  We provide a new combinatorial approach to studying the collection of
N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic
group. In particular, we show that for G the cyclic group of order p^n the
natural order on the collection of N-infinity-operads stands in bijection with
the poset structure of the (n+1)-associahedron. We further provide a lower
bound for the number of possible N-infinity-operads for any finite cyclic group
G.
