Chain enumeration, partition lattices and polynomials with only real roots

TL;DR

This paper proves that chain polynomials of certain posets, including partition lattices, have only real roots, introduces new convex polytopes with this property, and explores implications for face enumeration.

Contribution

It demonstrates the real-rootedness of chain polynomials for partition lattices and their analogues, and shows how this property is preserved under specific poset operations.

Findings

Partition lattice chain polynomials have only real roots

Real-rootedness is preserved under pyramid and prism operations

New convex polytopes with real-rooted chain polynomials are identified

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.