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arXiv · 2512.06273

Crystal skeleton polynomials with major index, charge and depth

Masato Kobayashi
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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.

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Masato Kobayashi. 2025-12-06. Crystal skeleton polynomials with major index, charge and depth. https://arxiv.org/abs/2512.06273

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