On the work performed by a transformation semigroup

A (partial) transformation α on the finite set {1,..., n} moves an element i of its domain a distance of {pipe}i-iα{pipe} units. The work w(α) performed by α is the sum of all of these distances. We derive formulae for the total work w(S)=∑α∈S w(α) performed by various semigroups S of (partial) transformations. One of our main results is the proof of a conjecture of Tim Lavers which states that the total work performed by the semigroup of all order-preserving functions on an n-element chain is equal to (n-1)22n-3.