## Kaliszewski, Ryan and Li, Huilan - The $(m, n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q, t$-symmetry

dmtcs:2500 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
The $(m, n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q, t$-symmetry

Authors: Kaliszewski, Ryan and Li, Huilan

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; skip; 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 $m=3$..

Source : oai:HAL:hal-01337775v1
Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Submitted on: November 21, 2016
Keywords: Dyck path,Catalan number,rank word,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]