Gaussian limiting distributions for the number of components in combinatorial structures

AbstractConsider the number of cycles in a random permutation or a derangement, the number of components in a random mapping or a random 2-regular graph, the number of irreducible factors in a random polynomial over a finite field, the number of components in a random mapping pattern. These random variables all tend to a limiting Gaussian distribution when the sizes of the random structures tend to infinity. Such results, some old and some new, are derived from two general theorems that cover structures decomposed into elementary “components” in either the labelled or the unlabelled case, when the generating function of components has a singularity of a logarithmic type. The proofs are constructed by combining the continuity theorem for characteristic functions with singularity analysis techniques based on Hankel contours