The symmetry of the $(m,n)$-Rational $q, t$-Catalan numbers for $m=3$

Ryan Kaliszewski, Huilan Li

Published 2014-12-10Version 1

We introduce a new statistics skip on rational $(3,n)$-Dyck paths and define a marked rank word for each path. 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. Then there is a natural bijection on the triples of statistics (area,skip,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the symmetry of $(m,n)$-rational $q, t$-Catalan numbers for $m=3$.