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Combining hook length formulas and BG-ranks for partitions via the Littlewood decomposition

Authors: Guo-Niu Han and Kathy Q. Ji
Journal: Trans. Amer. Math. Soc. 363 (2011), 1041-1060
MSC (2010): Primary 05A15, 05A17, 05A19, 11P81, 11P83
Published electronically: September 14, 2010
MathSciNet review: 2728596
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Abstract: Recently, the first author has studied hook length formulas for partitions in a systematic manner. In the present paper we show that most of those hook length formulas can be generalized and include more variables via the Littlewood decomposition, which maps each partition to its $ t$-core and $ t$-quotient. In the case $ t=2$ we obtain new formulas by combining the hook lengths and BG-ranks introduced by Berkovich and Garvan. As applications, we list several multivariable generalizations of classical and new hook length formulas, including the Nekrasov-Okounkov, the Han-Carde-Loubert-Potechin-Sanborn, the Bessenrodt-Bacher-Manivel, the Okada-Panova and the Stanley-Panova formulas.

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Additional Information

Guo-Niu Han
Affiliation: I.R.M.A., UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, F-67084 Strasbourg, France

Kathy Q. Ji
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China

Keywords: Hook length formulas, BG-ranks, integer partitions, Littlewood decomposition
Received by editor(s): March 10, 2009
Received by editor(s) in revised form: August 24, 2009
Published electronically: September 14, 2010
Additional Notes: The second author was partially supported by the PCSIRT Project of the Ministry of Education and the National Science Foundation of China under Grant No. 10901087.
Article copyright: © Copyright 2010 American Mathematical Society

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