Séminaire Lotharingien de Combinatoire, 85B.16 (2021), 12 pp.

Thomas McConville, Bruce Sagan and Clifford Smyth

On a Rank-Unimodality Conjecture of Morier-Genoud and Ovsienko

Abstract. Let α = (a,b,...) be a composition. Consider the associated poset F(α), called a fence, whose covering relations are

x1 ◁ x2 ◁ ... ◁ xa+1 ▷ xa+2 ▷ ... ▷ xa+b+1 ◁ xa+b+2 ◁ ... .

We study the associated distributive lattice L(α) consisting of all lower order ideals of F(α). 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(α) is rank unimodal. We show that if one of the parts of α is greater than the sum of the others, then the conjecture is true. We conjecture that L(α) 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.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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