Linear extension majority cycles for small ( n⩽9) partial orders

Abstract

Let ( X, P) denote a poset for which P is an asymmetric partial order on a finete set X of cardinality n. A linear extension of P is a linear order L on X for which P⊆ L. Let L ( P) be the family of all linear extensions of P, let L ( x, y) be the subset of L ( P) whose members have xLy and define binary relations M and M′ on X for the given partial order P by xMy if ∣ L(x, y))∣>∣ L(y, x)∣ , xM′y if x≠y and ∣ L(x, y)∣⩾∣ L(y, x)∣ . We refer to M as the linear extension majority (LEM) relation of ( X, P) and say that ( X, P) has an m-element LEM cycle when x 1 Mx 2 M… Mx m Mx 1 for some distinct x 1,…, x m ϵX. In addition, ( X, P) has an LEM quasi-cycle if there is a 3-element subset { x, y, z} of X for which xM′ yM′ zM′ M′ x and the equality part of M′ holds for exactly one pair in the triple. We show that ( X, P) never has an LEM cycle or an LEM quasi-cycle when n⩽8. For n = 9, there are exactly 5 nonisomorphic asymmetric partial orders with LEM cycles, and m = 3 for each LEM cycle. There are exactly 8 nonisomorphic asymmetric partial orders for n = 9 that have LEM quasi-cycles.