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

Han, Guo-Niu; Ji, Kathy

doi:10.1090/S0002-9947-2010-05191-8

Combining hook length formulas and BG-ranks for partitions via the Littlewood decomposition
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by Guo-Niu Han and Kathy Q. Ji

Trans. Amer. Math. Soc. 363 (2011), 1041-1060

DOI: https://doi.org/10.1090/S0002-9947-2010-05191-8

Published electronically: September 14, 2010

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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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