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arXiv · 2205.03796

Chain enumeration, partition lattices and polynomials with only real roots

Christos A. Athanasiadis·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. The chain polynomials of the partition lattices and their standard type $B$ analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen--Macaulay posets. As a result, new families of convex polytopes whose face lattices have real-rooted chain polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.

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Christos A. Athanasiadis, Katerina Kalampogia-Evangelinou. 2023-01-02. Chain enumeration, partition lattices and polynomials with only real roots. https://arxiv.org/abs/2205.03796

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