ON k-NONCROSSING PARTITIONS

In this paper we prove a duality between k-noncrossing partitions over [n] = {1, . . . , n} and k-noncrossing braids over [n − 1]. This duality is derived directly via (generalized) vacillating tableaux which are in correspondence to tangled-diagrams [6]. We give a combinatorial interpretation of the bijection in terms of the contraction of arcs of tangled-diagrams. Furthermore it induces by restriction a bijection between k-noncrossing, 2-regular partitions over [n] and k-noncrossing braids without isolated points over [n − 1]. Since braids without isolated points correspond to enhanced partitions this allows, using the results of [1], to enumerate 2-regular, 3-noncrossing partitions.