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Nov 12, 2014 at 20:59 comment added Dan Petersen You're right, they form a nonsymmetric operad. By the way, I think there are some things you might find useful in Nathalie Wahl's Ph.D. thesis.
Nov 12, 2014 at 20:51 vote accept Geoffroy Horel
Nov 12, 2014 at 20:44 comment added Geoffroy Horel Thanks, this answers my question. Also thanks for clarifying the fact that the $PB_n$ do not form an operad in groups. There is something that looks like an operadic composition but as you say there is no way to make the $S_n$ action work. It seems to me that there should still be a non-symmetric operad whose $n$-th space is $PB_n$. Of course that's not going to be a model for $\mathcal{D}_2$.
Nov 12, 2014 at 19:56 history answered Dan Petersen CC BY-SA 3.0