An identity involving $h$-polynomials of poset associahedra and type B Narayana polynomials

TL;DR

This paper establishes a new identity linking the $h$-polynomials of poset associahedra with type B Narayana polynomials, leading to novel combinatorial identities involving classical polynomials.

Contribution

It introduces a novel relation between $h$-polynomials of poset associahedra and type B Narayana polynomials, expanding understanding of their combinatorial structure.

Findings

Derived an explicit formula for $h$-polynomials of poset associahedra.

Established identities connecting Narayana, Eulerian, and stack-sorting polynomials.

Provided combinatorial interpretations and applications of the main identity.

Abstract

For any finite connected poset $P$, Galashin introduced a simple convex $(|P|-2)$-dimensional polytope $\mathscr{A}(P)$ called the poset associahedron. Let $P$ be a poset with a proper autonomous subposet $S$ that is a chain of size $n$. For $1\leq i \leq n$, let $P_i$ be the poset obtained from $P$ by replacing $S$ by an antichain of size $i$. We show that the $h$-polynomial of $\mathscr{A}(P)$ can be written in terms of the $h$-polynomials of $\mathscr{A}(P_i)$ and type B Narayana polynomials. We then use the identity to deduce several identities involving Narayana polynomials, Eulerian polynomials, and stack-sorting preimages.