Crossings and nestings in colored set partitions

Abstract

Chen, Deng, Du, Stanley, and Yan introduced the notion of k-crossings and k-nestings for set partitions, and proved that the sizes of the largest k-crossings and k-nestings in the partitions of an n-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an r-element set (which we call r-colored set partitions). In this con- text, a k-crossing or k-nesting is a sequence of arcs, all with the same color, which form a k-crossing or k-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that r-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in Nr, generalizing the correspondence between noncrossing (un- colored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-Mélou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.