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  "body": "\n## Summary\n\nThe order polynomial of a finite poset $P$ counts the [[order-preserving maps|0A7KV2WB7A]] from $P$ to a chain with $m$ elements, as a function of $m$.\nThis count is a polynomial in $m$ of degree $\\abs{P}$, and its leading coefficient is the number of [[linear extensions|W8DZ5KGGHN]] of $P$ divided by $\\abs{P}!$.\nIt was introduced by Richard Stanley, who proved a reciprocity theorem linking it to the count of strictly order-preserving maps.\n\n## Explanations\n\n### L1 human\n\nLabel the elements of a poset with numbers from $1$ to $m$ so that larger elements never get smaller labels.\nThe number of ways to do this depends on $m$, and it is always given by a polynomial in $m$.\nFor a set of $p$ elements with no relations, every labelling is allowed, so the count is $m^p$.\nFor a chain of $2$ elements, a labelling is a pair $a \\le b$, and there are $m(m+1)/2$ of them.\n",
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