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.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde