Let $A \subset \mathbb{Z}^d$ and let $|A|$ be its cardinality. Let $F_n(A)$ be the number of connected sets of $\mathbb{Z}^d$ having cardinality $n$ and intersecting $A$ in at least one site. Assume we know the function $F_n(0)$, where $0$ is the set containing only the origin.
For a connected set $A$, is there a qualitatively better upper bound than $$F_n(A) \leq |A| \cdot F_n(0),$$ which holds for $n$ large, $|A|$ large?
Comment In my opinion, $F_n(A)$ should be much smaller, as for each site of $A$, most of the ``large'' connected sets intersecting such a site will intersect the other sites of $A$ as well. Hence they should not be counted $|A|$ times.