Rigorous complexity bound for the inversion-based LRX algorithm

Prove that the inversion-based algorithm for decomposing permutations into the LRX generators has complexity at most n(n-1)/2+n/2.

Background

The authors present a second, empirically strongest algorithm for decomposing permutations into products of L, R, and X. Numerical experiments suggest a complexity bound of n(n-1)/2+n/2, which would improve the proved general upper bound discussed earlier. A rigorous proof of this sharper estimate is left unresolved.

References

Currently it is our top performing algorithm from the practical tests, we expect its complexity is bounded by $n(n-1)/2+n/2$, however it is not yet proved.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 5.4, “Upper bound on the diameter and the algorithms”