On the effective block size in Harper's theorem (Q1902351)

The paper is concerned with a problem in combinatorial enumeration. Let \(\sigma\) be a random set partition of \([n]= \{1, 2, \dots, n\}\) and \(X_n (\sigma)\) be the random variable marking the total number of blocks in \(\sigma\). Harper's theorem gives a central limit theorem for \(X_n (\sigma)\). The paper gives the effective size of the block size in Harper's theorem, that is, the minimal block size for which the conclusion of Harper's theorem is still maintained. The size is expressed as a quotient of the roots of two transcendental equations. That is, let \(u_n\) be the unique positive root of \(ze^z =n\) and \(r\) the unique root in \((0, 1)\) of \(z/2- \ln z- 1=0\), then the effective block size is \(u_n/r\).