Order polytope

Summary

The order polytope of a finite poset $P$ is the set of functions $f \colon P \to [0, 1]$ that are order-preserving, seen as points of $\R^P$. It is a convex polytope with integer vertices, namely the indicator functions of the upper sets of $P$. Its volume is the number of linear extensions of $P$ divided by $\abs{P}!$, and its lattice point counts are given by the order polynomial. It was introduced by Richard Stanley, together with the chain polytope of the poset.