Abstract
For fixed t=2 or 3, we investigate the statistical properties of {Yˆt(n)}, the sequence of random variables corresponding to the number of hook lengths divisible by t among the partitions of n. We characterize the support of Yˆt(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 Yˆ2(n) and Yˆ3(n) approximate continuous functions. Finally, we prove that although the mass functions fail to converge, the cumulative distribution functions of {Yˆ2(n)} and {Yˆ3(n)} converge pointwise to shifted Gamma distributions, completing a characterization initiated by Griffin–Ono–Tsai for t≥4.
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