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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 recentlyrealized that in~\cite[Section 7]{berele} there are few misprints and certainconfusion 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 carefullyrepeat these calculations, which lead to explicit values for the asymptotics of$S_{k,\ell}^{(2z)}(n).$

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