Partial rank symmetry of distributive lattices for fences

TL;DR

This paper proves partial symmetry in the rank distribution of distributive lattices associated with fences when the composition has an odd number of parts, using bijective methods, and extends known symmetry results to circular fences.

Contribution

It establishes a bijective proof of partial rank symmetry for distributive lattices of fences with odd parts and provides a bijective proof for rank symmetry in circular fences.

Findings

Coefficients of the rank generating function satisfy an interlacing condition.

Unimodality of the rank generating function is confirmed.

Partial symmetry holds for fences with an odd number of parts.

Abstract

Associated with any composition beta=(a,b,...) is a corresponding fence poset F(beta) whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... The distributive lattice L(beta) of all lower order ideals of F(beta) is important in the theory of cluster algebras. In addition, its rank generating function r(q;beta) is used to define q-analogues of rational numbers. Oguz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that r(q;beta) is unimodal. We show that, when beta has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of F(beta) of size k equals the number of filters of size k, when k is below a certain value. Our proof is completely bijective. Oguz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result as well. We end with some questions and conjectures raised by this work.