Largest 2-generated subsemigroups of the symmetric inverse semigroup

The symmetric inverse monoid I-n,, is the set of all partial permutations of an n-element set. The largest possible size of a 2-generated subsemigroup of I-n is determined. Examples of semigroups with these sizes are given. Consequently, if M(n) denotes this maximum, it is shown that M(n)vertical bar I-n vertical bar -> 1 as n -> infinity. Furthermore, we deduce the known fact that I-n embeds as a local submonoid of an inverse 2-generated subsemigroup of In+1.