Toward a Schurification of Parking Function Formulas via bijections with Young Tableaux

Abstract

This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schröder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over $m$-Schröder paths with a fix number of diagonal steps into Schur functions in the variables $q$ and $t$. Some results have been generalized to parking functions, and some bijections were made with standard Young tableaux giving way to partial combinatorial formulas in the basis $s_{\mu }(q,t)s_{\lambda }(X)$ for $\nabla (e_n)$ (respectively, ${\nabla }^m(e_n)$), when $\mu$ and $\lambda$ are hooks (respectively, $\mu$ is of length one). We also give an explicit algorithm that gives all the Schröder paths related to a Schur function $s_{\mu }(q,t)$ when $\mu$ is of length one. In a sense, it is a partial decomposition of Schröder paths into crystals.

The notation for the inverse of a permutation was changed, since it was also used for the notation for the inverse of a word, and typos were corrected. (36 pages, 38 figures)

References in corpus (2)

• Toric braids and $(m,n)$-parking functions
• Sweeping up Zeta