Polynomiality of some hook-length statistics

TL;DR

This paper proves a conjecture relating hook-length statistics of partitions to explicit formulas, revealing polynomiality properties and providing new exact combinatorial identities involving Young tableaux.

Contribution

It establishes a precise formula for a hook-length statistic conjectured by Okada, demonstrating polynomiality and deriving related identities.

Findings

Proved Okada's conjecture on hook-length statistics.

Derived explicit formulas involving sums over partitions.

Showed polynomial behavior of certain hook-length related statistics.

Abstract

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}(h_u^2 - i^2) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{r+1} \prod_{j=0}^{r} (n-j), where $f_{\lambda}$ is the number of standard Young tableaux of shape $\lambda$ and $h_u$ is the hook length of the square $u$ of the Young diagram of $\lambda$. We also obtain other similar formulas.