Mathematics – Combinatorics
Scientific paper
2012-03-26
Mathematics
Combinatorics
24 pages
Scientific paper
Several years ago, 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. The present work extends these results to $r$-colored set partitions, by which we mean set partitions whose arcs are labeled by an $r$-element set. A $k$-crossing or $k$-nesting in this context is a sequence or arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. To prove our extension, we produce a bijection from $r$-colored set partitions to certain sequences of $r$-partite partitions, which in the uncolored case specializes to a novel description of the map from set partitions to vacillating tableaux given by Chen et al. Among other applications, we explain how our construction implies recent results of Chen and Guo on colored matchings, and also an analogous symmetric joint distribution of crossings and nestings for colored rook placements in Young shapes, extending work of Krattenthaler. More concretely, we prove that the sequence counting 2-colored noncrossing set partitions is P-recursive, but conjecture that this fails for $r$-colored noncrossing partitions with $r>2$.
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