2018/11/19 by Mutafchiev, Ljuben, Savov, Mladen
#05A18 #60C05 #60F05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.07951
We study the asymptotic behavior of the maximal multiplicity Mn=Mn(σ) of the blocks in a set partition of [n]=\1,2,...,n\, assuming that σ is chosen uniformly at random from the set of all such partitions. Let W=W(n) be the unique positive root of the equation WeW=n and let fn be the fractional part of W(n). Furthermore, let Rn=W\lfloor W\rfloor/\lfloor W\rfloor ! and let ϑn=min\fn,1-fn\. We show that, over a subsequence \nk\k≥ 1, (Mnk-Rnk)/√Rnk converges weakly, as k→∞, to max\Z1,Z2-u\, where Z1 and Z2 are two independent copies of a standard normal random variable and either u=((1)/(2π))1/4limk→∞ϑnk\frac√(nk)log7/4nk∈ [0,∞) or u=∞. The proof uses the saddle point method. A comparison with the similar statistic for random integer partitions of n is also given.