Unexpected relations between parking function formulas, pattern avoiding permutations and the Robinson-Schensted algorithm.

Abstract: The Shuffle theorem of Carlsson and Mellit, states that $\nabla(e_n)$ is given by parking function formulas. These formulas are symmetric in the variables q and t. More preciously, for all n, $\nabla(e_n)$ can be seen as a $GL2×Sn-module$. In this talk we will put forth a partial expansion in terms of the irreducible bicharacters of these modules. Namely we will expand a subset of the parking function formulas as products of Schur functions in the variables q and t and the usual Schur functions in the variables $X={x1,x2,…}$. Part of these formulas are uncovered using a bijection between a subset of paths of area $0$ and standard Young tableaux that sends the dinv statistic to the major index. The Robinson-Schensted algorithm associates a pair of standard Young tableaux $(P,Q)$ to a given permutation. We will end by showing how the previous bijection is linked to the $Q$-tableau of some pattern avoiding permutations that is unrelated to the word of the parking function.

combinatorics

Audience: researchers in the topic

York University Applied Algebra Seminar