Edge partitions of the complete graph and a determinant-like function

Abstract. In this paper we prove the case dim(V3) = 3 of a conjecture from [6] about the exterior operad Λ 2 Vd . For this we introduce a collection of natural involutions on the set of homogeneous cycle-free d-partitions of the complete graph K2d, and show that these involutions correspond to the relations in Λ 2 Vd (2d + 1). When d = 3 this correspondence allows us to give an explicit description of a determinant-like map and to settle the above mentioned conjecture.