TITLE: Reference request: Goodwillie tower of the identity
QUESTION [11 upvotes]: The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor
$$
X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]})
$$
in which $W_j$ is a spectrum with (naive) $\Sigma_j$-action, $X^{[j]}$ is the $j$-fold smash power and ${}_{h\Sigma}$ denotes homotopy orbits. The underlying unequivariant homotopy type of the coefficient spectrum $W_j$ is that of a wedge of $(j-1)!$ spheres of dimension $S^{1-j}$. 
The restriction of the action to the subgroup $\Sigma_{j-1} \subset \Sigma_j$ is known to be given by the permutation action.
I have been told by several folks in the past that the $\Sigma_j$-action actually extends to an action by $\Sigma_{j+1}$.  The way this is somehow done, if I recall, is by considering a certain space of metric trees with $j+1$ leaves. 
Does anyone know of such a construction? Does anyone know a reference?

REPLY [5 votes]: Jean-Louis Loday told me about the extended action by $\Sigma_{j+1}$ in the fall of 1992, after an Oberwolfach talk I gave about the rank filtration of algebraic $K$-theory, where the $\Sigma_j$-representations given by the integral homology of the Goodwillie derivative  spectra $W_j$ played a role. I had shown that these representations were freely generated by $j$-fold Lie brackets, and Loday knew the connection to spaces of trees. Maybe you were there, too?