On a rank-unimodality conjecture of Morier-Genoud and Ovsienko

TL;DR

This paper investigates the rank unimodality of lattices derived from fence posets associated with compositions, providing conditions under which the conjecture holds and proposing stronger properties that these lattices may possess.

Contribution

It proves the conjecture for compositions with a dominant part and introduces stronger properties like nested chain decomposition and interlacing rank sequences.

Findings

The conjecture holds if one part exceeds the sum of the others.

Verified properties for compositions with up to three parts.

Extended results to d-divided posets, generalizing previous work.

Abstract

Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive lattice L(alpha) consisting of all lower order ideals of F(alpha). These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that L(alpha) is rank unimodal. We show that if one of the parts of alpha is greater than the sum of the others, then the conjecture is true. We conjecture that L(alpha) enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call d-divided posets, generalizing work of Claussen and simplifying a construction of Gansner.