Susanna Fishel ; Monica Vazirani

A bijection between (bounded) dominant Shi regions and core partitions
dmtcs:2848 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

https://doi.org/10.46298/dmtcs.2848
A bijection between (bounded) dominant Shi regions and core partitionsArticle
1 School of Mathematical and Statistical Sciences (Arizona, Tempe)
2 Department of Mathematics [Univ California Davis]
It is wellknown 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 ncores 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 $(mn1)$cores. The bijections are natural in the sense that they commute with the action of the affine symmetric group.