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  "body": "\n## Example\n\nLet $n \\ge 1$.\nThe fence $Z_n$ is the [[partially ordered set|0QV2JS9RJ3]] on $x_1, \\ldots, x_n$ with these covering relations:\n$$x_1 \u003c x_2 \u003e x_3 \u003c x_4 \u003e \\cdots.$$\nThat is, $x_i \u003c x_{i+1}$ for odd $i$ and $x_i \u003e x_{i+1}$ for even $i$, with $1 \\le i \u003c n$.\nIt is also called a zigzag poset.\nIts Hasse diagram is a path that goes up and down in turn.\n\nA [[linear extension|W8DZ5KGGHN]] of $Z_n$ is a bijection $f \\colon \\set{x_1, \\ldots, x_n} \\to \\set{1, \\ldots, n}$ with $f(x_1) \u003c f(x_2) \u003e f(x_3) \u003c \\cdots$.\nSo the linear extensions of $Z_n$ correspond to the alternating permutations of $\\set{1, \\ldots, n}$, and their number is the Euler zigzag number $E_n$.\nCheck for $n = 3$: the poset $x_1 \u003c x_2 \u003e x_3$ has the two linear extensions $x_1, x_3, x_2$ and $x_3, x_1, x_2$, and $E_3 = 2$.\n\n## Remarks\n\nThe number of down-sets of $Z_n$ is the Fibonacci number $F_{n+2}$, with $F_1 = F_2 = 1$.\nFor $n = 3$ the down-sets are $\\varnothing$, $\\set{x_1}$, $\\set{x_3}$, $\\set{x_1, x_3}$ and $\\set{x_1, x_2, x_3}$, which makes $5 = F_5$.\nEvery fence has height at most $2$.\n\n## Explanations\n\n### L1 human\n\nA fence is a zigzag: up, down, up, down.\nPutting its elements in a row that respects the zigzag is the same as writing the numbers $1$ to $n$ so that they go up, down, up, down.\nThis links a simple poset with a classical counting problem.\n",
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