The combinatorics of a tree-like functional equation for connected chord diagrams

TL;DR

This paper develops a tree-based framework for connected chord diagrams using Hopf algebra, leading to new enumeration results, bijections with triangulations, and conjectured relationships with other combinatorial structures.

Contribution

It introduces a novel tree-like functional equation approach for chord diagrams, generalizing previous models and establishing new combinatorial bijections and enumeration techniques.

Findings

Derived functional equations solved by weighted generating functions

Established a recursive bijection with disk triangulations

Connected diagram classes relate to Catalan intervals and permutations

Abstract

We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating functions of two different subsets of weighted connected chord diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle subdiagrams. These equations generalize the classic specification for increasing ordered trees and their solution uses a novel decomposition, simplifying and generalizing previous results. The resulting tree perspective on chord diagrams leads to new enumerative insights through the study of novel diagram classes. We present a recursive bijection between connected top-cycle-free diagrams with $n$ chords and triangulations of a disk with $n+1$ vertices, thereby counting the former. This connects to combinatorial maps, Catalan intervals, and uniquely sorted permutations, leading to new conjectured bijective relationships between diagram classes defined by forbidding graphical subdiagrams and imposing connectedness properties and a variety of other rich combinatorial objects. We conclude by exhibiting and studying a direct bijection between diagrams of size $n$ with a single terminal chord and diagrams of size $n-1$.