A study is made of the asymptotic behaviour of quantities of the form, where π is randomly chosen from the uniform distribution over the set of permutations of.Ucan always be decomposed into the sum of two uncorrelated parts, one degenerate and the other non-degenerate. When the non-degeneratepart dominates asymptotically, the limit law forUis typically nonn.al. When the degenerate part dominates, the limit law is sometimes normal and sometimes a quadratic form in correlated normal variables. Applications to random vertex colourings of graphs are discussed