Maximal degrees for Young diagrams in the \((k,l)\) hook (Q1272776)

For a given \(n\), let \(\lambda\) be a partition of \(n\) and denote by \(d_\lambda\) the number of standard (Young) tableaux of shape \(\lambda\). An open problem is : find the partition(s) \(\lambda\) for which \(d_\lambda\) is maximal. In this paper, this problem is addressed for the case where \(\lambda\) is inside the \((k,\ell)\)-hook, that is \(\lambda_{k+1}\leq \ell\). An approximate solution (as \(n\rightarrow\infty\)) is given, thus extending the `strip'-approximation found earlier by \textit{R. Askey} and \textit{A. Regev} [Maximal degrees for Young diagrams in a strip, Eur. J. Comb. 5, 189-191 (1984; Zbl 0555.05009)]. In the present solution, the lengths of the rows and columns of \(\lambda\) are determined by the zeros of the \(k\)th and \(\ell\)th Hermite polynomials.