Abstract
Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin. Athanasiadis a introduit la notion d'hyperplan de séparation pour une région dans l'arrangement de Shi et l'a utilisée pour généraliser les numéros de Narayana. Dans cet article, nous fixons un hyperplan dans l'arrangement de Shi pour le type A et calculons le nombre de régions dominantes qui ont l'hyperplan fixe pour mur de séparation, c'est-à-dire les régions où l'hyperplan soutient une facette de la région et sépare la région de l'origine.
Highlights
A hyperplane arrangement dissects its ambient vector space into regions
The regions in the extended Shi arrangement are enumerated by well-known sequences: all regions by the extended parking functions numbers, the dominant regions by the extended Catalan numbers, dominant regions with a given number of separating walls by the Narayana numbers
In this paper we study the extended Shi arrangement by fixing a hyperplane in it and calculating the number of regions for which that hyperplane is a separating wall
Summary
A hyperplane arrangement dissects its ambient vector space into regions. The regions have walls – hyperplanes which support facets of the region – and the walls may or may not separate the region from the origin. In this paper we study the extended Shi arrangement by fixing a hyperplane in it and calculating the number of regions for which that hyperplane is a separating wall. Athanasiadis (2005) generalized the Narayana numbers He showed they enumerated several types of objects; one of them was the number of dominant Shi regions with a fixed number of separating walls. This led us to investigate separating walls. We give a recursion for counting the regions which have other separating walls Hα,m in Section 4, by using generating functions
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