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

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We give a probalistic proof of the famous Meinardus' asymptotic formula for the number of weighted partitions with weakened one of the three Meinardus' conditions, and extend the resulting version of the theorem to other two classis types of decomposable combinatorial structures, which are called assemblies and selections. The results obtained are based on combining Meinardus' analytical approach with probabilistic method of Khitchine.