Let G be a finitely generated group with minimal generating set of size d. For each t ≥ d let Γ_t = Γ_t(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 g_i multiplied on left or right by g_j^±1, and all other g'_k are the same as the corresponding g_k.

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.