Fence (mathematics)

Short description: Partially ordered set with alternatingly-related elements

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⋯

a>b<c>d<e>f<h>i⋯

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:

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.[2]

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 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.[5] 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).

## Contents

* 1 Equivalent conditions
* 2 Notes
* 3 References
* 4 External links

## Equivalent conditions

The following conditions are equivalent for a poset P:[citation needed]

1. P is a disjoint union of zigzag posets.
2. If a ≤ b ≤ c in P, either a = b or b = c.
3. <∘<=∅, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
4. P has dimension at most one (defined analogously to the Krull dimension of a commutative ring).
5. Every element of P is either maximal or minimal.
6. The slice category Pos/P is cartesian closed.[lower-alpha 1]

The prime ideals of a commutative ring R, ordered by inclusion, satisfy the equivalent conditions above if and only if R has Krull dimension at most one.[citation needed]

## Notes

1. ↑ Here, Pos denotes the category of partially ordered sets.

## References

1. ↑ (André 1881).
2. ↑ (Gansner 1982) calls the fact that this lattice has a Fibonacci number of elements a “well known fact,” while (Stanley 1986) asks for a description of it in an exercise. See also (Höft Höft), (Beck 1990), and (Salvi Salvi).
3. ↑ (Valdes Tarjan).
4. ↑ (Currie Visentin); (Duffus Rödl); (Rutkowski 1992a); (Rutkowski 1992b); (Farley 1995).
5. ↑ (Gansner 1982).

* 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 .
* 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 .
* "Enumeration of order preserving maps", Order 9 (1): 15–29, 1992, doi:10.1007/BF00419036 .
* Farley, Jonathan David (1995), "The number of order-preserving maps between fences and crowns", Order 12 (1): 5–44, doi:10.1007/BF01108588 .
* 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 .
* Höft, Hartmut; Höft, Margret (1985), "A Fibonacci sequence of distributive lattices", Fibonacci Quarterly 23 (3): 232–237 .
* Kelly, David (1974), "Crowns, fences, and dismantlable lattices", Canadian Journal of Mathematics 26 (5): 1257–1271, doi:10.4153/cjm-1974-120-2 .
* Rutkowski, Aleksander (1992a), "The number of strictly increasing mappings of fences", Order 9 (1): 31–42, doi:10.1007/BF00419037 .
* 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 .
* Salvi, Rodolfo; Salvi, Norma Zagaglia (2008), "Alternating unimodal sequences of Whitney numbers", Ars Combinatoria 87: 105–117 .
* Enumerative Combinatorics, Wadsworth, Inc., 1986 Exercise 3.23a, page 157.
* Valdes, Jacobo (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". http://mathworld.wolfram.com/FencePoset.html.

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