Distribution of hooks in self-conjugate partitions

TL;DR

This paper proves that the distribution of t-hooks in self-conjugate partitions follows an asymptotically normal distribution, with explicit formulas for the mean and variance as the partition size grows large.

Contribution

It establishes the asymptotic normality of t-hook counts in self-conjugate partitions and provides explicit formulas for their mean and variance.

Findings

Number of t-hooks is asymptotically normal.

Explicit formulas for mean and variance of t-hooks.

Distribution descends from the unrestricted partition case.

Abstract

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 $\mu_t(n) \sim \frac{\sqrt{6n}}{\pi} + \frac{3}{\pi^2} - \frac{t}{2}+\frac{\delta_t}{4}$ and variance $\sigma_t^2(n) \sim \frac{(\pi^2 - 6) \sqrt{6n}}{\pi^3},$ where $\delta_t:=1$ if $t$ is odd, and is 0 otherwise.