P-partitions

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#P-partitions

The objects in the family of $P$-partitions — or rather generating functions of $P$-partitions — are indexed by labeled posets. These functions, $\pPartition_{(P,w)}(\xvec),$ are quasisymmetric and positive in the fundamental quasisymmetric basis. It is a rich family, and includes the skew Schur functions, the elementary and complete homogeneous symmetric functions as well as the fundamental quasisymmetric functions.

For a historical overview of the theory of $P$-partitions, see I. Gessel’s survey [Ges16].

N. R. T. Lesnevich and P. R. W. McNamara study when differences of $P$-partition generating functions are positive in the fundamental quasisymmetric basis [LM22]. This gives a systematic framework for positivity comparisons between labeled-poset generating functions.

P. Nadeau and V. Tewari extend flagged $P$-partitions to a setting naturally expressed in back stable quasisymmetric functions [NT23]. Their framework also introduces forest polynomials and gives a signed multiplicity-free expansion for monomials in the slide basis. J. Li, F. Liu, and G. Xin introduce operators extending $P$-partition generating functions by two-rowed plane partitions [LLX24]. Their formulas give explicit generating functions for several classes, including skew plane partitions and ladder poset extensions. S. Mitrovic defines a combinatorial Hopf algebra of posets whose character gives a poset symmetric function related to the Redei–Berge symmetric function [Mit25]. The paper studies expansions in natural bases of $\spaceQSym$ and $\spaceSym$ and gives a decomposition method for posets. R. P. Stanley associates to every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ a quasisymmetric function $F_P$ encoding the flag $f$-vector, equivalently the flag $h$-vector [Sta96]. When $F_P$ is symmetric, the poset is called flag-symmetric; for $q$-primary lattices, its Schur coefficients recover values of Kostka polynomials. The same paper also discusses connections with representations of the symmetric group and its Hecke algebra. J. B. Lewis and E. Marberg introduce enriched set-valued $P$-partitions [LM21]. As an application, they construct a $K$-theoretic analogue of Stembridge’s Hopf algebra of peak quasisymmetric functions; its symmetric part is generated by the shifted stable Grothendieck polynomials of T. Ikeda and H. Naruse.

#Definition

Let $P$ be a poset and let $w$ be a labeling of $P.$ A $(P,w)$-partition is a map $f:P \to \setP$ such that \[x \lt_P y \implies f(x) \leq f(y)\] and \[x \lt_P y \text{ and } w(x)\gt w(y) \implies f(x) \lt f(y).\]

The $(P,w)$-partition generating function $\pPartition_{(P,w)}(\xvec)$ is then defined as \[\pPartition_{(P,w)}(\xvec) = \sum_{\text{$(P,w)$-partitions } f} \prod_{y \in P} x_{f(y)}.\] When the labeling $w$ is natural, we simply write $\pPartition_{P}(\xvec),$ as this function is then independent of the choice of labeling.

See also enriched P-partitions, where peaks are used instead of descents. The analogues of fundamental quasisymmetric functions are the peak quasisymmetric functions.

Example (A small $P$-partition).

Let $P$ be the three-element poset with cover relations $1\lt_P 3$ and $2\lt_P 3,$ with the natural labeling $w(i)=i:$ \[\begin{array}{c} 3\\[-2pt] /\quad\backslash\\[-2pt] 1\quad 2 \end{array}\] The two linear extensions are $123$ and $213.$ Their descent sets are $\emptyset$ and $\{1\},$ respectively. Therefore the fundamental quasisymmetric expansion is \[\pPartition_P(\xvec) =\gessel_{3,\emptyset}(\xvec)+\gessel_{3,\{1\}}(\xvec) =\gessel_{(3)}(\xvec)+\gessel_{(1,2)}(\xvec).\]

Conjecture (Stanley, [Sta72]).

The function $\pPartition_{(P,w)}(\xvec)$ is symmetric if and only if $(P,w)$ describes the weak and strict inequalities in a skew Young diagram, required for generating skew semistandard Young tableaux.

In other words, every symmetric $\pPartition_{(P,w)}(\xvec)$ is a skew Schur function.

See [Statement 3.11, McN06] for possible generalizations of this conjecture.

#The order polynomial

Proposition (See [Ch. 7.19, Sta01]).

Let $P$ be a poset on $[n].$ The order polynomial $\Omega_P(m)$ is defined by \[\Omega_P(m) \coloneqq \left| \left\{ \tau:P\to [m] : x \leq_P y \implies \tau(x)\leq \tau(y) \right\}\right|,\] that is, $\Omega_P(m)$ counts the order-preserving maps from $P$ to $[m].$ This is a polynomial in $m$ of degree $n.$ If $\mathcal{L}(P)$ denotes the set of linear extensions of $P,$ then \[\sum_{m\geq 1} \Omega_P(m) x^m = \frac{\sum_{\pi \in \mathcal{L}(P)} x^{1+\des(\pi)}}{(1-x)^{n+1}},\] where $\des(\pi)=|\DES(\pi)|$ is the number of descents of $\pi.$ Equivalently, for the order polytope $\mathcal{O}(P),$ \[E_{\mathcal{O}(P)}(m)=\Omega_P(m+1) \qquad \text{and} \qquad \operatorname{Ehr}(\mathcal{O}(P);x) = \frac{\sum_{\pi \in \mathcal{L}(P)} x^{\des(\pi)}}{(1-x)^{n+1}}.\] In particular, the $h^*$-polynomial of $\mathcal{O}(P)$ is the $P$-Eulerian polynomial.

For Ferrers posets arising from skew Young diagrams, the linear extensions are standard Young tableaux, so this specializes to the following identity.

Proposition (See [Ch. 7.19, Sta01]).

We have \[\sum_{n\geq 0} |\schurS_{\lambda/\mu}(1^{n+1})| t^n = \frac{ \sum_{T \in \SYT(\lambda/\mu)} t^{\des(T)} } {(1-t)^{1+|\lambda/\mu|}}.\]

Remark

$h^*$-polynomial of $\gtp_{a^b}$ is the descent-generating polynomial of rectangular standard Young tableaux, $\sum_{T \in \SYT(a^b)} z^{\des(T)}.$ This follows from [Eqs. (7.96), (7.108), Sta01], together with the standard bijection between GT-patterns for rectangular SSYTs and rectangular plane partitions.

#Fundamental quasisymmetric expansion

Let $(P,w)$ be a labeled poset on $n$ elements. The Jordan–Hölder set of a labeled poset is defined as \[\mathcal{L}(P,w) \coloneqq \{\sigma\in\symS_n:\sigma^{-1}\circ w\text{ is order-preserving}\}.\]

The expansion of $\pPartition_{(P,w)}(\xvec)$ in the fundamental quasisymmetric basis is given by \[\pPartition_{(P,w)}(\xvec) = \sum_{\pi \in \mathcal{L}(P,w)} \gessel_{n,\DES(\pi)}(\xvec),\] where $\DES(\pi)$ is the descent set of $\pi.$ For a reference of this result, see [Eq. (7.95), Sta01].

#Quasisymmetric powersum expansion

P. Alexandersson and R. Sulzgruber [AS19] show that $\pPartition_P(\xvec)$ is positive in the quasisymmetric powersum basis.

Theorem (Alexandersson and Sulzgruber (2019), [AS19]).

Let $P$ be a poset on $n$ elements. A surjection $f:P \to [k]$ has type $\alpha \vDash n$ if the cardinality of $f^{-1}(j)$ is $\alpha_j,$ for $j=1,\dotsc,k.$ Let $\mathcal{O}_{\alpha}^{\ast}(P)$ be the set of order-preserving surjections $P \to [k]$ of type $\alpha,$ such that each subposet $f^{-1}(j) \subseteq P$ has a unique minimal element. Then \[\pPartition_P(\xvec) = \sum_{\alpha\vDash n} \frac{\qPsi_{\alpha}(\xvec)}{z_{\alpha}}\left| \opsurj_{\alpha}^{\ast}(P) \right|.\]

F. Aliniaeifard, V. Wang, and S. v. Willigenburg [AWW23] take a related point of view in their work on $P$-partition power sums. They use weighted labeled $P$-partition generating functions to define combinatorial power sum bases for QSym. These bases refine the ordinary power sum symmetric functions, have nonnegative monomial expansions, and satisfy shuffle-product and deconcatenation-coproduct rules.

As a consequence, whenever the symmetric function $f$ can be expressed as a non-negative linear combination of $\pPartition_P(\xvec),$ it is necessarily positive in the power-sum basis. In hindsight, it is rather remarkable that this property was not discovered earlier.

Thus ordinary $P$-partition generating functions are one bridge between fundamental-positive quasisymmetric functions and quasisymmetric powersum positivity. The same definition has two useful nearby variants on this site: enriched $P$-partitions, where descents are replaced by peak data, and flagged $(P,w)$-partitions, which refine the ordinary family and expand positively in the fundamental slide basis.

#Murnaghan–Nakayama rule

The result by Alexandersson and Sulzgruber is later generalized to all $(P,w)$-partition generating functions by R.I. Liu and M. Weselcouch [Thm. 6.9, LW20], where a signed rule is proved. This rule coincides with the classical Murnaghan–Nakayama rule for (skew) Schur functions whenever $(P,w)$ is a skew diagram.

Question (See [LW20]).

Suppose $P_1$ and $P_2$ are posets such that $\pPartition_{(P_1,w_1)}(\xvec) = \pPartition_{(P_2,w_2)}(\xvec),$ and $P_1$ is series-parallel. Does it follow that $P_2$ is series-parallel?

#Not Lorentzian

The $P$-partition generating functions $\pPartition_{(P,w)}(\xvec)$ are in general not Lorentzian, nor even normalized Lorentzian, not even for naturally labeled $P.$ For example, the poset on three vertices with relations $2\leq 1 \geq 3$ (forming a $V$) gives a generating function that is not normalized Lorentzian.

#Flagged $(P,w)$-partitions

S. Assaf and N. Bergeron [AB19] consider flagged $(P,w)$-partitions. They show that these are positive in the fundamental slide basis.

Bibliography

[AS19]Per Alexandersson and Robin Sulzgruber. P-partitions and p-positivity. International Mathematics Research Notices, 2021(14):10848–10907, July 2019.

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[AWW23]Farid Aliniaeifard, Victor Wang and Stephanie Willigenburg. ${P}$-partition power sums. European Journal of Combinatorics, 110:103688, May 2023.

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[AB19]Sami Assaf and Nantel Bergeron. Flagged $(\mathcal{ P },ρ)$-partitions. arXiv:1904.06630, 2019.

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[Ges16]Ira Gessel. A historical survey of P-partitions. The mathematical legacy of Richard Stanley, 100:169–188, 2016.

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[LM22]Nathan R. T. Lesnevich and Peter R. W. McNamara. Positivity among $P$-partition generating functions. Annals of Combinatorics, 26(1):171–204, 2022.

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[LM21]Joel Brewster Lewis and Eric Marberg. Enriched set-valued P-partitions and shifted stable Grothendieck polynomials. Mathematische Zeitschrift, 299(3–4):1929–1972, 2021.

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[LLX24]Jingxuan Li, Feihu Liu and Guoce Xin. On ${P}$-partitions Extended by Two-Rowed Plane Partitions. arXiv:2412.03971, 2024.

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[LW20]Ricky Ini Liu and Michael Weselcouch. P-partitions and quasisymmetric power sums. International Mathematics Research Notices, 2021(16):12707–12747, February 2020.

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[Mit25]Stefan Mitrovic. Linear function of a poset. arXiv:2504.20975, 2025.

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[NT23]Philippe Nadeau and Vasu Tewari. ${P}$-partitions with flags and back stable quasisymmetric functions. arXiv:2303.09019, 2023.

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The symmetric functions catalog

#P-partitions

#Definition

#The order polynomial

#Fundamental quasisymmetric expansion

#Quasisymmetric powersum expansion

#Murnaghan–Nakayama rule

#Not Lorentzian

#Flagged \((P,w)\)-partitions

Bibliography