Analytic combinatorics of chord and hyperchord diagrams with k crossings
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European Commisision's projectCOUNTGRAPH - Enumeration of discrete structures: algebraic, analytic, probabilistic and algorithmic methods for enriched planar graphs and planar maps (EC-FP7-630749)
EXPLOREMAPS - Combinatorial methods, from enumerative topology to random discrete structures and compact data representations. (EC-FP7-208471)
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.
CitationPilaud, 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.