Let Ys,n denote the number of part sizes ≧ s in a random and uniform partition of the positive integer n that are counted without multiplicity. For s = λ(6n)1/2/π + o(n1/4), 0 ≦ λ < ∞, as n → ∞, we establish the weak convergence of Ys,n to a Gaussian distribution in the form of a central limit theorem. The mean and the standard deviation are also asymptotically
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 − c0n2/3)/c1n1/3 log1/2n converges weakly, as n → ∞, to the standard normal distribution, where c0 = ζ(2)/ [2ζ(3)]2/3, c1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σj=1∞j−s.