2011/06/19 by Karola Mészáros, Meszaros, Karola
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1106.3774
The number of regions of the type An-1 Shi arrangement in Rn is counted by the intrinsically beautiful formula (n+1)n-1. First proved by Shi, this result motivated Pak and Stanley as well as Athanasiadis and Linusson to provide bijective proofs. We give a description of the Athanasiadis-Linusson bijection and generalize it to a bijection between the regions of the type Cn Shi arrangement in Rn and sequences a1a2...an, where ai ∈ \-n, -n+1,..., -1, 0, 1,..., n-1, n\, i ∈ [n]. Our bijections naturally restrict to bijections between regions of the arrangements with a certain number of ceilings (or floors) and sequences with a given number of distinct elements. A special family of posets, whose antichains encode the regions of the arrangements, play a central role in our approach.