The symmetric functions catalog

An overview of symmetric functions and related topics

2021-04-29

P-partitions

The elements 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 Ira Gessel's survey, [Ges16].

Definition

Let $P$ be a poset and $w$ 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

\[ \pPartition_{(P,w)}(\xvec) = \sum_{f\in \text{$(P,w)$-partition }} \prod_{y \in P} x_{f(y)}. \]

When the labeling $w$ is order-preserving, 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 analog of fundamental quasisymmetric functions are the peak quasisymmetric functions.

Fundamental quasisymmetric expansion

Let $(P,w)$ be a labeled poset on $n$ elements. The Jordan–Hölder set of a labeled poset is defined as

\begin{equation*} \mathcal{L}(P,w) \coloneqq\{\sigma\in\symS_n:\sigma^{-1}\circ w\text{ is order-preserving}\}. \end{equation*}

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

Alexandersson and Sulzgruber [AS19b] show that $\pPartition_P(x)$ is positive in the quasisymmetric powersum basis.

Theorem (Alexandersson and Sulzgruber (2018), [AS19b]).

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

\[ K_P(\xvec) = % \sum_{\alpha\vDash n} % \frac{\qPsi_{\alpha}(\xvec)}{z_{\alpha}}\left| \mathcal{L}_{\alpha}^{\ast}(P,w) \right| % = \sum_{\alpha\vDash n} \frac{\qPsi_{\alpha}(\xvec)}{z_{\alpha}}\left| \opsurj_{\alpha}^{\ast}(P) \right|. \]

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

Murnaghan–Nakayama rule

The result by Alexandersson and Sulzgruber is later generalized to all $(P,w)$-partition by R. 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.

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

Flagged $(P,w)$-partitions

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

References