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

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

2011-07-24

arxiv.org/abs/1107.4754v1

Mathematics

Probability

Scientific paper

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

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