Poset Associahedra and Stack-sorting

TL;DR

This paper explores the structure of poset associahedra, a class of convex polytopes related to finite posets, providing combinatorial insights and linking them to stack-sorting theory, with implications for polynomial root properties.

Contribution

It offers a combinatorial interpretation of the $h$-vector for certain poset associahedra, connecting them to stack-sorting and proving real-rootedness of related polynomials.

Findings

Combinatorial interpretation of the $h$-vector for specific poset associahedra

Connection established between poset associahedra and stack-sorting of permutations

Proof of real-rootedness for certain $h$-polynomials

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.