Edge partitions of the complete graph and a determinant like function

TL;DR

This paper proves a specific case of a conjecture related to exterior operads by analyzing edge partitions of complete graphs, introducing involutions, and describing a determinant-like map.

Contribution

It introduces involutions on cycle-free partitions of complete graphs and uses them to explicitly describe a determinant-like function, settling a conjecture for the case d=3.

Findings

Established the case d=3 of the conjecture.

Defined involutions corresponding to relations in the operad.

Provided an explicit description of a determinant-like map.

Abstract

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.