Ryan Kaliszewski ; Huilan Li

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)

https://doi.org/10.46298/dmtcs.2500
The $(m, n)$rational $q, t$Catalan polynomials for $m=3$ and their $q, t$symmetry
Authors: Ryan Kaliszewski ^{1}; Huilan Li ^{1}
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Ryan Kaliszewski;Huilan Li
1 Department of mathematics [Philadelphie]
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 $m=3$..