The fibration sequences of the lattice path operad

We introduce a notion of an operad of complexity $m$, for $m \geq 1$. Operads of complexity $1$ are monoids in the category of $\mathbb{N}$-indexed collections, with monoidal product given by the Day convolution, and operads of complexity $2$ are non-symmetric operads. In general, we prove that the operad for operads of complexity $m$ is a suboperad of the $m$-th stage filtration of the lattice path operad introduced by Batanin and Berger. Finally, we exhibit fibration sequences involving this new notion, extending the results of Turchin and Dwyer-Hess.