Chain enumeration and non-crossing partitions

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

“…These theorems result from our formulae for the (combinatorial) decomposition numbers upon appropriate summations. Subsequently, it is shown that the corresponding formulae imply all known enumeration results on non-crossing partitions and generalised non-crossing partitions, plus several new ones; see Corollaries 12,14,[16][17][18][19] and the accompanying remarks. Section 9 presents the announced computational proof of the F = M (ex-)Conjecture for type D n , based on our formula in Corollary 19 for the rank-selected chain enumeration in the poset of generalised non-crossing partitions of type D n , while Section 10 addresses Conjecture 3.5.13 from [1], showing that it does not hold in general since it fails in type D n .…”

“…The subject has been enriched by Armstrong through the introduction of his generalised non-crossing partitions for reflection groups in [1]. In the symmetric group case, these reduce to the m-divisible non-crossing partitions of Edelman [18], while they produce new combinatorial objects already for the reflection groups of type B n . Again, these generalised non-crossing partitions possess numerous beautiful properties and seem to relate to several other objects of combinatorics and algebra, most notably to the generalised cluster complex of Fomin and Reading [19] (cf.…”

Enumerative Properties of NC (B) (p,q)

“…These follow immediately by extending the bijection given in Proposition 4.2 to count multichains in N C (B) (p, q), similar to Theorem 3.2 of [7] and Proposition 7 of [11]. …”

Noncrossing Partitions in Surprising Locations

“…First defined and studied by Germain Kreweras in 1972 [33], it caught the imagination of combinatorialists beginning in the 1980s [20], [21], [22], [23], [29], [37], [39], [40], [45], and has come to be regarded as one of the standard objects in the field. In recent years it has also played a role in areas as diverse as lowdimensional topology and geometric group theory [9], [12], [13], [31], [32] as well as the noncommutative version of probability [2], [3], [35], [41], [42], [43], [49], [50].…”