The complete graph operad

A fundamental sequence of operads is that of the little n-cubes operads, originally due to Boardman-Vogt and May. It led to the notion of an E_n-operad, as one equivalent to the little n-cubes operad. There is no really structural definition of this notion, but since the introduction of the little n-cubes operads, many very different types of E-n-operads have been discovered. One of these is a quite elementary and seemingly rather central one:  the complete graphs operad, originally proposed by Berger and later modified in a paper by Brun, Fiedorwiecz, and Vogt. I will present this operad and provide a (new) proof that actually is an E_n-operad.