The $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q,t$-symmetry

Abstract

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q,t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area,skips,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q,t$-symmetry of $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$.