A recursive approach for the enumeration of the homomorphisms from a poset $P$ to the chain $C_3

Let ${\cal H}(P,C_3)$ be the set of order homomorphisms from a poset $P$ to the chain $C_3 = 1 < 2 < 3$. We develop a recursive approach for the calculation of the cardinality of ${\cal H}(P,C_3)$, and we apply it on several types of posets, including $P = C_3 \times C_3 \times C_k$ and $P = {\cal H}(C_k, C_3)$; for the latter poset $P$, we derive a direct formula for $\# {\cal H} ( P, C_3 )$.