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

${\displaystyle a<b>c<d>e<f>h<i\cdots }$

or

${\displaystyle a>b<c>d<e>f<h>i\cdots }$

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.^[1] The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are:

${\displaystyle 1,1,2,4,10,32,122,544,2770,15872,101042.}$

(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.^[2]

A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.^[3]

Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.^[4]

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.^[5] For instance, Q(2,9) has the elements and relations

${\displaystyle a>b>c<d>e>f<g>h>i.}$

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

Equivalent conditions

The following conditions are equivalent for a poset P:^{[citation needed]}

1. P is a disjoint union of zigzag posets.
2. If a ≤ b ≤ c in P, either a = b or b = c.
3. ${\displaystyle <\circ <\;=\emptyset }$, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
4. P has dimension at most one (defined analogously to the Krull dimension of a commutative ring).
5. Every element of P is either maximal or minimal.
6. The slice category Pos/P is cartesian closed.^[a]

The prime ideals of a commutative ring R, ordered by inclusion, satisfy the equivalent conditions above if and only if R has Krull dimension at most one.^{[citation needed]}