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?



Reference request: Goodwillie tower of the identity
This is done by identifying the partition complex of the set with j elements with the space of fully grown trees with j+1 leaves. Reference: "Partition complexes, duality and integral tree representations" by Alan Robinson, Algebr. Geom. Topol. 4. See specifically proposition 2.7 and corollary 2.8. <http://msp.org/agt/2004/4-2/agt-v4-n2-p12-s.pdf>


