Meinardus' theorem on weighted partitions: Extensions and a probabilistic proof

AbstractThe number cn of weighted partitions of an integer n, with parameters (weights) bk, k⩾1, is given by the generating function relationship ∑n=0∞cnzn=∏k=1∞(1−zk)−bk. Meinardus (1954) established his famous asymptotic formula for cn, as n→∞, under three conditions on power and Dirichlet generating functions for the sequence bk. We give a probabilistic proof of Meinardus' theorem with weakened third condition and extend the resulting version of the theorem from weighted partitions to other two classic types of decomposable combinatorial structures, which are called assemblies and selections