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