Solution
= Solution
On an $n$-sample, each coordinate partition has at most $(n+1)^{m-1}$ placements, hence all $p$ partitions have at most $(n+1)^{p(m-1)}$ configurations. For each fixed configuration, the sign class from a <vector space> of dimension $1+p(m-1)$ has at most $(n+1)^{1+p(m-1)}$ patterns. Multiplication gives
$$
\boxed{s(H_{F_3},n)\le(n+1)^{1+2p(m-1)}.}
$$