Title: Weighted generating functions for weighted chord diagrams
|Affiliation:||University of Waterloo|
|Zoom:||Contact Karen Yeats|
Motivated by the universal property of the Connes-Kreimer Hopf algebra of rooted trees and Hopf subalgebras arising from so-called combinatorial Dyson-Schwinger equations, we introduce a class of two-variable recursive functional equations involving Hochschild 1-cocycle operators. An instance of this equation has been studied in the context of quantum field theory and found to be solved by an expansion over connected chord diagrams. We extend and generalize this line of work to show that these equations are solved by weighted generating functions for certain classes of connected weighted chord diagrams. We then look towards explaining why chord diagrams appear in the solutions by proving that the 1-cocycle property is equivalent to a differential equation related to Stein's recurrence for the number of connected chord diagrams and discuss how this work relates to other combinatorial objects, including weighted ordered trees and Stirling permutations.