Summary: 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 [D. Armstrong et al., Eur. J. Comb. 41, 205–220 (2014; Zbl 1297.05024)], proved recently by P. Johnson [“Lattice points and simultaneous core partitions”, Preprint, arXiv:1502.07934], says that the average size of an \((s,t)\)-core, when \(s\) and \(t\) are coprime positive integers, is \(\frac{1}{24}(s-1)(t-1)(s+t-1)\). Armstrong also conjectured that the same formula gives the average size of a self-conjugate \((s,t)\)-core; this was proved by W. Y. C. Chen et al. [Proc. Am. Math. Soc. 144, No. 4, 1391–1399 (2016; Zbl 1331.05030)].
In the present paper, we develop the ideas from the author’s paper [J. Comb. Theory, Ser. A 118, No. 5, 1525–1539 (2011; Zbl 1232.05235)], 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.