Fence

Example

Let $n \ge 1$. The fence $Z_n$ is the partially ordered set on $x_1, \ldots, x_n$ with these covering relations: $$x_1 < x_2 > x_3 < x_4 > \cdots.$$ That is, $x_i < x_{i+1}$ for odd $i$ and $x_i > x_{i+1}$ for even $i$, with $1 \le i < n$. It is also called a zigzag poset. Its Hasse diagram is a path that goes up and down in turn.

A linear extension of $Z_n$ is a bijection $f \colon \set{x_1, \ldots, x_n} \to \set{1, \ldots, n}$ with $f(x_1) < f(x_2) > f(x_3) < \cdots$. So the linear extensions of $Z_n$ correspond to the alternating permutations of $\set{1, \ldots, n}$, and their number is the Euler zigzag number $E_n$. Check for $n = 3$: the poset $x_1 < x_2 > x_3$ has the two linear extensions $x_1, x_3, x_2$ and $x_3, x_1, x_2$, and $E_3 = 2$.