Polynomiality of some hook-length 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_{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 $f_{\lambda }$ 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.

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

References in corpus (1)

• An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths

Cited by in corpus (10)

• Theorems, Problems and Conjectures
• Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions
• Difference operators for partitions and some applications
• Asymptotic results for Representation Theory
• Multiplication theorems for self-conjugate partitions
• Difference operators for partitions under the Littlewood decomposition
• Polynomiality of shifted Plancherel averages and content evaluations
• Modular Nekrasov-Okounkov formulas
• New hook-content formulas for strict partitions
• Asymptotics of Some Plancherel Averages via Polynomiality Results