Asymptotics of Young tableaux in the $(k,\ell)$ hook

TL;DR

This paper clarifies and refines the asymptotic analysis of sums related to Young tableaux in the $(k,\, ext{ell})$ hook, correcting previous misprints and providing explicit asymptotic formulas.

Contribution

It provides a detailed and corrected derivation of the asymptotics of $(k,\ell)$ hook sums, resolving previous ambiguities and errors in the literature.

Findings

Explicit asymptotic formulas for $S_{k,\ell}^{(2z)}(n)$

Clarification of previous misprints and notation issues

Enhanced understanding of Young tableaux asymptotics in the $(k,\ell)$ hook

Abstract

The asymptotics of the "$(k,\ell)$ hook" sums $S_{k,\ell}^{(2z)}(n)$ (see~\eqref{the.sums.1}) were calculated in~\cite{berele}. It was recently realized that in~\cite[Section 7]{berele} there are few misprints and certain confusion with regard to the notations, so that the precise asymptotics of $S_{k,\ell}^{(2z)}(n)$ is not clear. Here we add more details and carefully repeat these calculations, which lead to explicit values for the asymptotics of $S_{k,\ell}^{(2z)}(n).$