Some combinatorial properties of hook lengths, contents, and parts of partitions

[摘要] The main result of this paper is a generalization of a conjecture of Guoniu Han, originally inspired by an identity of Nekrasov and Okounkov. Our result states that if F is any symmetric function (say over ℚ) and if $$Phi_n(F)=frac{1}{n!}sum_{lambdavdash n}f_lambda^2F(h_u^2:uinlambda),$$ where h u denotes the hook length of the square u of the partition λ of n and f λ is the number of standard Young tableaux of shape λ, then Φ n (F) is a polynomial function of n. A similar result is obtained when F(h u 2:u∈λ) is replaced with a function that is symmetric separately in the contents c u of λ and the shifted parts λ i +n−i of λ.

[发布机构] Springer