Facets of the m-generalized cluster complex and regions in the m-extended Catalan arrangement of type Aₙ
Fishel, Susanna · Kallipoliti, Myrto · Tzanaki, Eleni
Original · EN
In this paper we present a bijection ωₙ between two well known families of Catalan objects: the set of facets of the m-generalized cluster complex Δᵐ(Aₙ) and the set of dominant regions in the m-Catalan arrangement Catᵐ(Aₙ), where m>₀. In particular, ωₙ bijects the facets containing the negative simple root -α to dominant regions having the hyperplane {v∈ V<v,α>=m} as separating wall. As a result, ωₙ restricts to a bijection between the set of facets of the positive part of Δᵐ(Aₙ) and the set of bounded dominant regions in Catᵐ(Aₙ). The map ωₙ is a composition of two bijections in which integer partitions in an m-staircase shape come into play.
English translation
This paper has no Arabic translation yet. Be the first: it takes a few seconds, and the result is stored for every future reader.