Let G be a finitely generated group with minimal generating set of size d. For each t ≥ d let Γt = Γt(G) be the graph with vertex set V consisting of all generating t-tuples of elements of G and with edges 1) if for some distinct i and j, g'i is gi multiplied on left or right by gj±1, and all other g'k are the same as the corresponding gk.
Following work by Graham and Diaconis I examine connectivity properties of these graphs when G is abelian and when G is a small symmetric group. (For instance, |V (Γ3(Σ4))| = 10,080!!). Pictures will be provided free of charge.
I will relate the size and connectivity properties of these graphs to classic counting problems of Phillip Hall.