We confirm the speculation that the distribution of t-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length t among the size n self-conjugate partitions is asymptotically normally distributed with mean μₜ(n) ~ √6nπ + 3/π² - t/2 and variance σₜ²(n) ~ (π² - 6) √6nπ³.
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Craig et al. (2024) studied this question.
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