On simultaneous s-cores/t-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.