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  "body": "\n## Summary\n\nThe order polytope of a finite poset $P$ is the set of functions $f \\colon P \\to [0, 1]$ that are [[order-preserving|0A7KV2WB7A]], seen as points of $\\R^P$.\nIt is a convex polytope with integer vertices, namely the indicator functions of the upper sets of $P$.\nIts volume is the number of [[linear extensions|W8DZ5KGGHN]] of $P$ divided by $\\abs{P}!$, and its lattice point counts are given by the [[order polynomial|VDBDJZWPEA]].\nIt was introduced by Richard Stanley, together with the chain polytope of the poset.\n\n## Explanations\n\n### L1 human\n\nGive each element of a poset a real number between $0$ and $1$, so that larger elements get numbers at least as large.\nAll such assignments fill a convex region in space, one coordinate per element.\nFor two unrelated elements you get the whole unit square.\nFor a chain $x \u003c y$ you get the triangle where $0 \\le f(x) \\le f(y) \\le 1$, which is half of the square.\n",
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