Results and conjectures on simultaneous core partitions

An nn-core partition is an integer partition whose Young diagram contains no hook lengths equal to nn. We consider partitions that are simultaneously aa-core and bb-core for two relatively prime integers aa and bb. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type AA). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type CC, and use abacus diagrams to unite the discussion of these two sets of objects.In particular, we prove that 2n2n- and (2mn+1)(2mn+1)-core partitions correspond naturally to dominant alcoves in the mm-Shi arrangement of type  CnCn, generalizing a result of Fishel–Vazirani for type  AA. We also introduce a major index statistic on simultaneous nn- and (n+1)(n+1)-core partitions and on self-conjugate simultaneous 2n2n- and (2n+1)(2n+1)-core partitions that yield qq-analogs of the Coxeter–Catalan numbers of type  AA and type  CC.We present related conjectures and open questions on the average size of a simultaneous core partition, qq-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,tq,t-Catalan numbers.