2022/03/29 by Κωνσταντίνος Παναγιώτου, Konstantinos Panagiotou, Panagiotou, Konstantinos +2
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.CO #math.PR #msc:05A16 #msc:60C05
paper · pdf · doi:10.48550/arxiv.2203.15543
arxiv created 2022/03/29 · arxiv updated 2022/03/30
Consider a non-negative sequence cn = h(n) ⋅ nα-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-xky)-ck, 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 xnyN 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.