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Hook Length Formulas for Partitions and Plane Trees

Guo-Niu Han (Strasbourg)

This web site is devoted to my recent works on hook length formulas for partitions and plane trees.

My papers on hook length formulas

[H01] Discovering hook length formulas by an expansion technique, Electronic J. Combinatorics, 15(1), #R133, 2008, 41 pages. [download | | ps | | pdf | |]

[H02] New hook length formulas for binary trees, Combinatorica, 2008, 4 pages. [download | | ps | | pdf | |]

[H03] Yet another generalization of Postnikov's hook length formula for binary trees, SIAM J. Discrete Math, 23, 2009, pp. 661-664. [download | | ps | | pdf | |]

[H04] Some conjectures and open problems on partition hook lengths, Experimental Mathematics, 18, 2009, pp. 97-106. [download | | ps | | pdf | |]

[H05] The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'Institut Fourier, 2009, 29 pages. [download | | ps | | pdf | |]

[H05'] An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, 35 pages, 2008. [Abstract and Download] (unpublished, arXiv:0804.1849 [math.CO])

[H05''] A generalized hook formula for partitions via t-cores and its applications, 13 pages, 2008. (unpublished, draft version)

[H06] (with Ken Ono) Hook lengths and 3-cores, Annals of Combin., 2008, 7 pages. [download | | ps | | pdf | |]

[H07] Hook lengths and shifted parts of partitions, The Ramanujan Journal, 2009, 9 pages. [download | | ps | | pdf | |]

[H08] (with Ch. Bessenrodt) Symmetry distribution between hook length and part length for partitions, Discrete Mathematics, 309, 2009, pp. 6070-6073. [download | | ps | | pdf | |]

[H09] (with Kathy Q. Ji) Combining hook length formulas and BG-ranks for partitions via the Littlewood decomposition, Trans. Amer. Math. Soc. 2009, 24 pages. [download | | ps | | pdf | |]

Maple programs for verifying the formulas and conjectures

HookExp.mpl - Document [1]

Last news and comments

  • [2008.05.01] Laura Yang has generalized hook length formulas for binary trees in [2] and [3] to k-ary trees, arXiv:0805.0109 [math.CO]

  • [2008.05.04] Mihai Cipu has proved Conjecture 5.2 in [4]. Private communication.

  • [2008.05.05] Richard Stanley has found an elementary, but not bijective, proof of the marked hook formula in [4] and [5']. Private communication.

  • [2008.05.07] Bruce Sagan has found probabilistic proofs of hook length formulas for binary trees in [2], arXiv:0805.0817 [math.CO]

  • [2008.05.08] Why papers [5'] and [5''] remain unpublished ?

  • [2008.05.10] Richard Stanley can prove the k=2 case of Conjecture 3.1 in [4]. Private communication.

  • [2008.06.18] Richard Stanley proves Conjecture 3.1 in [4]. Private communication.

  • [2008.07.01] Answer the Quiz.

  • [2008.07.02] Richard Stanley proved and generalized Conjecture 3.1 in [4] arXiv:0807.0383 [math.CO]

  • [2008.07.11] Gil Kalai wrote some comments about my recent works on hook length formula in his blog, Powers of Euler Products and Han's Marked Hook Formula

  • [2008.07.17] Paper [4] is cited by Tewodros Amdeberhan, Differential operators, shifted parts, and hook lengths, arXiv:0807.2473 [math.CO]

  • [2008.07.21] Emily Clader, Yvonne Kemper, Matt Wage, Lacunarity of certain partition theoretic generating functions (arising from Han's generalization of the Nekrasov-Okounkov formula) Proceedings of the American Mathematical Society, 137< 2009, pages 2959-2968.

  • [2008.07.21] Ameya Velingker, An exact formula for the coefficients of Han's generating function Annals of Combinatorics, accepted for publication.

  • [2008.08.04] Kevin Carde, Joe Loubert, Aaron Potechin, Adrian Sanborn, Proof of Han's Hook Expansion Conjecture

  • D. Collins and S. Wolfe, Congruences for Han's generating function, Involve, 2 (2009), pages 225-236.

  • G. Panova, Proof of a conjecture of Okada, arxiv:0811.3463, 2008

  • W. Y.C. Chen, O. X.Q. Gao and P. L. Guo, Hook Length Formulas for Trees by Han's Expansion, Electr. J. Combin.16(1) 2009, Research Paper R62, 16 pages

  • Heesung Shin, Jiang Zeng, An involution for symmetry of hook length and part length of partitions, to appear in Discrete. Math., 2009, 9 pages

  • Amitai Regev, Doron Zeilberger, A Multi-Set Identity for Partitions, arXiv:0909.3459v2, 2009, 4 pages

  • Niklas Eriksen, Combinatorial proofs for some forest hook length identities, 2009, 5 pages

  • G. Olshanski, Plancherel Averages: Remarks on a paper by Stanley, arXiv:0905.1304, 16 pages, 2009

    • Quiz

    Triple mixed hook formula: What is the numerical value of the following expression ? 0, 1 ,2, 3 or 1001 ?

    where t ranges over all binary trees with 1001 vertices, hv is the hook length for trees and hu is the hook length for partitions.

    Answer : click here ...

    Back to List of Papers - Send comments to Email - First version: 2008/04/04 - Last update: 2010/01/10