Partial Rank Symmetry of Distributive Lattices for Fences

Abstract

Associated with any composition \(\beta =(a,b,\ldots )\) is a corresponding fence poset \(F(\beta )\) whose covering relations are $$\begin{aligned} x_1\lhd x_2 \lhd \ldots \lhd x_{a+1}\rhd x_{a+2}\rhd \ldots \rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots . \end{aligned}$$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. Kantarcı Oğuz 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. Kantarcı Oğuz 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.