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

Vincent Pilaud (1) , Juanjo Rué (2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 Institute of Mathematics [Berlin]

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.

Mots clés: Quasi-planar configurations; Chord diagrams; Analytic combinatorics; Generating functions

Domaines: Combinatoire [math.CO]

Vincent Pilaud, Juanjo Rué. Analytic combinatorics of chord and hyperchord diagrams with $k$ crossings. Advances in Applied Mathematics, 2014, 57, pp.60-100. ⟨10.1016/j.aam.2014.04.001⟩. ⟨hal-01276788⟩