Fence (mathematics)

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 a downward orientations for every upward one and b 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).

References

• André, Désiré (1881), "Sur les permutations alternées", J. Math. Pures Appl., (Ser. 3), 7: 167–184.
• Beck, István (1990), "Partial orders and the Fibonacci numbers", Fibonacci Quarterly, 28 (2): 172–174, MR 1051291.
• Currie, J. D.; Visentin, T. I. (1991), "The number of order-preserving maps of fences and crowns", Order, 8 (2): 133–142, doi:10.1007/BF00383399, hdl:10680/1724, MR 1137906, S2CID 122356472.
• Duffus, Dwight; Rödl, Vojtěch; Sands, Bill; Woodrow, Robert (1992), "Enumeration of order preserving maps", Order, 9 (1): 15–29, doi:10.1007/BF00419036, MR 1194849, S2CID 84180809.
• Farley, Jonathan David (1995), "The number of order-preserving maps between fences and crowns", Order, 12 (1): 5–44, doi:10.1007/BF01108588, MR 1336535, S2CID 120372679.
• Gansner, Emden R. (1982), "On the lattice of order ideals of an up-down poset", Discrete Mathematics, 39 (2): 113–122, doi:10.1016/0012-365X(82)90134-0, MR 0675856.
• Höft, Hartmut; Höft, Margret (1985), "A Fibonacci sequence of distributive lattices", Fibonacci Quarterly, 23 (3): 232–237, MR 0806293.
• Kelly, David; Rival, Ivan (1974), "Crowns, fences, and dismantlable lattices", Canadian Journal of Mathematics, 26 (5): 1257–1271, doi:10.4153/cjm-1974-120-2, MR 0417003.
• Rutkowski, Aleksander (1992a), "The number of strictly increasing mappings of fences", Order, 9 (1): 31–42, doi:10.1007/BF00419037, MR 1194850, S2CID 120965362.
• Rutkowski, Aleksander (1992b), "The formula for the number of order-preserving self-mappings of a fence", Order, 9 (2): 127–137, doi:10.1007/BF00814405, MR 1199291, S2CID 121879635.
• Salvi, Rodolfo; Salvi, Norma Zagaglia (2008), "Alternating unimodal sequences of Whitney numbers", Ars Combinatoria, 87: 105–117, MR 2414008.
• Stanley, Richard P. (1986), Enumerative Combinatorics, Wadsworth, Inc. Exercise 3.23a, page 157.
• Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), "The Recognition of Series Parallel Digraphs", SIAM Journal on Computing, 11 (2): 298–313, doi:10.1137/0211023.

External links

• Weisstein, Eric W. "Fence Poset". MathWorld.