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In this paper we prove the case dim(V3) = 3 of a conjecture from [6] about the exterior operad Λ S 2 V d . For this we introduce a collection of natural involutions on the set of homogeneous cycle-free d-partitions of the complete graph K 2d , and show that these involutions correspond to the relations in Λ S 2 V d (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.

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