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.

中文翻译：

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关于随机集分区中块大小的最大多重性

我们假设[ n ] = {1,2，…，n }的集合分区σ中的块大小的最大多重性M _n  =  M _n（σ）的渐近行为，假设σ是从所有此类分区的集合。众所周知，对于大的n，随机集分区的块通常大小为W  =  W（n），而We ^W  =  n。我们表明，在子序列{ ň _ķ } _{ķ  ≥1}适当归一化的自然数序列中的，当k → ∞时收敛至，其中Z ₁和Z ₂是标准正态随机变量的独立副本。子序列{ Ñ _ķ } _{ķ  ≥1} ，其中所观察到的弱收敛，并且数量ü取决于小数部分˚F _Ñ函数w ^（Ñ）。特别是，我们确定。最大多重性M _n的行为与n的整数分区的相似统计量形成鲜明对比。还给出了对该现象的启发式解释。