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Using methods from Analytic Combinatorics, we study the families of perfectmatchings, partitions, chord diagrams, and hyperchord diagrams on a disk with aprescribed number of crossings. For each family, we express the generatingfunction of the configurations with exactly $k$ crossings as a rationalfunction of the generating function of crossing-free configurations. Usingthese expressions, we study the singular behavior of these generating functionsand derive asymptotic results on the counting sequences of the configurationswith precisely $k$ crossings. Limiting distributions and random generators arealso studied.

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