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.
CitacióPilaud, V., Rue, J. Analytic combinatorics of chord and hyperchord diagrams with k crossings. "Advances in applied mathematics", 07 Maig 2014, vol. 57, p. 60-100.