Partial Rank Symmetry of Distributive Lattices for Fences

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 x_{a+1}\rhd x_{a+2}\rhd \rhd x_{a+b+1}\lhd x_{a+b+2}\lhd . \end{aligned}$$ The distributive lattice $$L(\beta of all lower order ideals important in the theory cluster algebras. In addition, its rank generating function $$r(q;\beta used to define q-analogues rational numbers. Kantarcı Oğuz and Ravichandran recently showed that coefficients satisfy an interlacing condition, proving conjecture McConville, Smyth, Sagan, which turn implies previous Morier-Genoud Ovsienko unimodal. We show that, when $$ has odd number parts, then polynomial also partially symmetric: size k equals filters k, below certain value. Our proof completely bijective. introduced circular version fences proved, using algebraic techniques, for such symmetric. give bijective this result, as well. end some questions conjectures raised by work.