The Young diagram of the partition \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_m)\) of the positive integer \(n\) is a left-justified array of square cells with rows of lengths \(\lambda_1,\lambda_2,\dots,\lambda_m\). Note that the integers \(\lambda_j\) must satisfy \(\lambda_1+\lambda_2+\dots+\lambda_m=n\) and \(\lambda_1\ge\lambda_2\ge\dots\ge\lambda_m>0\). The hook length of a cell in a Young diagram of \(\lambda\) is the sum of the number of cells to the right of it in its row, the number of cells below it in its column, and one (to account for the cell itself). For a fixed integer \(t\ge 2\), let \(\mathcal{H}_t(\lambda)\) be the multiset of the hook lengths of the cells of \(\lambda\) which are divisible by \(t\). Furthermore, let \(Y_t(n)=\mid\mathcal{H}_t(\lambda)\mid\). In this paper the authors investigate some statistical properties of the sequence of random variables \(\{\hat{Y}_t(n)\}\) especially in the cases \(t=2,3\).
From the authors’ abstract: “We characterize the support of \(\hat{Y}_t(n)\) and show, in accordance with empirical observations, that the support is vanishingly small for large \(n\). Moreover, we demonstrate that the nonzero values of the mass functions of \(\hat{Y}_2(n)\) and \(\hat{Y}_3(n)\) approximate continuous functions. Finally, we prove that although the mass functions fail to converge, the cummulative distribution functions of \(\{\hat{Y}_2(n)\}\) and \(\{\hat{Y}_2(n)\}\) converge pointwise to shifted Gamma distributions, completing a characterization initiated by Griffin-Ono-Tsai for \(t\ge 4\).”