On Unimodality for Linear Extensions of Partial Orders

R. Rivest has recently proposed the following intriguing conjecture: Let x* denote an arbitrary fixed element in an n-element partially ordered set P, and for each k in {1, 2, , n let Nk be the number of order-preserving maps from P onto {1, 2,..., n} that map x* into k. Then the sequence N1,'' ', N,, is unimodal. This note proves the conjecture for the special case in which P can be covered by two linear orders. It also generalizes this result for P that have disjoint components, one of which can be covered by two linear orders. 1. Introduction. Given a finite partially ordered set (P, <), where < is asymmetric, we say that an injection A from P into the set Z of integers is a linear extension of P if, for all x, y P, x < y :::),A (x) < A (y). We shall presume that P has n elements and, in the main part of the paper, restrict ourselves to bijections A :P-> In]-{1, 2,..., n}. Generalizations are discussed later. Let x* be an arbitrary fixed element in P. For each k [n], define Nk to be the