We consider partitions of the positive integer n whose parts satisfy the following condition. For a given sequence of non-negative numbers {b k } k 1 , a part of size k appears in exactly b k possible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest part X n . Let D(s) = ∞ k=1 b k k −s , s = σ + iy, be the Dirichlet generating series of the weights b k . Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions as n → ∞. Using the Meinardus scheme of conditions, we prove that X n , appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equals e −e −t , −∞ < t < ∞). This limit theorem extends some known results on particular types of partitions and on the Bose-Einstein model of ideal gas.
We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τn = τn(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τn − c0n 2/3 )/c1n
We study the asymptotic behavior of the maximal multiplicity µ n = µ n (λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πµ n /(6n) 1/2 converges weakly to max j X j /j as n → ∞, where X 1 , X 2 , . . . are independent and exponentially distributed random variables with common mean equal to 1.
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