Some Monotonicity Properties of Partial Orders

A fundamental quantity which arises in the sorting of n numbers $a_1$, $a_2$,..., $a_n$ is Pr($a_i$ > $a_j$ | P), the probability that $a_i$ > $a_j$ assuming that all linear extensions of the partial order P are equally likely. In this paper we establish various properties of Pr($a_i$ > $a_j$ | P) and related quantities. In particular, it is shown that Pr($a_i$ > $b_j$ | P'') $\geq$ Pr($a_i$ > $b_j$ | P), if the partial order P consists of two disjoint linearly ordered sets A = {$a_1$ > $a_2$ > ... > $a_m$}, B = {$b_1$ > $b_2$ > ... > $b_n} and P'' = P $\cup$ {any relations of the form $a_k$ > $b_l$}. These inequalities have applications in determining the complexity of certain sorting-like computations.