Enumerative gadget phenomena for (4, 1)-adinkras

Adinkras are combinatorial objects developed to study supersymmetry representations. Gates et al. introduced the gadget as a function of pairs adinkras, obtaining some mysterious results for $(n=4, k=1)$ adinkras with computer-aided computation. Specifically, very few values actually appear, suggesting great deal symmetry in these objects. In this paper, we compute gadgets symbolically and explain observed phenomena group theory combinatorics. Guided by work, give suggestions generalizations other $n$ $k$ parameters.