I was wondering the same and came across your question. You may find joy in this paper by Ali Assem Mahmoud and Karen Yeats
"Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions", Journal of Integer Sequences vol. 25, 2022
I was wondering the same and came across your question. You may find joy in this paper by Ali Assem Mahmoud and Karen Yeats
"Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions", Journal of Integer Sequences vol. 25, 2022