Two classes of posets with real-rooted chain polynomials

TL;DR

This paper identifies two new classes of posets with real-rooted chain polynomials, extending previous results and demonstrating properties like log-concavity and unimodality in related permutation descent enumerators.

Contribution

It introduces two novel classes of posets with real-rooted chain polynomials, generalizing prior work and analyzing permutation descent polynomials.

Findings

All rank-selected subposets of Cohen-Macaulay simplicial posets have real-rooted chain polynomials.

Noncrossing partition lattices associated with finite Coxeter groups have real-rooted chain polynomials.

Permutation descent enumerators are real-rooted, log-concave, and unimodal.

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.