If I'm not mistaken, this simple Sage code computes $F(n)$:
def F(n): P = Posets.IntegerPartitions(n) return sum( len(P.closed_interval(p,P.top())) for p in P ) The values of $F(n)$ for $n=0,1,\dots,20$ are:
1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294