Symmetric $q,t$ Catalan polynomials

Abstract: The $q, t$-Catalan functions $C_n(q,t)$, an $q, t$- analogue of Catalan numbers, were first introduced in connection with Macdonald polynomials and Garsia–Haiman’s theory of diagonal harmonics [1996] as certain rational functions in $q$ and $t$. Haglund [2003] and shortly after that, Haiman announced two combinatorial interpretations of $C_n(q,t)$ as a weighted sum over all Dyck paths. An open problem related to these polynomials is a combinatorial proof of its symmetry in $q$ and $t$.

We define two symmetric $q,t$ Catalan polynomials on Dyck paths and provide proof of the symmetry by establishing an involution on plane trees. We also give a combinatorial proof of a result by Garsia et al. regarding parking functions and the number of connected graphs. This is joint work with Joseph Pappe and Anne Schilling.

combinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

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ARCSIN - Algebra, Representations, Combinatorics and Symmetric functions in INdia

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