Partitions of the complete hypergraph K63 and a determinant-like function

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