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

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.
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https://hal.inria.fr/hal-02353375
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Submitted on : Thursday, November 7, 2019 - 11:54:09 AM
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Vincent Pilaud, Juanjo Rué. Analytic combinatorics of chord and hyperchord diagrams with $k$ crossings. 25th International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'14), Jun 2014, Paris, France. pp.339-350. ⟨hal-02353375⟩

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