Fence (mathematics)

File:Zigzag poset.svg

The Hasse diagram of a six-element fence.

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:

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

or

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

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. 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 Template:Mvar downward orientations for every upward one and Template:Mvar 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).

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• Exercise 3.23a, page 157.
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