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arXiv · 2307.04839

Two classes of posets with real-rooted chain polynomials

Christos A. Athanasiadis·Theo Douvropoulos·Katerina Kalampogia-Evangelinou
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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.

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Christos A. Athanasiadis, Theo Douvropoulos, Katerina Kalampogia-Evangelinou. 2023-07-10. Two classes of posets with real-rooted chain polynomials. https://arxiv.org/abs/2307.04839

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