  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 $\#
{\cal H} ( P, C_3 )$. The approach gives raise to new ways to calculate
Dedekind numbers.
