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