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 λ: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 λ is λ(b)−λ(a). The average relational distance of λ, denoted , is the average of these distances over all relations in P. We show that we can find a linear extension of P that maximises in polynomial time. Furthermore, we show that this maximum is at least , and this bound is extremal.