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. ticular, we prove that 2 n - and ( 2 m n + 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 index statistic on simultaneous n - and ( n + 1 ) -core partitions and on self-conjugate simultaneous 2 n - and ( 2 n + 1 ) -core partitions that yield q -analogs of the Coxeter–Catalan numbers of type  A and type  C . sent 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.