Enumeration of injective partial transformations (Q1117933)

The paper contains the following two theorems. If \(r_ n\) denotes the number of injective partial transformations of \(\{1,2,...,n\}\), then the exponential generating function is \[ \sum^{\infty}_{n=0}\frac{r_ n}{n!}x^ n=e^{x/(1-x)}/(1-x). \] This result makes a connection to Laguerre polynomials, since the exponential generating function of Laguerre polynomials evaluated at 1 is \(e^{x/(1-x)}\). The number of injective partial transformations f for which \(f(a)\leq a\) if \(a\in Domain(f)\), is the Bell number \(B_{n+1}\).