Greene's theorem (posets)

Statement

Let

$P$ be a finite partially ordered set (poset).
$l_{k}$ be the maximum size of a union of $k$ chains
$m_{k}$ be the maximum size of a union of $k$ antichains

Observe $l_{1} \leq l_{2} \leq \dots$ and similarly for the $m_{i}$ 's. Then, we claim that

$λ (P) = (l_{1}, l_{2} - l_{1}, l_{3} - l_{2}, \dots)$ (see lambda(poset)) and
$μ (P) = (m_{1}, m_{2} - m_{1}, m_{3} - m_{2}, \dots)$
are partitions and they are conjugate to each other.

Proof