Crossings and nesting in tangled-diagrams

TL;DR

This paper introduces a bijection between certain tableaux and tangled-diagrams, demonstrating the equinumerosity of k-noncrossing and k-nonnesting tangled-diagrams, and provides enumeration results.

Contribution

It generalizes previous constructions to establish a bijection and proves the equality in counts of k-noncrossing and k-nonnesting tangled-diagrams.

Findings

Number of k-noncrossing tangled-diagrams equals the number of k-nonnesting ones.

Enumeration formulas for tangled-diagrams are derived.

A bijection with generalized vacillating tableaux is established.

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.