Some limit theorems with respect to constrained permutations and partitions
In this paper, it is shown that for fixed positive integers \(k\) and \(m\) the number of permutations on \(n\) elements where the number of cycles in the permutations is \(k\) (\(k\)-permutations for short) is asymptotically equal (for \(n\rightarrow \infty\)) to the number of \(k\)-permutations on \(n\) elements such that all the \(k\) cycles contain at least \(m\) elements. This property also holds for the number of partitions of \(n\) elements into \(k\) subsets and the number of partitions of \(n\) elements into \(k\) subsets such that each block contains at least \(m\) elements. Finally, the exact asymptotics for the number of partitions of \(n\) elements such that each block contains at least \(m\) elements was deduced by using the classical method of Hayman.
scientific article; zbMATH DE number 6677802