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A bijection between (bounded) dominant Shi regions and core partitions

Abstract : It is well-known that Catalan numbers $C_n = \frac{1}{ n+1} \binom{2n}{n}$ 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 $(mn+1)$-cores. We also modify our construction to produce a bijection between bounded dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn-1)$-cores. The bijections are natural in the sense that they commute with the action of the affine symmetric group.
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Susanna Fishel, Monica Vazirani. A bijection between (bounded) dominant Shi regions and core partitions. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.283-294. ⟨hal-01186276⟩

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