  We investigate the combinatorial data arising from the classification of
equivariant homotopy commutativity for cyclic groups of order $G=C_{p_1 \cdots
p_n}$ for $p_i$ distinct primes. In particular, we will prove a structural
result which allows us to enumerate the number of $N_\infty$-operads for
$C_{pqr}$, verifying a computational result.
