Crossings and Nestings in Tangled Diagrams

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 knoncrossing 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.