University of Minnesota Combinatorics Seminar: 2022

Date: 04/22/2022

Speaker: Sergi Elizalde

Partial Rank Symmetry of Distributive Lattices for Fences
Given a composition, the corresponding fence poset $F$ consists of chains whose lengths are given by the parts the composition, where adjacent chains share their maximum or their minimum element, in an alternating fashion. The distributive lattice of lower order ideals of $F$ plays a role in the theory of cluster algebras, and its rank generating function $r(q)$ is used to define $q$-analogues of rational numbers. Oguz and Ravichandran recently showed that the coefficients of $r(q)$ 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 $r(q)$ is unimodal. In this talk we show that, when the composition has an odd number of parts, the polynomial is also partially symmetric: the number of ideals of $F$ 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. This is joint work with Bruce Sagan.

Partial Rank Symmetry of Distributive Lattices for Fences

Given a composition, the corresponding fence poset $F$ consists of chains whose lengths are given by the parts the composition, where adjacent chains share their maximum or their minimum element, in an alternating fashion. The distributive lattice of lower order ideals of $F$ plays a role in the theory of cluster algebras, and its rank generating function $r(q)$ is used to define $q$-analogues of rational numbers.

Oguz and Ravichandran recently showed that the coefficients of $r(q)$ 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 $r(q)$ is unimodal. In this talk we show that, when the composition has an odd number of parts, the polynomial is also partially symmetric: the number of ideals of $F$ 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.

This is joint work with Bruce Sagan.