Crossings and Nestings in Colored Set Partitions (original) (raw)

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Keywords: Colored set partitions, crossings, nestings, vacillating tableaux, matchings, 2-Motkin paths, D-finite generating functions, colored permutations, colored tangled diagrams

Abstract

Chen, Deng, Du, Stanley, and Yan introduced the notion of kkk-crossings and kkk-nestings for set partitions, and proved that the sizes of the largest kkk-crossings and kkk-nestings in the partitions of an nnn-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an rrr-element set (which we call rrr-colored set partitions). In this context, a kkk-crossing or kkk-nesting is a sequence of arcs, all with the same color, which form a kkk-crossing or kkk-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 rrr-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in mathbbNr\mathbb{N}^rmathbbNr, generalizing the correspondence between noncrossing (uncolored) 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.

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How to Cite

Marberg, E. (2013). Crossings and Nestings in Colored Set Partitions. The Electronic Journal of Combinatorics, 20(4), #P6. https://doi.org/10.37236/3163