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 |i − iα| 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.