Analytic combinatorics of chord and hyperchord diagrams with crossings
arXiv:1307.6440 · doi:10.1016/j.aam.2014.04.001
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 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 crossings. Limiting distributions and random generators are also studied.
51 pages, 24 figures, 5 tables