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For a given sequence of weights (non-negative numbers), we considerpartitions of the positive integer n. Each n-partition is selected uniformly atrandom from the set of all such partitions. Under a classical scheme ofassumptions on the weight sequence, which are due to Meinardus (1954), we showthat the largest part in a random weighted partition, appropriately normalized,converges weakly, as n tends to infinity, to a random variable having theextreme value (Gumbel's) distribution. This limit theorem extends some knownresults on particular types of integer partitions and on the Bose-Einsteinmodel of ideal gas.

The Size of the Largest Part of Random Weighted Partitions of Large Integers