Abstract
In this paper we present, for any integer d, a description of the set of hooks in a d-symbol. We then introduce generalized hook length functions for a d-symbol, and prove a general result about them, involving the core and quotient of the symbol. We list some applications, for example to the well-known hook lengths in integer partitions. This leads in particular to a generalization of a relative hook formula for the degree of characters of the symmetric group discovered by G. Malle and G. Navarro in Trans. Am. Math. Soc. 363, 6647–6669, 2011.
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Malle, G., Navarro, G.: Blocks with equal height zero degrees. Trans. Am. Math. Soc. 363, 6647–6669 (2011)
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Bessenrodt, C., Gramain, JB. & Olsson, J.B. Generalized hook lengths in symbols and partitions. J Algebr Comb 36, 309–332 (2012). https://doi.org/10.1007/s10801-011-0338-9
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DOI: https://doi.org/10.1007/s10801-011-0338-9