On simultaneous -cores/-cores

In this paper, the authors investigate the question of when a partition of n ∈ 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 ′ , r ) -cores.