Abstract: We call a set S in N^d, d>1, a lower set if for any x in S all the points in N^d with each coordinate being less or equal to that of x also belong to S. Equivalently, one can think of a d-dimensional lower set as of a union of unit cubes such that in each direction any cube leans either on another one or on the coordinate hyperplane.

We will be interested in estimating the number p_d(n) of d-dimensional lower sets of cardinality n. The two-sided inequality n^(1-1/d)<log p_d(n)< C(d)n^(1-1/d) is known to be true whenever n is large enough in terms of d. We show that if d is sufficiently small with respect to n, then C does not depend on d, which means that log p_d(n) is up to an absolute constant equal to n^(1-1/d).