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

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.

Tubings, chord diagrams, and Dyson--Schwinger equations

“…contributions [4,6]. The chord diagram expansion also singled out certain parameters on rooted connected chord diagrams, the terminal chords, see below, which had not had substantial study in the past 2 but are interesting in their distribution [4], and correspond to reasonable parameters on other combinatorial objects such as vertices in bridgeless combinatorial maps [5], and lead to interesting enumerative questions and results [1,45]. These expansions rejuvenated the pure combinatorial study of chord diagrams.…”

“…The Hopf algebra H introduced so far is not yet sufficiently general to capture a Dyson-Schwinger equation of type (1), because it contains only a single cocycle B + . We generalize it to a family of Hopf algebras H(D), where the elements of D are called decorations.…”

“…this extracts the same coefficient. The form ( 14) is more physically natural (see Section 2.4) since it appears in the Dyson-Schwinger equation (1). Nonetheless, we will make use of the form (13) in the proof of our main theorem.…”

“…The bijection θ behaves nicely on the special classes of tubings considered earlier in Section 3, including ladders and leaf tubings, as we now demonstrate. We first require several additional notions picking out the associated classes of chord diagrams (see for example [1] for full formal definitions and further background on these notions).…”

“…In this paper we will investigate a certain notion of tubings on rooted trees and how this notion of tubings on the one hand relates to rooted connected chord diagrams, and on the other hand lets us give series solutions to a wide class of Dyson-Schwinger equations in quantum field theory. A third leg of the triangle relating this notion is the connection between rooted connected chord diagrams and Dyson-Schwinger equations which has been developed over the last decade in order to give a theory of chord diagram expansions of Dyson-Schwinger equations [1,2,3,4,5,6].…”