An Extremal Problem for Finite Topologies and Distributive Lattices

Let (rI , r, ,...) be a sequence of non-negative integers summing to n. We determine under what conditions there exists a finite distributive lattice L of rank n with ri join-irreducibles of rank i, for all i = 1,2,.. . . When I. exists, we give explicit expressions for the greatest number of elements L can have of any given rank, and for the greatest total number of elements L can have. The problem is also formulated in terms of finite topological spaces.