Nekrasov-Okounkov hook length formula

introduction

• expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory

$$ \prod_{n=1}^{\infty}(1-x^n)^{\beta-1}=\sum_n\sum_{\lambda \vdash n}\sum_{v\in H(\lambda)}(1-\frac{\beta}{h_v^2})x^{|\lambda|} $$ $h_v$ is the hook of the box $v$ in the Young tableau of $\lambda$.

memo

• http://mathoverflow.net/questions/200951/nekrasov-okounkov-hook-length-formula

articles

• Han, Guo-Niu, and Huan Xiong. “Difference Operators for Partitions and Some Applications.” arXiv:1508.00772 [math], August 4, 2015. http://arxiv.org/abs/1508.00772.
• Pétréolle, Mathias. “A Nekrasov-Okounkov Type Formula for C.” arXiv:1505.01295 [math], May 6, 2015. http://arxiv.org/abs/1505.01295.
• Han, Guo-Niu. ‘The Nekrasov-Okounkov Hook Length Formula: Refinement, Elementary Proof, Extension and Applications’. arXiv:0805.1398 [math], 9 May 2008. http://arxiv.org/abs/0805.1398.