Sign variation and descents

TL;DR

This paper studies the topological and combinatorial properties of certain posets related to sign variation, proving they are partitionable and connecting their $h$-vectors to Eulerian numbers of type D, with implications for algebraic combinatorics.

Contribution

The paper proves that the order complexes of these posets are partitionable and provides a new interpretation of their $h$-vectors, linking them to Eulerian numbers of type D.

Findings

$ ext{Delta}_{n,m}$ is partitionable for the studied posets.

The $h$-vector of $ ext{Delta}_{n,m}$ is nonnegative and interpretable.

When } m=n-1$, the $h$-vector entries are the Eulerian numbers of type D.

Abstract

For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let $\Delta_{n,m}$ be the order complex of $P_{n,m}$. A previous result from the third author shows that $\Delta_{n,m}$ is Cohen-Macaulay over $\mathbb{Q}$ whenever $m$ is even or $m = n-1$. Hence, it follows that the $h$-vector of $\Delta_{n,m}$ consists of nonnegative entries. Our main result states that $\Delta_{n,m}$ is partitionable and we give an interpretation of the $h$-vector when $m$ is even or $m = n-1$. When $m = n-1$ the entries of the $h$-vector turn out to be the new Eulerian numbers of type $D$ studied by Borowiec and M\l otkowski in [{\em Electron. J. Combin.}, 23(1):Paper 1.38, 13, 2016]. We then combine our main result with Klee's generalized Dehn-Sommerville relations to give a geometric proof of some facts about these Eulerian numbers of type $D$.