Rank Polynomials of Fence Posets are Unimodal

TL;DR

This paper proves that rank polynomials of fence posets are unimodal, introduces circular fence posets to strengthen the result, and explores symmetry and homomesy properties related to these structures.

Contribution

It establishes unimodality of rank polynomials for fence posets and extends the analysis to circular fence posets, providing new insights and conjectures.

Findings

Rank polynomials of fence posets are unimodal.

Circular fence posets have symmetric rank polynomials.

Many homomesy results extend to circular fence posets.

Abstract

We prove a conjecture of Morier-Genoud and Ovsienko that says that rank polynomials of the distributive lattices of lower ideals of fence posets are unimodal. We do this by introducing a related class of circular fence posets and proving a stronger version of the conjecture due to McConville, Sagan and Smyth. We show that the rank polynomials of circular fence posets are symmetric and conjecture that unimodality holds except in some particular cases. We also apply the recent work of Elizalde, Plante, Roby and Sagan on rowmotion on fences and show many of their homomesy results hold for the circular case as well.