ARITIES OF PERMUTATION-GROUPS - WREATH-PRODUCTS AND K-SETS

We introduce an invariant of finite permutation groups celled the arit y which is well known to model theorists but has not been examined fro m an algebraic point of view. There are few cases in which this invari ant is known explicitly. We analyze the behavior of this invariant in power representations of wreath products. We compute it exactly for th e action of the symmetric group on n letters on the set of k-sets from an n-element set, and we estimate it rather closely for symmetric pow ers of these actions. In the case k = I we formulate an explicit combi natorial conjecture which would pin down the values exactly in all cas es. (C) 1996 Academic Press, Inc.