Polynomiality of some hook-length statistics

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{λ\vdash n} f_λ^2 \sum_{u \in λ} \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 $f_λ$ is the number of standard Young tableaux of shape $λ$ and $h_u$ is the hook length of the square $u$ of the Young diagram of $λ$. We also obtain other similar formulas.

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