Introdu~on Let %n denote the set of injective, partial self-maps on a set of n elements (this notation comes from [2]). If further, this set is linearly ordered, let Se, denote the subset of order decreasing injective partial self-maps. In this paper we compute the generating functions of r, = l%!,J and b, = I9l,J. It turns out that these enumerative problems are closely related to Bell numbers, Sterling numbers of the second kind and Laguerre polynomials; known to many as some of the prettiest gems of combinatorial theory ([l] p. 67, 91, 116 and 232). It is initially very surprising that there should be such a direct correspondence between partitions and order decreasing injective partial maps. The motivation for this problem originally came from 121, where it was made clear that we are (here) dealing with the most familiar interesting case in the recursive enumeration of generalized Bruhat cells on algebraic monoids. To be more specific, let M = M&) and let Bn be identified with the set of 0, 1 matrices with at most one nonzero entry in each row and column. Then there is exactly one element of 3,, in each two-sided B-orbit on M (where B is the upper triangular group). Se, corresponds to the set of orbits of upper triangular matrices. Notice that %& is an inverse semigroup under composition of partial functions, usually referred to as the symmetric inverse semigroup on n letters. In fact, %!,, plays the same role in inverse semigroup theory as does the symmetric group in group theory. It appears tha t there are many other infinite families of algebraic monoids which yield enumer:tri>c prt3tilerns similar to the one solved here. *Suppxted by NSERC.