The limiting distribution of the trace of a random plane partition

Abstract

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ n = τ n (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 (τ n − c 0 n 2/3)/c 1 n 1/3 log1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c 0 = ζ(2)/ [2ζ(3)]2/3, c 1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σ =1 ∞ j −s .