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 @l: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 @l is @l(b)-@l(a). The average relational distance of @l, denoted dist"P(@l), is the average of these distances over all relations in P. We show that we can find a linear extension of P that maximises dist"P(@l) in polynomial time. Furthermore, we show that this maximum is at least 13(|X|+1), and this bound is extremal.