Average relational distance in linear extensions of posets

We consider a natural analogue of the graph linear arrangement problem for posets. Let P = (X,≺) be a poset that is not an antichain, and let : X → [n] be an order-preserving bijection, that is, a linear extension of P . For any relation a ≺ b of P , the distance between a and b in is (b)− (a). The average relational distance of , denoted distP ( ), is the average of these distances over all relations in P . We show that we can find a linear extension of P that maximises distP ( ) in polynomial time. Furthermore, we show that this maximum is at least 1 3(∣X∣+ 1), and this bound is extremal.