Distribution of hooks in self-conjugate partitions

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 $\hat{I}{¼}_t(n)\sim \frac{\sqrt{6n}}{\ddot{I}}€+\frac{3}{\ddot{I}{€}^2}-\frac{t}{2}+\frac{\hat{I}{´}_t}{4}$ and variance $\ddot{I}{ƒ}_t^2(n)\sim \frac{(\ddot{I}{€}^2-6)\sqrt{6n}}{\ddot{I}{€}^3},$ where $\hat{I}{´}_t:=1$ if $t$ is odd, and is 0 otherwise.

Corrected one formula based on referee's comment