The combinatorics of \(N_\infty\) operads for \(C_{qp^n}\) and \(D_{p^n}\)

Abstract

We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct \(N_{\infty} \) operads for dihedral groups \(D_{p^n}\), \(p \gt 2\) prime, and cyclic groups \(C_{qp^n}\), \(p \neq q\) prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful \(N_\infty\) operads for these groups.