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

A weighted partition of the positive integer n is a multiset of size n whose decomposition into a union of disjoint components (parts) satisfies the following condition: for a given sequence of non-negative numbers {bk}k≥1, a part of size k appears in exactly one of bk possible types. Assuming that a weighted partition of n is selected uniformly at random from the set of all such partitions, we study the limiting distribution of the largest part size Xn as n → ∞. Under certain fairly general assumptions on the Dirichlet generating series D(s) = ∑∞ k=1 bkk −s, s = σ + iy, G. Meinardus, Math. Z. 59(1954), 388398, has obtained the asymptotic of the total number of weighted partitions of n. We assume that Meinardus conditions hold and prove that Xn, appropriately normalized, converges weakly to a Gumbel distribution.