  In this paper we prove the case $dim(V_3)=3$ of a conjecture about the
exterior operad ${\Lambda}^{S^2}_{V_d}$. 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 ${\Lambda}^{S^2}_{V_d}(2d+1)$. When $d=3$ this correspondence
allows us to give an explicit description of a determinant-like map and to
settle the above mentioned conjecture.
