  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 C_n,
generalizing a result of Fishel--Vazirani for type A. We also introduce a major
statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate
simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues 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.
