$\newcommand{\Pois}{\mathit{Pois}}\newcommand{\Comm}{\mathit{Comm}}\newcommand{\Lie}{\mathit{Lie}}$For my thesis, I'm trying to understand the Poisson operad (I'll call it $\Pois$) and its homotopy counterpart $\Pois_\infty$.

In Loday-Valette, it is stated that $\Pois=\Comm\circ \Lie$, the operad structure relying on the existence of a distributive law $\lambda:\Comm\circ \Lie\rightarrow \Lie\circ \Comm$. Somewhere on the internet, I've read that $\Pois_\infty$ can be defined in a similar way, as the composition $E_\infty \circ \Lie$ (if I remember correctly).

I was wondering if there's any reference for this (if it is correct). Also, can we define $\Pois_n:=E_n\circ \Lie$? What if we use the cofibrant replacement $\Lie_\infty$ (the $L_\infty$ operad) instead of $Lie$, do we still have an operad structure? I assume by homotopy theory of operads that using $\Lie_\infty$ is not a problem, but I'm not so sure.

EDIT: As @Connor Malin rightfully points out, I'm using a different notation than usual. Here $\Pois_n$ means a Poisson structure up to homotopy, while normally it means a Poisson structure with Lie brackets of degree $n$.

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