Poset Associahedra and Stack-sorting

Son Nguyen, Andrew Sack This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-2034835 and National Science Foundation Grants No. DMS-1954121 and DMS-2046915. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Abstract

For any finite connected poset $P$ , Galashin introduced a simple convex $(|P|-2)$ -dimensional polytope $\mathscr{A}(P)$ called the poset associahedron. For a certain family of posets, whose poset associahedra interpolate between the classical permutohedron and associahedron, we give a simple combinatorial interpretation of the $h$ -vector. Our interpretation relates to the theory of stack-sorting of permutations. It also allows us to prove real-rootedness of some of their $h$ -polynomials.

1 Introduction

For a finite connected poset $P$ , Galashin introduced the poset associahedron $\mathscr{A}(P)$ (see [galashin2021poset]). The faces of $\mathscr{A}(P)$ correspond to tubings of $P$ , and the vertices of $\mathscr{A}(P)$ correspond to maximal tubings of $P$ ; see Section LABEL:sec:poset-ass for the definitions. $\mathscr{A}(P)$ can also be described as a compactification of the configuration space of order-preserving maps $P\rightarrow\mathbb{R}$ . Many polytopes can be described as poset associahedra, including permutohedra and associahedra. In particular, when $P$ is the claw poset, i.e. $P$ consists of a unique minimal element $0$ and $n$ pairwise-incomparable elements, then $\mathscr{A}(P)$ is the $n$ -permutohedron. On the other hand, when $P$ is a chain of $n+1$ elements, i.e. $P=C_{n+1}$ , then $\mathscr{A}(P)$ is the associahedron $K_{n+1}$ . Among many different combinatorial interpretations for the $h$ -vector $(h_{0},h_{1},\ldots,h_{n-1})$ of $K_{n+1}$ , we want to recall the following interpretation: $h_{i}$ counts the number of 231-avoiding permutations with exactly $i$ descents.

Stack-sorting is a function $s:\mathfrak{S}_{n}\rightarrow\mathfrak{S}_{n}$ which attempts to sort the permutations $w$ in $\mathfrak{S}_{n}$ in linear time, not always sorting them completely (see definition in Section LABEL:subsec:stack-sorting). A permutation $w\in S_{n}$ is stack-sortable if $s(w)=12\ldots n$ . It is well-known that stack-sortable permutations are exactly 231-avoiding permutations. Thus, we have an alternative interpretation for the $h$ -vector of $\mathscr{A}(C_{n+1})$ : $h_{i}$ counts the number of permutations in $s^{-1}(12\ldots n)$ with exactly $i$ descents.

The focus of our paper is the posets $A_{n,k}=C_{n+1}\oplus A_{k}$ where $A_{k}$ is the antichain of $k$ elements. In particular, $A_{0,k}$ is a claw poset, and $A_{n,0}$ is the chain $C_{n+1}$ . Surprisingly, the $h$ -vector of $\mathscr{A}(A_{n,k})$ is also counted by descents of stack-sorting preimages. Let $\mathfrak{S}_{n,k}=\{w~{}|~{}w\in\mathfrak{S}_{n+k},w_{i}=i~{}\text{for all}~{}i>k\}$ , we prove the following generalization of the above classic result.

Theorem LABEL:thm:A_n,k-h-vector.

Let $h=(h_{0},h_{1},\ldots,h_{n+k-1})$ be the $h$ -vector of $\mathscr{A}(A_{n,k})$ . Then $h_{i}$ counts the number of permutations in $s^{-1}(\mathfrak{S}_{n,k})$ with exactly $i$ descents.

An immediate corollary of Theorem LABEL:thm:A_n,k-h-vector is $\gamma$ -nonnegativity of $\mathscr{A}(A_{n,k})$ . In particular, we have the following result by Bränden.

Theorem LABEL:thm:ss-preimage-gamma ([branden2008actions]).

For $A\subseteq\mathfrak{S}_{n}$ , we have

\sum_{\sigma\in s^{-1}(A)}x^{\operatorname{des}(\sigma)}=\sum_{m=0}^{\lfloor\frac{n-1}{2}\rfloor}\dfrac{|\{\sigma\in s^{-1}(A)~{}:~{}\text{peak}(\sigma)=m\}|}{2^{n-1-2m}}x^{m}(1+x)^{n-1-2m},

where $\text{peak}(\sigma)$ is the number of index $i$ such that $\sigma_{i-1}<\sigma_{i}>\sigma_{i+1}$ .

Thus, we have the following corollary.

Corollary LABEL:cor:A_n,k-gamma-nonnegative.

The $\gamma$ -vector of $\mathscr{A}(A_{n,k})$ is nonnegative.

In addition, in the process of proving Theorem LABEL:thm:A_n,k-h-vector, we find the size of $s^{-1}(\mathfrak{S}_{n,k})$ in terms of $k!$ and the Catalan convolution $C_{n}^{(k)}$ , which will be introduced in Section LABEL:subsec:catalan-convolution.