The combinatorial objects and parameters above, except for extremal ones, lead to univariate and bivariate generating functions, given in Table 1 below. Finding C n , k goes through a parameterization of the functional equation of C . The resulting configuration is called a chord diagram . A diagram is connected if no set of chords can be separated from the remaining chords by a line. A component is a maximal connected subdiagram.
Similar computations yield the variance.
Let k denote the number of chord crossings in a chord diagram, and let I n be the set of all diagrams of size n . Theorem 3 Let X n be the random variable equal to the value of k taken over the set of chord diagrams I n of size n endowed with the uniform probability distribution. Discrete Mathematics , vol. Analytic combinatorics of chord diagrams. A foundation for computer science.