Fence (mathematics)

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

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. 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.

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

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

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