Some Monotonicity Properties of Partial Orders

A fundamental quantity which arises in the sorting of n numbers app**.9an is Pr(ai < aj \ P) , the probability that ai < a. assuming J that all linear extensions of the partial order P are equally likely. In this paper we establish various properties of Pr(ai < aj 1 P) and related quantities. In particular, it is shown that Pr(ai < bj 1 P') 2 Pr(ai < bj 1 P) , if the partial order P consists of two disjoint linearly ordered sets A= {al< a2 < . . . < am] , B = {bl < b2 < . . . < b ] and n P' = PU {any relations of the form ak < ba) . These inequalities have applications in determining the complexity of certain sorting-like computations.