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; 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 $m=3$.. Nous introduisons une nouvelle statistique, le skip, sur les chemins de $(3,n)$-Dyck rationnels et définissons le mot de rang marqué pour chaque chemin quand $n$ n’est pas un multiple de 3. Si un triplet valide de statistiques (aire, skip, dinv) est donné, nous avons un algorithme pour construire le mot de rang marqué correspondant au triplet. En considérant tous les triplets valides, nous donnons une formule explicite pour les polynômes de $q; t$-Catalan $(m,n)$- rationnels quand $m=3$. Enfin, il existe une bijection naturelle sur les triplets de statistiques (aire, skip, dinv) qui échange les statistiques aires et dinv en conservant le skip. Ainsi, nous prouvons la $q; t$-symétrie des polynômes de $q; t$-Catalan $(m,n)$-rationnels pour $m=3$..
Highlights
In the early 1990’s Garsia and Haiman introduced an important sum of rational functions in q, t, the classic q, t-Catanlan polynomial Cn(q, t), which has since been shown to have interpretations in terms of algebraic geometry and representation theory
It is well-known that the classic q, t-Catalan polynomial is
Though the symmetry problem of the classic q, t-Catalan polynomials has been solved through the use of Macdonald polynomials, no bijective proof has been found, where the bijection means an involution on the set of
Summary
In the early 1990’s Garsia and Haiman introduced an important sum of rational functions in q, t, the classic q, t-Catanlan polynomial Cn(q, t), which has since been shown to have interpretations in terms of algebraic geometry and representation theory. The number of full cells between an (m, n)-Dyck path Π and the main diagonal is denoted area(Π). We use this triple to construct an explicit formula for C3,n(q, t). The formula and the uniqueness of the triple allows us to find a unique bijection on (3, n)-Dyck paths which switches the area and dinv statistics while fixing skip.
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More From: Discrete Mathematics & Theoretical Computer Science
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