Young's lattice

Example

Young's lattice $\mathbb{Y}$ is the set of all integer partitions $\lambda = (\lambda_1 \ge \lambda_2 \ge \cdots)$, including the empty partition. It is ordered by $\lambda \le \mu$ when $\lambda_i \le \mu_i$ for all $i$, that is, when the Young diagram of $\lambda$ fits inside that of $\mu$.

It is a distributive lattice. The meet and join are $(\lambda \wedge \mu)_i = \min(\lambda_i, \mu_i)$ and $(\lambda \vee \mu)_i = \max(\lambda_i, \mu_i)$, the intersection and union of the diagrams. Both are again weakly decreasing sequences, and $\min$ and $\max$ distribute over each other.

It is a graded poset with rank function $\rho(\lambda) = \abs{\lambda} = \lambda_1 + \lambda_2 + \cdots$. Indeed, $\mu$ covers $\lambda$ exactly when the diagram of $\mu$ is that of $\lambda$ with one more box.