A new notion of transitivity for groups and sets of permutations

Let $\Omega=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $\Omega$ is the integer partition $\lambda=(\lambda_1,...,\lambda_k)$ defined by $\lambda_i = |P_i|$. Let G be a group of permutations acting on $\Omega$. For a fixed partition $\lambda$ of n, we say that G is {\em $\lambda$-transitive} if G has only one orbit when acting on partitions P of shape $\la$. A corresponding definition can also be given when G is just a set. For example, if $\lambda=(n-t,1,...,1)$, then a $\lambda$-transitive group is the same as a t-transitive permutation group and if $\lambda=(n-t,t)$, then we recover the t-homogeneous permutation groups. In this paper, we use the character theory of the symmetric group $S_n$ to establish some structural results regarding $\lambda$-transitive groups and sets. In particular, we are able to generalize a theorem of Livingstone and Wagner about t-homogeneous groups. We survey the relevant examples coming from groups. While it is known that a finite group of permutations can be at most 5-transitive unless it contains the alternating group, we show that it is possible to construct a non-trivial t-transitive set of permutations for each positive integer t. We also show how these ideas lead to a split basis for the association scheme of the symmetric group.