Determine symmetry of inversion number under reversing a dijoin

Determine whether the inversion number of a dijoin is symmetric for all tournaments; that is, prove or disprove that inv(D_1 \to D_2) = inv(D_2 \to D_1) for every pair of tournaments D_1 and D_2.

Background

The paper’s constructions show that the inversion number of a dijoin can be substantially smaller than the sum of the inversion numbers of its parts. They also indicate that the order and direction in which the two tournaments are joined may affect how their transitive orders intertwine in a decycling matrix.

The authors state that Wang, Yang, and Lu conjectured symmetry, but explain that their own construction makes the issue unclear and that they suspect asymmetry may occur for sufficiently large tournaments. The question is equivalently whether reversing both constituent tournaments leaves the inversion number of the dijoin unchanged.

References

Theorem \ref{T1} raises the related interesting question of whether the inversion number of a dijoin is symmetric, that is, whether $inv(D_1 \to D_2) = inv(D_2 \to D_1)$ holds for all tournaments $D_1$ and $D_2$. This was conjectured to be true by Wang, Yang and Lu.

— On the failure of additivity of inversion numbers for dijoins  (2610.13798 - Behague et al., 9 Oct 2026) in Section 5, “Concluding Remarks,” immediately before Conjecture 5.1