On simultaneous ss-cores/tt-cores

In this paper, the authors investigate the question of when a partition of n∈Nn∈N is an ss-core and also a tt-core when ss and tt are not relatively prime. A characterization of all such s/ts/t-cores is given, as well as a generating function dependent upon the polynomial generating functions for s/ts/t-cores when ss and tt are relatively prime. Furthermore, characterizations and generating functions are given for s/ts/t-cores which are self-conjugate and also for (e,r)/(e′,r)(e,r)/(e′,r)-cores.