On the maximal multiplicity of block sizes in a random set partition. Issue 3 (9th October 2019) - Mutafchiev, Ljuben R.; Savov, Mladen
Abstract : 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, M n k, appropriately normalized, converges weakly, as k → ∞, to max { Z 1, Z 2 − u }, where Z 1 and Z 2 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 lim k → ∞ 1 / ( 2 π ) 1 / 4 min { f n k, 1 − f n k } n k / log 7 / 4 n k = u ∈ [ 0, ∞ ) ∪ { ∞ } . 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.