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

Author(s)
Stanley, Richard P.
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Abstract
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
URI
http://hdl.handle.net/1721.1/60871
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

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