Crystal skeleton polynomials with major index, charge and depth

TL;DR

This paper introduces crystal skeleton polynomials to enhance understanding of combinatorial objects like Young tableaux and their algebraic expansions, connecting various statistics and revisiting key theorems to improve existing formulas.

Contribution

It develops a new family of polynomials called crystal skeleton polynomials, linking combinatorial statistics and algebraic expansions, and revisits foundational theorems for improved results.

Findings

Organized calculus of crystal skeleton polynomials

Connected major index, charge, depth, and inversions with RSK

Improved Gessel's expansion of Schur functions

Abstract

We introduce a new family of polynomials, crystal skeleton polynomials, to better understand enumeration of standard Young tableaux, quasi-Yamanouchi tableaux and interactions with Gessel's expansion of a Schur function, quasi-crystals and crystal skeletons as Maas-Gari\'{e}py introduced in 2023. After developing calculus of those polynomials, we organize thoughts on major index, charge, depth, inversions with RSK correspondence and a bivariate factorial. Also, we revisit the theorem on internal zeros of fake degree polynomials by Billey--Konvalinka--Swanson (2020). These results altogether improve Gessel's expansion.