Linear Extension Majority Cycles in Height-1 Orders.

For points x and y in an poset (X, >) let x > sub p y mean that more linear extensions of the poset have x above y than y above x . It has been known for some time that > sub p can have cycles when the height of the poset is two or more. Moreover, the smallest posets with a > sub p cycle have nine points and heights of 2, 3 and 4. We show here that height-1 posets can also have > sub p cycles. Our smallest example for this phenomenon has 15 points.