Combinatorics and distributions of partial injections

We obtain several combinatorial results about chains, cycles and orbits of the elements of the symmetric inverse semigroup ${\text{IS}}_n$ and the set $T_n$ of nilpotent elements in ${\text{IS}}_n$. We also get some estimates for the growth of $\mathrm{\mid }{\text{IS}}_n\mathrm{\mid }$ and $\mathrm{\mid }{T}_n\mathrm{\mid }$, and study random products of elements from ${\text{IS}}_n$.