A bijection between dominant Shi regions and core partitions

It is well-known that Catalan numbers C n = 1 n + 1 ( 2 n n ) count the number of dominant regions in the Shi arrangement of type A , and that they also count partitions which are both n -cores as well as ( n + 1 ) -cores. These concepts have natural extensions, which we call here the m -Catalan numbers and m -Shi arrangement. In this paper, we construct a bijection between dominant regions of the m -Shi arrangement and partitions which are both n -cores as well as ( m n + 1 ) -cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric group.