On the size of inverse semigroups given by generators

The size of the transformation semigroup of a reversible deterministic finite automaton with n states, or equivalently, of a semigroup given by generators of injective partial functions on n objects, had remained unexplored in the case where the set of generators is a pair. We show that in this case, the maximal size is attained by a semigroup generated by a permutation that satisfies a property depending on n and a partial injective mapping whose domain and image both have size n−1. Moreover, we give precise formulas in terms of n for this maximal size.