Two classes of posets with real-rooted chain polynomials

arXiv2307.04839MaRDI QIDQ6443235

Theo Douvropoulos, [[Person:5886250|Author name not available (Why is that?)]], Christos A. Athanasiadis

Publication date: 10 July 2023

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 with this property, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to irreducible finite Coxeter groups, are presented here. 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 unimodal, and a good estimate for the location of the peak is deduced.

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Combinatorics of partially ordered sets (06A07) Real polynomials: location of zeros (26C10) Combinatorial aspects of simplicial complexes (05E45)

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