Crossings and nesting in tangled-diagrams

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 {\it 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.