Ryan Kaliszewski ; Huilan Li
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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)
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https://doi.org/10.46298/dmtcs.2500
The $(m, n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q, t$-symmetryArticle
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 $m=3$..