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The number c n of weighted partitions of an integer n, with parameters (weights) b k , k 1, is given by the generating function relationship ∞ n=0 c n z n = ∞ k=1 (1 − z k ) −b k . Meinardus (1954) established his famous asymptotic formula for c n , as n → ∞, under three conditions on power and Dirichlet generating functions for the sequence b k . 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.

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We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [5] for the above class of processes. The method used goes back to A.Khintchine.

No abstract