Enumeration of Chord Diagrams without Loops and Parallel Chords

TL;DR

This paper counts specific types of chord diagrams, relating them to Hamiltonian paths in octahedrons and shapes, providing generating functions and recurrence relations for both labelled and unlabelled cases.

Contribution

It introduces enumeration methods for chord diagrams without loops and parallel chords, linking them to Hamiltonian paths and shapes, with explicit generating functions and recurrence relations.

Findings

Enumeration formulas for labelled diagrams

Recurrence relations for unlabelled diagrams

Connection to Hamiltonian paths in octahedrons

Abstract

We enumerate chord diagrams without loops and without both loops and parallel chords. We show that the former ones describe Hamiltonian paths in $n$-dimensional octahedrons. The latter ones are also known as shapes. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations.