  The study of core partitions has been very active in recent years, with the
study of $(s,t)$-cores - partitions which are both $s$- and $t$-cores - playing
a prominent role. A conjecture of Armstrong, proved recently by Johnson, says
that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive
integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the
same formula gives the average size of a self-conjugate $(s,t)$-core; this was
proved by Chen, Huang and Wang.
  In the present paper, we develop the ideas from the author's paper [J.
Combin. Theory Ser. A 118 (2011) 1525-1539] studying actions of affine
symmetric groups on the set of $s$-cores in order to give variants of
Armstrong's conjectures in which each $(s,t)$-core is weighted by the
reciprocal of the order of its stabiliser under a certain group action.
Informally, this weighted average gives the expected size of the $t$-core of a
random $s$-core.
