Arities of Permutation Groups: Wreath Products andk-Sets

We introduce an invariant of finite permutation groups called the arity which is well known to model theorists but has not been examined from an algebraic point of view. There are few cases in which this invariant is known explicitly. We analyze the behavior of this invariant in power representations of wreath products. We compute it exactly for the action of the symmetric group onnletters on the set ofk-sets from ann-element set, and we estimate it rather closely for symmetric powers of these actions. In the casek=1 we formulate an explicit combinatorial conjecture which would pin down the values exactly in all cases.