Partial rank symmetry of distributive lattices for fences
Abstract
Associated with any composition beta=(a,b,...) is a corresponding fence poset F(beta) whose covering relations are x1 < x2 < ... < xa+1 > xa+2 > ... > xa+b+1 < xa+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.