# partition lattice

The partition lattice (or lattice of partitions) $\Pi_{n}$ is the lattice of set partitions (http://planetmath.org/Partition) of the set $[n]=\{1,\dots,n\}$. The partial order on $\Pi_{n}$ is defined by refinement, setting $x\leq y$ if any only if each cell of $x$ is contained in a cell of $y$.

If $n<3$, then $\Pi_{n}$ is a chain. But $\Pi_{3}$ is not even a distributive lattice: