QuasiElementary	1984	★★★★★	PositiveIn Monomial symmetric functions [Ges84]
Monomial slide polynomials	All	Schur	2016	★★★★★	Contains Monomial quasisymmetric functions [AS17] StableLimit Monomial quasisymmetric functions [AS17] PositiveIn Monomial symmetric functions [AS17]
Monomial symmetric functions	Sym	Elementary	1700	★★★★★	RefinedBy Monomial quasisymmetric functions [Ges84]
Multi-Macdonald P polynomials	Other	Schur	2019	★★★★★
Multi-Schur functions	All	Schur	2023	★★★★★	Generalizes Hook Schur polynomials [Iwa23] Generalizes Schur polynomials (flagged) [Iwa23]
Möbius symmetric functions of graphs	Sym	Other	2026	★★★★★
Non-commutative unicellular LLT polynomials	WQSym	Other	2019	★★★★★
Non-symmetric elementary polynomials	All	Elementary	2019	★★★★★	Generalizes Elementary symmetric functions [AS19] PositiveIn Key polynomials [Ass18] [AS19]
Non-symmetric q-Whittaker polynomials	All	Schur	1995	★★★★★	Contains q-Whittaker functions [Opd95] SpecializesTo Non-symmetric elementary polynomials [AS19] PositiveIn Key polynomials [AS17] [Ass18] [AG18] PositiveIn Demazure atoms [MNS26]
Noncommutative complete homogeneous functions	NSym	Elementary	1995	★★★★★	PositiveIn Noncommutative power sums (Psi) [BDHM+20] PositiveIn Noncommutative power sums (Phi) [BDHM+20] NSymQSymDual Monomial quasisymmetric functions [GKLL+95]
Noncommutative power sums (Phi)	NSym	Elementary	1995	★★★★★	NSymQSymDual Powersum quasisymmetric functions (Phi) [BDHM+20]
Noncommutative power sums (Psi)	NSym	Elementary	1995	★★★★★	NSymQSymDual Powersum quasisymmetric functions (Psi) [BDHM+20]
Noncommutative Schur functions	NSym	Schur	1995	★★★★★
Odd Schur functions	OSym	Schur	2012	★★★★★
Orthogonal Schur polynomials	Other	Schur	1971?	★★★★★	TransformsTo Symplectic Schur polynomials [Ham97]
P-Grothendieck polynomials	Sym	Schur	2023	★★★★★	KTheoreticAnalogueOf Schur's $P$ functions [Haw23] PositiveIn Schur's $P$ functions [Haw23]
P-partitions	QSym	QuasiElementary	1984	★★★★★	Contains Skew Schur polynomials [Ges84] Contains Elementary symmetric functions [Ges84] Contains Complete homogeneous symmetric functions [Ges84] Contains Gessel quasisymmetric functions [Ges84] PositiveIn Gessel quasisymmetric functions [Sta01] PositiveIn Powersum quasisymmetric functions (Psi) [AS19]
Parking function symmetric functions	Sym	Other	1994	★★★★★	PositiveIn Power-sum symmetric functions [Sta01] PositiveIn Schur polynomials [Sta01] PositiveIn Complete homogeneous symmetric functions [Sta01]
Peak quasisymmetric functions	PQSym	Quasi	1997	★★★★★	PositiveIn Gessel quasisymmetric functions [Ste97]
Peak Young quasisymmetric Schur functions	PQSym	Schur	2025	★★★★★	Contains Schur's $Q$ functions [Sea25] PositiveIn Peak quasisymmetric functions [Sea25]
Permuted basement Macdonald E polynomials	All	Schur	2011	★★★★★	Contains Macdonald E polynomials [Fer11] SpecializesTo Permuted-basement $t$-atoms [AS19] SpecializesTo Permuted-basement atoms [AS19] PositiveIn Demazure atoms [MNS26]
Permuted-basement $t$-atoms	All	Schur	2019	★★★★★	Contains Demazure $t$-atoms [AS19] SpecializesTo Permuted-basement atoms [AS19]
Permuted-basement atoms	All	Schur	2019	★★★★★	Contains Demazure atoms [AS19] Contains Key polynomials [AS19]
Petrie symmetric functions	Sym	Other	1992	★★★★★	Contains Elementary symmetric functions [DW92] PositiveIn Monomial symmetric functions [DW92]
Power-sum symmetric functions	Sym	Elementary	1700	★★★★★	PositiveIn Powersum quasisymmetric functions (Psi) [BDHM+20] PositiveIn Powersum quasisymmetric functions (Phi) [BDHM+20]
Powersum quasisymmetric functions (p)	QSym	QuasiElementary	2021	★★★★★	PositiveIn Monomial quasisymmetric functions [AWW21]
Powersum quasisymmetric functions (Phi)	QSym	QuasiElementary	2017	★★★★★	PositiveIn Monomial quasisymmetric functions [BDHM+20]
Powersum quasisymmetric functions (Psi)	QSym	QuasiElementary	2017	★★★★★	PositiveIn Monomial quasisymmetric functions [BDHM+20]
q-Whittaker functions	All	Schur	1995	★★★★★	SpecializesTo Schur polynomials [Mac88] PositiveIn Schur polynomials [Ass18] [AG18]
Quantum double Grothendieck polynomials	All	Schubert	2006	★★★★★
Quantum Double Schubert polynomials	All	Schubert	2000	★★★★★
Quantum Schubert polynomials	All	Schubert	1997	★★★★★	SpecializesTo Schubert polynomials [FGP97]
Quasi-key polynomials	All	Schur	2018	★★★★★	PositiveIn Demazure atoms [Sea19] PositiveIn Fundamental slide polynomials [AS18] StableLimit Quasisymmetric Schur functions [AS18]
Quasisymmetric hook Schur functions	QSym	Schur	2018	★★★★★
Quasisymmetric Macdonald E polynomials	QSym	Schur	2019	★★★★★	SpecializesTo Quasisymmetric Schur functions [CHMM+19]
Quasisymmetric Macdonald J polynomials	QSym	Schur	2022	★★★★★
Quasisymmetric Schur $Q$-functions	PQSym	Schur	2015	★★★★★	Generalizes Schur's $Q$ functions [JL15] PositiveIn Peak quasisymmetric functions [Og19] PositiveIn Peak Young quasisymmetric Schur functions [CNO25] [SS26]
Quasisymmetric Schur functions	QSym	Schur	2011	★★★★★	PositiveIn Demazure atoms [HLMW11] PositiveIn Gessel quasisymmetric functions [HLMW11] PositiveIn Monomial quasisymmetric functions [HLMW11] TransformsTo Young quasisymmetric Schur functions [LMW13] TransformsTo Row-strict quasisymmetric Schur functions [MR13]
Restricted Schur polynomials	Other		2007	★★★★★
Ribbon Schur polynomials	Sym	Schur	1901	★★★★★	PositiveIn Schur polynomials [Sag01] PositiveIn Gessel quasisymmetric functions [Ges84] SignedIn Complete homogeneous symmetric functions [Mou23]
Row-strict dual immaculate Schur functions	QSym	Schur	2022	★★★★★	PositiveIn Gessel quasisymmetric functions [NSWV+23] TransformsTo Dual immaculate Schur functions [NSWV+23]
Row-strict extended Schur functions	QSym	Schur	2023	★★★★★	TransformsTo Extended Schur functions [Dau24] [NSWV+23]
Row-strict quasisymmetric Schur functions	QSym	Schur	2013	★★★★★	PositiveIn Gessel quasisymmetric functions [MR13] TransformsTo Quasisymmetric Schur functions [MR13]
Row-strict Young quasisymmetric Schur functions	QSym	Schur	2013	★★★★★	PositiveIn Gessel quasisymmetric functions [MN15] TransformsTo Young quasisymmetric Schur functions [MN15]
Schubert polynomials	All	Schubert	1982	★★★★★	Contains Schur polynomials (flagged) [LS82] [RS95] Contains Schur polynomials [LS82] StableLimit Stanley symmetric functions [Sta84] PositiveIn Key polynomials [LS90] PositiveIn Quasi-key polynomials [Sea19] PositiveIn Fundamental slide polynomials [AS17] PositiveIn Forest polynomials [NT23]
Schur functions in noncommuting variables	NCSym	Schur	2022	★★★★★
Schur multiple zeta functions	NA	Schur	2018	★★★★★
Schur polynomials	Sym		1841	★★★★★	PositiveIn Demazure atoms [LS90] PositiveIn Generalized Demazure atoms [LS90] PositiveIn Gessel quasisymmetric functions [Ges84] PositiveIn Monomial symmetric functions [Mac95] PositiveIn Hall--Littlewood P [Mac95] PositiveIn Quasisymmetric Schur functions [HLMW11] PositiveIn Row-strict quasisymmetric Schur functions [MR13]
Schur polynomials (flagged)	All	Schur	1982	★★★★★	Contains Schur polynomials [LS82] Contains Skew Schur polynomials [GV89] [Wac85] PositiveIn Key polynomials [RS95]
Schur's $P$ functions	Sym	Schur	1911	★★★★★	PositiveIn Schur polynomials [Ste89] TransformsTo Schur's $Q$ functions [Sch11]
Schur's $Q$ functions	Sym	Schur	1911	★★★★★	PositiveIn Schur polynomials [Ste89]
Schur-S polynomials	All	Schur	?	★★★★★
Shifted Jack polynomials	ShiftedSym	Schur	1994	★★★★★	SpecializesTo Jack P polynomials [OO97] SpecializesTo Shifted Schur polynomials [Sah94]
Shifted Schur polynomials	ShiftedSym	Schur	1996	★★★★★	SpecializesTo Schur polynomials [OO96]
Skew affine Schur functions	Sym	Schur	2006	★★★★★	Contains Affine Schur functions [Lam06] Contains Cylindric Schur polynomials [Lam06] PositiveIn Affine Schur functions [LM08] PositiveIn Schur polynomials [Lam06]
Skew affine Stanley symmetric functions	Sym	Schubert	2006	★★★★★	Contains Affine Stanley symmetric functions [Lam06]
Skew key polynomials	All	Schur	2019	★★★★★	Contains Key polynomials [AW19] PositiveIn Key polynomials [AW19] PositiveIn Demazure atoms [AW19] [LS90]
Skew Schubert polynomials	All	Schubert	2003	★★★★★	Contains Schubert polynomials [LS03] PositiveIn Schubert polynomials [LS03]
Skew Schur polynomials	Sym	Schur	1901	★★★★★	Contains Schur polynomials [Sch01] Contains Ribbon Schur polynomials [Sch01] PositiveIn Gessel quasisymmetric functions [Ges84] PositiveIn Monomial symmetric functions [Sag01] PositiveIn Schur polynomials [Sag01]
Stable Grothendieck polynomials	Sym	Schubert	1994	★★★★★	Contains Grassmann Grothendieck polynomials [FK94] KTheoreticAnalogueOf Stanley symmetric functions [FK94] PositiveIn Grassmann Grothendieck polynomials [BKST+07] SignedIn Schur polynomials [Las01]
Stanley symmetric functions	Sym	Schubert	1984	★★★★★	PositiveIn Schur polynomials [MS15]
Symplectic $Q$ functions	Other	Schur	2020	★★★★★
Symplectic Schur polynomials	Other	Schur	1971?	★★★★★	TransformsTo Orthogonal Schur polynomials [Ham97]
Thom polynomials	SuperSym	Other	1956	★★★★★	PositiveIn Hook Schur polynomials [PW07]
Toric Schur polynomials	Sym		2005	★★★★★	Contains Skew Schur polynomials [Pos05] PositiveIn Schur polynomials [Pos05]
Transformed Hall--Littlewood	Sym	Schur	1961	★★★★★	PositiveIn Schur polynomials [Mac95]
Tutte symmetric functions	Sym		1998	★★★★★	SignedIn Complete multipartite basis [CS21]
Twisted Schubert polynomials	All	Schubert	2019	★★★★★
Type $B/C$ Demazure atoms	All	Schur	2009	★★★★★
Type $B/C$ key polynomials	All	Schur	2009	★★★★★
Type $C$ universal characters	Sym	Schur	2022	★★★★★
Type B/C Stanley symmetric functions	Sym	Schubert	1996	★★★★★	PositiveIn Schur's $P$ functions [Lam96] PositiveIn Schur polynomials [HPS17]
Type B/C/D Schubert polynomials	All	Schubert	1996	★★★★★
Unicellular LLT polynomials	Sym	Schur	2005	★★★★★	PositiveIn Schur polynomials [Bla16] PositiveIn $k$-Schur polynomials [Mil19] PositiveIn Elementary symmetric functions [AS20]
Universal Schubert polynomials	All	Schubert	1999	★★★★★	SpecializesTo Schubert polynomials [Ful99]
Weighted-bond chromatic symmetric functions	Sym	Other	2026	★★★★★	Contains Möbius symmetric functions of graphs [DW26]
Whittaker functions	Sym	Schur	1967	★★★★★
Witt symmetric functions	Sym	Elementary	1996	★★★★★
Wreath Macdonald P polynomials	Other	Schur	2002	★★★★★
Wreath product Schur polynomials	Other	Schur	2009	★★★★★
Young quasisymmetric Schur functions	QSym	Schur	2013	★★★★★	PositiveIn Gessel quasisymmetric functions [LMW13] PositiveIn Monomial quasisymmetric functions [LMW13] TransformsTo Quasisymmetric Schur functions [LMW13] TransformsTo Row-strict Young quasisymmetric Schur functions [MN15]
Zonal symmetric functions	Sym	Schur	1963	★★★★★