Polynomiality of some hook-length statistics

Details Polynomiality of some hook-length statistics Polynomiality of some hook-length statistics

2008-11-21

arxiv.org/abs/0811.3463v4

Mathematics

Combinatorics

Published in The Ramanujan Journal; minor edits, title changed (formerly "Proof of a conjecture of Okada")

Scientific paper

10.1007/s11139-011-9332-z

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.

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