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

Abstract

We study the asymptotic behavior of the maximal multiplicity Mn = Mn(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 WeW = n. We show that, over subsequences {nk}k ≥ 1 of the sequence of the natural numbers, Mnk, appropriately normalized, converges weakly, as k→∞, to maxfalse{Z1,Z2−ufalse}, where Z1 and Z2 are independent copies of a standard normal random variable. The subsequences {nk}k ≥ 1, where the weak convergence is observed, and the quantity u depend on the fractional part fn of the function W(n). In particular, we establish that limk→∞1false/false(2πfalse)1false/4minfalse{fnk,1−fnkfalse}nkfalse/normallog7false/4nk=u∈false[0,∞false)∪false{∞false}. The behavior of the largest multiplicity Mn is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.