Linear extensions of semiorders: a maximization problem

Fishburn, P.C. and W.T. Trotter, Linear extensions of semiorders: A maximization problem, Discrete Mathematics 103 (1992) 25-40. We consider the problem of determining which partially ordered sets on n points with k pairs in their ordering relations have the greatest number of linear extensions. The posets that maximize the number of linear extensions for each hxed (n, k), 0 G k G (;), are semiorders. However, except for special cases, it appears difficult to say precisely which semiorders solve the problem. We give a complete solution for k G n, a nearly complete solution for k = n + 1, and comment on a few other cases. Let (n, >0) denote the set n = (1, 2, . . . , n} partially ordered by an irreflexive and transitive relation >,, c n2, and let e(n, >J = I{ (n, >*): >* is an irreflexive, transitive and complete (a # b j a >* b or b >* a) relation in n2 that includes >,,} 1, be the number of linear extensions of (n, >O). We consider the problem of determining the posets that maximize e(n, Bo) when >0 has exactly k ordered pairs in n2. That is, given 0 c k c (;) and letting p(n, k) = {(n, >o): PO1 = k), 0, k) = pg=j eh >d, Elsevier Science Publishers B.V. 26 P.C. Fishburn. W.T. Trotter our aim is to characterize the members of P(n, k) for which e(n, >,J = e(n, k). We are also interested in the values of the e(n, k). The extreme cases for k are sufficiently restricted to make their answers obvious: e(n, 0) = n!, e(n, 1) = n!/2, c(n, (n2) 1) = 2, e(n, (8) = 1, with poset diagrams in Fig. l(a)-(d) respectively. A little more effort shows that e(n, 2) = n!/3 and e(n, (;) 2) = 4, see Fig. l(e)-(f). Other cases tend to be far from obvious, and we resolve only a small number of them. They are summarized at the end of this introduction. We refer to a poset (n, >,-J in P(n, k) as a realizer of e(n, k) if e(n, >,J = e(n, k). Our search for realizers is greatly aided by a theorem of Trotter [5] which says that every realizer is a semiorder. Recall that (n, >J is a semiorder (Lute [2]) if, for all a, b, x, y E n, U>~X and b>,y j a>,y or b>ox, a>ox>ob j a>,y or yBob. The only poset of Fig. 1 that is not a semiorder is the suboptimal poset in (e).