Average relational distance in linear extensions of posets

We consider a natural analogue of the graph linear arrangement problem for posets. Let P=(X,≺)P=(X,≺) be a poset that is not an antichain, and let λ:X→[n]λ:X→[n] be an order-preserving bijection, that is, a linear extension of PP. For any relation a≺ba≺b of PP, the distance between aa and bb in λλ is λ(b)−λ(a)λ(b)−λ(a). The average relational distance of λλ, denoted distP(λ), is the average of these distances over all relations in PP. We show that we can find a linear extension of PP that maximises distP(λ) in polynomial time. Furthermore, we show that this maximum is at least 13(|X|+1), and this bound is extremal.