Results and conjectures on simultaneous core partitions

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