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

TL;DR

This paper develops a recursive method to classify and count distinct $N_infty$ operads for specific finite groups, revealing rich combinatorial structures.

Contribution

It introduces a recursive approach to construct transfer systems and computes the number of homotopically distinct $N_infty$ operads for dihedral and cyclic groups.

Findings

Calculated the number of $N_infty$ operads for $D_{p^n}$ and $C_{qp^n}$ groups.

Revealed intricate combinatorial patterns in the structure of these operads.

Displayed combinatorics of meaningful $N_infty$ operads within these groups.

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 $D_{p^n}$, $p > 2$ 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.