Some asymptotic bijections
Let \(S_ n\supseteq S_ n'\), \(T_ n\supseteq T_ n'\) with \(| S_ n| \sim | S_ n'|\) and \(| T_ n\sim | T_ n'|\) as \(n\to \infty\). If there exist bijections \(\Phi_ n\) from \(S_ n'\) to \(T_ n'\) then \(\Phi_ n\) is called an asymptotic bijection from \(S_ n\) to \(T_ n\). Using this idea, the authors verify the partition formula: \(B_ r=e^{-1}\sum^{\infty}_{r=0}k^ r/k!\) for the number of partitions of an r-set, as well as \(I_ r=e^{- 1/2}r!\sum^{\infty}_{k=0}\left( \begin{matrix} 2k\\ r\end{matrix} \right)/r!2^ k\), the number of involutions on an r-set.
scientific article; zbMATH DE number 3882433