Partitions of the complete hypergraph $$K_6^3$$ and a determinant-like function

In this paper, we introduce a determinant-like map $$\mathrm{det}^{S^3}$$ and study some of its properties. For this, define graded vector space $$\Lambda ^{S^3}_V$$ that has similar properties with the exterior algebra _V$$ GSC-operad ^{S^2}_V$$ from Staic. When $$\mathrm{dim}(V_2)=2$$ , show $$\mathrm{dim}_k(\Lambda ^{S^3}_{V_2}[6])=1$$ which gives existence uniqueness . We also give an explicit formula for as sum over certain 2-partitions complete hypergraph $$K_6^3$$