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Linear extension majority cycles in height-1 orders. (English) Zbl 0708.06002

Let x, y be elements of the finite poset X, and write \(x>_ py\) to indicate that more linear extensions of the poset have x above y than y above x. Earlier work by Fishburn showed that when the height of X is \(\geq 2\) then \(>_ p\) can have a cycle. It has recently been shown by Gehrlein and Fishburn that the smallest poset having a \(>_ p\)-cycle has 9 elements, and that there are exactly 5 nonisomorphic 9-element posets having \(>_ p\)-cycles. The current work addresses the case of posets of height 1. If A denotes the set of maximal non-isolated elements, and B the set of minimal non-isolated elements, it is shown that any such cycle must lie entirely within A or entirely within B. If #A\(=3\), then there is no cycle within A. If #A\(=4\) the smallest poset of height 1 admitting a \(>_ p\)-cycle has 15 members. If #A\(=3\), the question is left open as to whether B can contain a cycle.
Reviewer: M.F.Janowitz

MSC:

06A06 Partial orders, general
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[1] P. C.Fishburn (1974) On the family of linear extensions of a partial order, J. Combin. Theory 17, 240-243. · Zbl 0288.06001 · doi:10.1016/0095-8956(74)90030-6
[2] P. C.Fishburn (1976) On linear extension majority graphs of partial orders, J. Combin. Theory 21, 65-70. · Zbl 0324.06002 · doi:10.1016/0095-8956(76)90028-9
[3] P. C.Fishburn (1986) Proportional transitivity in linear extensions of ordered sets, J. Combin. Theory Ser. B 41, 48-60. · Zbl 0566.06002 · doi:10.1016/0095-8956(86)90027-4
[4] B.Ganter, G.H?fner, and W.Poguntke (1987) On linear extensions of ordered sets with a symmetry, Discrete Math. 63, 153-156. · Zbl 0607.06001 · doi:10.1016/0012-365X(87)90005-7
[5] W. V. Gehrlein and P. C. Fishburn (1989) Linear extension majority cycles for small (n?9) partial orders, Comput. Math. Appl. (in press). · Zbl 0708.06003
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