  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
$S_{k,\ell}^{(2z)}(n).$
