Crystal skeleton polynomials with major index, charge and depth

2025/12/06 by Kobayashi, Masato
Mathematics · #2020 Primary:05E05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Secondary:05E10

paper · doi:10.48550/arxiv.2512.06273

openalex publication_date 2025/12/06 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

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-Gariépy 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.

Citations

• Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions
• Quasicrystal Structure of Fundamental Quasisymmetric Functions, and Skeleton of Crystals
• Enumerating quasi-Yamanouchi tableaux of Durfee size two
• From symmetric fundamental expansions to Schur positivity

Related

• Crystal skeletons: Combinatorics and axioms
• Quasicrystal Structure of Fundamental Quasisymmetric Functions, and Skeleton of Crystals
• Crystals and Schur P-positive expansions
• Polynomial bases: positivity and Schur multiplication
• Crystals for shifted key polynomials