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

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 $D_{p^{n}}$ , $p > 2$ prime, and cyclic groups $C_{q p^{n}}$ , $p \neq q$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $N_{\infty}$ operads for these groups.

The combinatorics of $N_{\infty}$ operads for $C_{q p^{n}}$ and $D_{p^{n}}$

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The combinatorics of $N_{\infty}$ operads for $C_{q p^{n}}$ and $D_{p^{n}}$

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The combinatorics of N∞N∞​ operads for CqpnCqpn​ and DpnDpn​

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The combinatorics of N∞N∞​ operads for CqpnCqpn​ and DpnDpn​

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The combinatorics of $N_{\infty}$ operads for $C_{q p^{n}}$ and $D_{p^{n}}$

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The combinatorics of $N_{\infty}$ operads for $C_{q p^{n}}$ and $D_{p^{n}}$

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