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

##### Author(s)

Stanley, Richard P.
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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
$$\Phi_n(F)=\frac{1}{n!}\sum_{\lambda\vdash n}f_\lambda^2F(h_u^2:u\in\lambda),$$
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.

##### Date issued

2009-10##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Ramanujan Journal

##### Publisher

Springer

##### Citation

Stanley, Richard. “Some combinatorial properties of hook lengths, contents, and parts of partitions.” The Ramanujan Journal 23.1 (2010): 91-105-105.

Version: Author's final manuscript

##### ISSN

1382-4090

1572-9303