Distributive lattices of small width, II: A problem from Stanleyʹs 1986 text Enumerative Combinatorics

In Richard P. Stanleyʹs 1986 text, Enumerative Combinatorics, the following problem is posed: Fix a natural number k. Consider the posets P of cardinality n such that, for 0 < i < n , P has exactly k order ideals (down-sets) of cardinality i. Let f k ( n ) be the number of such posets. What is the generating function ∑ f 3 ( n ) x n ? s paper, the problem is solved.