Two classes of posets with real-rooted chain polynomials

TL;DR

The paper proves real-rooted chain polynomials for two broad poset families: rank-selected subposets of Cohen-Macaulay simplicial posets and noncrossing partition lattices of finite Coxeter groups. It introduces and analyzes $A^T_n(x)$, the descent-enumerator polynomials, proving their real-rootedness and interlacing properties, and extends results to $r$-colored permutations. Consequences include unimodality and explicit peak localization for these polynomials, with a symmetric, nonnegative real-rooted decomposition for ${\rm NC}_W$. The approach blends interlacing techniques, flag-vector identities, and explicit combinatorial interpretations, contributing to a broader understanding of real-rootedness in doubly Cohen–Macaulay-like contexts.

Abstract

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of posets, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to finite Coxeter groups, are shown to have this property. The first result generalizes one of Brenti and Welker. As a special case, the descent enumerator of permutations of the set $\{1, 2,\dots,n\}$ which have ascents at specified positions is shown to be real-rooted, hence log-concave and unimodal, and a good estimate for the location of the peak is deduced.

Theorems & Definitions (27)

• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Example 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• proof