Asymptotics of Young tableaux in the strip, the $d$-sums

The asymptotics of the “strip” sums $S_{\mathrm{\ell }}^{(\text{al})}(n)$ and of their $d$-sums generalizations $T_{d,ds}^{(\text{al})}(dm)$ (see Definition~\ref{definition1}) were calculated in~\cite{regev}. It was recently noticed that when $d>1$ there is a certain confusion about the relevant notations in~\cite{regev}, and the constant in the asymptotics of these $d$-sums $T_{d,ds}^{(\text{al})}(dm)$ seems to be off by a certain factor. Based on the techniques of~\cite{regev} we again calculate the asymptotics of the $d$-sums $T_{d,ds}^{(\text{al})}(dm)$. We do it here carefully and with complete details. This leads to Theorem~\ref{d.sum222} below, which replaces Corollary 4.4 of~\cite{regev} in the cases $d>1$.