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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a finite [[partially ordered set|0QV2JS9RJ3]] with $p$ elements.\nFor an integer $m \\ge 1$, let $[m] = \\set{1, \\ldots, m}$ with the usual order.\nLet $\\Omega(P, m)$ be the number of [[order-preserving|0A7KV2WB7A]] maps $f \\colon P \\to [m]$, that is, maps with $f(x) \\le f(y)$ whenever $x \\le y$.\nThe *order polynomial* of $P$ is the unique polynomial $\\Omega(P, t)$ in $t$ that takes the value $\\Omega(P, m)$ at every integer $m \\ge 1$.\n\n## Remarks\n\nSuch a polynomial exists, and it has degree $p$ with leading coefficient $e(P)/p!$, where $e(P)$ is the number of [[linear extensions|W8DZ5KGGHN]] of $P$.\n\nExample: for the chain $x_1 \u003c x_2 \u003c \\cdots \u003c x_p$, $\\Omega(P, m) = \\binom{m + p - 1}{p}$.\nCheck: an order-preserving map is a weakly increasing sequence $f(x_1) \\le \\cdots \\le f(x_p)$ in $[m]$, that is, a multiset of size $p$ from $m$ values; the leading coefficient is $1/p!$, and $e(P) = 1$.\n\nNon-example: the count $m^p$ of all maps $P \\to [m]$ is not the order polynomial of the chain when $p \\ge 2$.\nIt also counts maps that break the order, such as $f(x_1) = 2$, $f(x_2) = 1$.\n\nThe strict order polynomial $\\Omega^\\circ(P, m)$ counts the maps with $x \u003c y \\Rightarrow f(x) \u003c f(y)$.\nStanley's reciprocity theorem says $\\Omega^\\circ(P, t) = (-1)^p \\, \\Omega(P, -t)$.\n",
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