In Richard P. Stanley's 1986 text, {\sl 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 $\sum f_k(n) x^n$?

We solve this problem. (Joint work with Ryan Klippenstine.)