Some combinatorial properties of hook lengths, contents, and parts of partitions
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 $$\Phi_n(F)=\frac{1}{n!}\sum_{\lambda\vdash n}f_\lambda^2F(h_u^2:u\in\lambda),$$ w...
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| Format: | Article |
| Language: | en_US |
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Springer
2011
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| Online Access: | http://hdl.handle.net/1721.1/60871 https://orcid.org/0000-0003-3123-8241 |
| _version_ | 1826208155257798656 |
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| author | Stanley, Richard P. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Stanley, Richard P. |
| author_sort | Stanley, Richard P. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T14:01:23Z |
| format | Article |
| id | mit-1721.1/60871 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:01:23Z |
| publishDate | 2011 |
| publisher | Springer |
| record_format | dspace |
| spelling | mit-1721.1/608712022-09-28T17:46:18Z Some combinatorial properties of hook lengths, contents, and parts of partitions Stanley, Richard P. Massachusetts Institute of Technology. Department of Mathematics Stanley, Richard P. Stanley, Richard P. 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. National Science Foundation (U.S.) (Grant No.0604423) 2011-02-01T13:43:05Z 2011-02-01T13:43:05Z 2009-10 Article http://purl.org/eprint/type/JournalArticle 1382-4090 1572-9303 http://hdl.handle.net/1721.1/60871 Stanley, Richard. “Some combinatorial properties of hook lengths, contents, and parts of partitions.” The Ramanujan Journal 23.1 (2010): 91-105-105. https://orcid.org/0000-0003-3123-8241 en_US http://dx.doi.org/10.1007/s11139-009-9185-x Ramanujan Journal Attribution-Noncommercial-Share Alike 3.0 Unported http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Springer MIT web domain |
| spellingShingle | Stanley, Richard P. Some combinatorial properties of hook lengths, contents, and parts of partitions |
| title | Some combinatorial properties of hook lengths, contents, and parts of partitions |
| title_full | Some combinatorial properties of hook lengths, contents, and parts of partitions |
| title_fullStr | Some combinatorial properties of hook lengths, contents, and parts of partitions |
| title_full_unstemmed | Some combinatorial properties of hook lengths, contents, and parts of partitions |
| title_short | Some combinatorial properties of hook lengths, contents, and parts of partitions |
| title_sort | some combinatorial properties of hook lengths contents and parts of partitions |
| url | http://hdl.handle.net/1721.1/60871 https://orcid.org/0000-0003-3123-8241 |
| work_keys_str_mv | AT stanleyrichardp somecombinatorialpropertiesofhooklengthscontentsandpartsofpartitions |