Fence (mathematics)



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:
 * $$acehbdfi \cdots$$

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. The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are:
 * $$1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042.$$

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
 * $$a>b>ce>fh>i.$$

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