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  "body": "\n## Example\n\nLet $n \\ge 3$.\nThe crown $S_n$ is the [[partially ordered set|0QV2JS9RJ3]] on $2n$ elements $a_1, \\ldots, a_n, b_1, \\ldots, b_n$ whose only strict relations are $a_i \u003c b_j$ for all $i \\ne j$.\nIt is also called the standard example of dimension $n$, because its [[order dimension|JC4J24MF93]] is $n$.\nSome authors use the name crown for a different poset of height $2$, whose comparability graph is a cycle of length $2n$.\n\nLower bound.\nFor each $i$, the elements $a_i$ and $b_i$ are incomparable, so a realizer needs a [[linear extension|W8DZ5KGGHN]] $L$ with $b_i$ before $a_i$.\nSuppose one linear extension $L$ has $b_i$ before $a_i$ and $b_j$ before $a_j$ with $i \\ne j$.\nSince $a_i \u003c b_j$ and $a_j \u003c b_i$, $L$ would contain the cycle $a_i, b_j, a_j, b_i, a_i$, which is impossible.\nSo each linear extension reverses at most one pair, and at least $n$ linear extensions are needed.\n\nUpper bound.\nFor each $i$, let $L_i$ be the following order.\nFirst come the $a_j$ with $j \\ne i$, in increasing order of $j$, then $b_i$, then $a_i$.\nLast come the $b_j$ with $j \\ne i$, in increasing order of $j$.\nEach $L_i$ is a linear extension, and every incomparable pair appears in both relative orders among $L_1, \\ldots, L_n$, so these $n$ orders form a realizer.\n\n## Remarks\n\nThe [[comparability graph|1P4830BH6Z]] of $S_n$ is the crown graph: the complete bipartite graph $K_{n,n}$ with a perfect matching removed.\nThe crown $S_n$ has height $2$ and width $n$, so it shows that the dimension of a poset of height $2$ can be as large as one likes.\n\n## Explanations\n\n### L1 human\n\nDraw $n$ points in a bottom row and $n$ points in a top row.\nJoin each bottom point to every top point except the one just above it.\nThis poset needs $n$ different total orders to describe it exactly, because each total order can account for only one missing link.\n",
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