Hook-content formulae for symplectic and orthogonal tableaux

TL;DR

This paper extends Stanley's hook-content formula to symplectic and orthogonal tableaux by deriving new enumeration formulas using specialized Schur functions, building on prior work by El Samra and King.

Contribution

It introduces new hook-content formulas for symplectic and orthogonal tableaux, generalizing Stanley's classical results to these types.

Findings

Derived formulas for symplectic tableaux counts

Extended Stanley's hook-content formula to new settings

Connected to prior work by El Samra and King

Abstract

By considering the specialisation $s_{\lambda}(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $\lambda$ in terms of two properties of the boxes in the diagram for $\lambda$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $\lambda$-tableaux.