2024-06-14

## 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.

**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 semi-standard 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.

### 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 (2019), [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

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

### Not Lorentzian

The $P$-partition generating functions $\pPartition_{(P,w)}(\xvec)$ are in general is not Lorentzian (or normalized Lorentzian), not even for naturally labeled $P.$ For example, the poset on three vertices with relations $2\leq 1 \geq 3$ (forming a $V$), is not normalized Lorentzian.

## 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

- [AB19] Sami Assaf and Nantel Bergeron. Flagged $(\mathcal{ p },ρ)$-partitions. arXiv e-prints, 2019.
- [AS19b] Per Alexandersson and Robin Sulzgruber. P-partitions and P-positivity. International Mathematics Research Notices, 2021(14):10848–10907, July 2019.
- [Ges16] Ira Gessel. A historical survey of P-partitions. In The Mathematical Legacy of Richard Stanley. Amer. Math. Soc., 2016.
- [LW20] Ricky Ini Liu and Michael Weselcouch. P-partitions and quasisymmetric power sums. International Mathematics Research Notices, 2021(16):12707–12747, February 2020.
- [McN06] Peter McNamara. Cylindric skew Schur functions. Advances in Mathematics, 205(1):275–312, September 2006.
- [Sta01] Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First edition, 2001.
- [Sta72] Richard P. Stanley. Ordered structures and partitions. Memoirs of the American Mathematical Society, American Mathematical Society, 1972.