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. Il est bien connu que les nombres de Catalan $C_n = \frac{1}{ n+1} \binom{2n}{n}$ comptent non seulement le nombre de régions dominantes dans le Shi arrangement de type $A$ mais aussi les partitions qui sont à la fois $n$-cœur et $(n+1)$-cœur. Ces concepts ont des extensions naturelles, que nous appelons ici les nombres $m$-Catalan et le $m$-Shi arrangement. Dans cet article, nous construisons une bijection entre régions dominantes du $m$-Shi arrangement et les partitions qui sont à la fois $n$-cœur et $(nm+1)$-coeur. Nous modifions également notre construction pour produire une bijection entre régions dominantes bornées du $m$-Shi arrangement et les partitions qui sont à la fois $n$-coeur et $(mn-1)$-cœur. Ces bijections sont naturelles dans le sens où elles commutent avec l'action du groupe affine symétrique.
Highlights
Let ∆ be the root system of type An−1, with Weyl group W, and let m be a positive integer
Let Snm be the arrangement of hyperplanes Hα,k = {x | α | x = k for − m + 1 ≤ k ≤ m and α ∈ ∆+}
We will modify the construction to give a direct bijection between bounded dominant regions and partitions which are simultaneously n-cores and-cores
Summary
Let ∆ be the root system of type An−1, with Weyl group W , and let m be a positive integer. In Fishel and Vazirani (2010, 2009), the authors constructed and analyzed bijections between certain regions of Snm and certain n-cores. In this extended abstract, we summarize the results from both papers. We will modify the construction to give a direct bijection between bounded dominant regions and partitions which are simultaneously n-cores and (mn − 1)-cores. We characterize alcove walls in terms of addable and removable boxes
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