Crystal skeleton polynomials with major index, charge and depth

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.