On the maximal multiplicity of block sizes in a random set partition

We study the asymptotic behavior of the maximal multiplicity M_n = M_n(σ) of the block sizes in a set partition σ of n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with We^W = n. We show that, over subsequences {n_k}_{k ≥ 1} of the sequence of the natural numbers, , appropriately normalized, converges weakly, as k→∞, to , where Z₁ and Z₂ are independent copies of a standard normal random variable. The subsequences {n_k}_{k ≥ 1}, where the weak convergence is observed, and the quantity u depend on the fractional part f_n of the function W(n). In particular, we establish that . The behavior of the largest multiplicity M_n is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.