In mathematics, a **fence**, also called a **zigzag poset**, is a partially ordered set in which the order relations form a path with alternating orientations:

*a* <

*b* >

*c* <

*d* >

*e* <

*f* >

*h* <

*i* ...

or

*a* >

*b* <

*c* >

*d* <

*e* >

*f* <

*h* >

*i* ...

A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences.

A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century. The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are

1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042 (sequence

A001250 in the OEIS).

The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.

A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.

Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.

An **up-down poset** *Q*(*a*,*b*) is a generalization of a zigzag poset in which there are *a* downward orientations for every upward one and *b* total elements. For instance, *Q*(2,9) has the elements and relations

*a* >

*b* >

*c* <

*d* >

*e* >

*f* <

*g* >

*h* >

*i*.

In this notation, a fence is a partially ordered set of the form *Q*(1,*n*).