Crossings and nestings in colored set partitions

TL;DR

This paper generalizes the concepts of crossings and nestings in set partitions to r-colored set partitions, demonstrating symmetric distributions, establishing bijections with lattice paths, and proving P-recursiveness of certain enumeration sequences.

Contribution

It introduces the notion of r-colored set partitions with colored arcs, extending symmetry results and establishing bijections with paths, along with proving P-recursiveness of related counting sequences.

Findings

Largest crossings and nestings have symmetric joint distribution in colored set partitions.

r-colored noncrossing set partitions correspond to paths in ^r, generalizing known bijections.

The sequence counting noncrossing 2-colored set partitions is P-recursive.

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 \emph{$r$-colored set partitions}). In this context, 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 $\NN^r$, generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-M\'elou 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.