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Description

Catalan numbers are known to count many mathematical objects. (See Richard Stanley's ``Enumerative Combinatorics" or http://math.mit.edu/~rstan/ec/catalan.pdf and http://math.mit.edu/~rstan/ec/catadd.pdf for a list of over 150 different combinatorial interpretations.) Some of the more well-known include triangulations of an $n+2$-gon or ways of closing up $n$ pairs of parentheses. In particular, the $n$-th Catalan number counts dominant regions in the Shi arrangement (of type $A_{n-1}$) and partitions that are both $n$-cores and $n+1$-cores. This fits into a more general framework, considering the $m$-Shi arrangement and partitions that are both $n$-cores and $mn+1$-cores. In joint work with Susanna Fishel, we give a bijective proof of this result, (given necessary definitions along the way) using the techniques of J. Anderson.