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On simultaneous \(s\)-cores/\(t\)-cores. (English) Zbl 1189.05019

Summary: The authors investigate the question of when a partition of \(n\in \mathbb N\) is an \(s\)-core and also a \(t\)-core when \(s\) and \(t\) are not relatively prime. A characterization of all such \(s/t\)-cores is given, as well as a generating function dependent upon the polynomial generating functions for \(s/t\)-cores when \(s\) and \(t\) are relatively prime. Furthermore, characterizations and generating functions are given for \(s/t\)-cores which are self-conjugate and also for \((e,r)/(e{^{\prime}},r\))-cores.

MSC:

05A17 Combinatorial aspects of partitions of integers
05A15 Exact enumeration problems, generating functions
11P81 Elementary theory of partitions
05E05 Symmetric functions and generalizations
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References:

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