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 Schr\"oder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over $m$-Schr\"oder 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 Schr\"oder 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 Schr\"oder paths into crystals.