Geodesic
A bijection between (bounded) dominant Shi regions and core partitions
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010) Cet article a éte moissonné depuis la source Episciences

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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. 
@article{DMTCS_2010_special_259_a43,
     author = {Fishel, Susanna and Vazirani, Monica},
     title = {A bijection between (bounded) dominant {Shi} regions and core partitions},
     journal = {Discrete mathematics & theoretical computer science},
     year = {2010},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     doi = {10.46298/dmtcs.2848},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2848/}
}
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DO  - 10.46298/dmtcs.2848
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%0 Journal Article
%A Fishel, Susanna
%A Vazirani, Monica
%T A bijection between (bounded) dominant Shi regions and core partitions
%J Discrete mathematics & theoretical computer science
%D 2010
%V DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
%U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2848/
%R 10.46298/dmtcs.2848
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%F DMTCS_2010_special_259_a43
Fishel, Susanna; Vazirani, Monica. A bijection between (bounded) dominant Shi regions and core partitions. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi: 10.46298/dmtcs.2848

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