Partial Rank Symmetry of Distributive Lattices for Fences

ociated with any composition β =(a,b,… ) is a corresponding fence poset F(β ) whose covering relations are x_1 x_2 … x_a+1 x_a+2… x_a+b+1 x_a+b+2… . The distributive lattice L(β ) of all lower order ideals of F(β ) is important in the theory of cluster algebras. In addition, its rank generating function r(q;β ) 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;β ) is unimodal. We show that, when β has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of F(β ) 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.