  Let $L(m,n)$ denote Young's lattice consisting of all partitions whose Young
diagrams are contained in the $m\times n$ rectangle. It is a well-known result
that the poset $L(m,n)$ is rank symmetric, rank unimodal, and Sperner. A direct
proof of this result by finding an explicit order matching of $L(m,n)$ is an
outstanding open problem. In this paper, we present an explicit order matching
$\varphi$ for $L(3,n)$ by several different approaches, and give chain tableau
version of $\varphi$ that is very helpful in finding patterns. It is surprise
that the greedy algorithm and a recursive knead process also give the same
order matching. Our methods extend for $L(4,n)$.
