Expansive Multisets: Asymptotic Enumeration

Abstract

Consider a non-negative sequence cn = h(n) • n α−1 • ρ −n , where h is slowly varying, α > 0, 0 < ρ < 1 and n ∈ N. We investigate the coefficients of G(x, y) = k≥1 (1 − x k y) −c k , which is the bivariate generating series of the multiset construction of combinatorial objects. By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size n with N components, that is, the coefficient of x n y N in G(x, y), asymptotically as n → ∞ and for all ranges of N . Our results reveal a phase transition in the structure of the counting formula that depends on the ratio n/N and that demonstrates a prototypical passage from a bivariate local limit to an univariate one.