Random Set Partitions: Asymptotics of Subset Counts

We study the asymptotics of subset counts for the uniformly random partition of the set [n]. It is known that typically most of the subsets of the random partition are of sizer, withrer=n. Confirming a conjecture formulated by Arratia and Tavaré, we prove that the counts of other subsets are close, in terms of the total variation distance, to the corresponding segments of a sequence {Zj} of independent, Poisson (rj/j!) distributed random variables. DeLaurentis and Pittel had proved that the finite–dimensional distributions of a continuous time process that counts the typical size subsets converge to those of the Brownian Bridge process. Combining the two results allows to prove a functional limit theorem which covers a broad class of the integral functionals. Among illustrations, we prove that the total number of refinements of a random partition is asymptotically lognormal.