Abstract:  A tangled-diagram over [n] = {1, ..., n} is a graph of degree less than two whose vertices 1, ..., n are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Generalizing the construction of Chen et.al. we prove a bijection between generalized vacillating tableaux with less than k rows and k-noncrossing tangled-diagrams and study their crossings and nestings. We show that the number of k-noncrossing and k-nonnesting tangled-diagrams are equal and enumerate tangled-diagrams.

  AMS Classification: 

  Keywords:  tangled-diagram, partition, matching, crossing, nesting, vacillating tableau.

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