A Poisson * Geometric Convolution Law for the Number of Components in Unlabelled Combinatorial Structures

Given a class of combinatorial structures C, we consider the quantity N(n,m), the number of multiset constructions P (of C) of size n having exactly m C-components. Under general analytic conditions on the generating function of C, we derive precise asymptotic estimates for N(n,m), as n→∞ and m varies through all possible values (in general 1 ≤ m ≤ n). In particular, we show that the number of C-components in a random (assuming a uniform probability measure) P-structure of size n obeys asymptotically a convolution law of the Poisson and the geometric distributions. Applications of the results include random mapping patterns, polynomials in finite fields, parameters in additive arithmetical semigroups, etc. This work develops the “additive” counterpart of our previous work on the distribution of the number of prime factors of an integer [20].