Transposon diameter

Prove that the diameter of the Cayley graph generated by block transpositions (transposons) is ceil((n+1)/2), apart from the stated exceptional cases n=13 and n=15.

Background

The paper recalls a conjecture from the genome-rearrangement literature concerning the diameter of the Cayley graph generated by block transpositions. The proposed formula has explicit exceptions at n=13 and n=15.

References

Diameter of transposons is $\left\lceil \tfrac{n+1}{2}\right\rceil$ for $n\ne 13,15$.

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 14, subsection “Transposons (block transpositions)”