Analytic combinatorics of chord and hyperchord diagrams with $k$ crossings

TL;DR

This paper applies analytic combinatorics to derive generating functions, asymptotic counts, and distribution results for families of chord and hyperchord diagrams with a fixed number of crossings.

Contribution

It provides explicit rational function expressions for generating functions of diagrams with exactly k crossings, extending combinatorial enumeration techniques.

Findings

Explicit rational generating functions for diagrams with k crossings

Asymptotic formulas for counting such diagrams

Analysis of limiting distributions and random generation methods

Abstract

Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly $k$ crossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with precisely $k$ crossings. Limiting distributions and random generators are also studied.