Facets of the Generalized Cluster Complex and Regions in the Extended Catalan Arrangement of Type $A$

Abstract

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 ∆ m (An) and the set of dominant regions in the m-Catalan arrangement Cat m (An), where m ∈ N >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 ∆ m (An) and the set of bounded dominant regions in Cat m (An). The map ωn is a composition of two bijections in which integer partitions in an m-staircase shape come into play.

Bijections between generalized Catalan families of types A and C

“…For each m-Dyck path, we ignore its first step (5) and we rewrite its step sequence as λ n−1 ≤ ⋯ ≤ λ 1 with 0 ≤ λ i ≤ (n − 1 − i)m. Theorem 3.4 shows that there exists a unique way to construct a Shi tableau of a dominant region in R m + (A n−1 ), whose i-th row has coordinates which sum up to λ i , for all 1 ≤ i ≤ n − 1 (see Figure 9). A0 B0 4 3 2 0 3 2 1 2 1 1 Figure 9: For m = 3, the bijection of Theorem 3.4 maps the Dyck path (1,3,6,9) to the region Shi tableau depicted on the right.…”

“…Each of the bijections Bj 1 ,Bj 2 can stand on its own and the sections presenting them (Section 2 and 3 respectively) can be read independently. However, in the setting of dominant regions in Shi m (A n ), there exists previous work [9] which reveals a connection between the two bijections. Unifying previous and current results, we have the following commutative diagram: where FKT 1 and FKT 2 are bijections given in [9].…”

“…We note that the second bijection is based on an idea in [16] while the first, which is the main contribution of this work, relies on the two different ways to view the representative alcove of a region in Shi m (Φ): its Shi tableau [9] and its abacus diagram [2,11]. Each of the bijections Bj 1 ,Bj 2 can stand on its own and the sections presenting them (Section 2 and 3 respectively) can be read independently.…”

On Floors and Ceilings of the $k$-Catalan Arrangement

“…One reason why floors and ceilings of dominant regions are interesting is that they give a more refined enumeration of the dominant regions of the k-Catalan arrangement of Φ that corresponds to refined enumerations of other objects counted by the Fuß-Catalan number Cat For the special case where Φ is of type A n−1 , more is known. For example, there is an explicit bijection between the set of dominant regions of the k-Catalan arrangement of Φ and the set of facets of the cluster complex of Φ [FKT13]. There is also an enumeration of those dominant regions that have a fixed hyperplane as a floor [FTV13].…”

“…Similarly, the number of bounded dominant regions of the k-Catalan arrangement of Φ that have exactly j ceilings of height k equals the (n − j)-th entry of the h-vector of the For the special case where Φ is of type A n−1 , more is known. For example, there is an explicit bijection between the set of dominant regions of the k-Catalan arrangement of Φ and the set of facets of the cluster complex of Φ [FKT13]. There is also an enumeration of those dominant regions that have a fixed hyperplane as a floor [FTV13].…”