Diameters of binary-string Schreier graphs

Determine whether the diameters of the Schreier graphs for the action of S_{2n} on binary strings with n zeros and n ones lie between n^2/6 and n^2/5.

Background

The paper also examines the Schreier graph obtained from the LRX action on binary strings containing equal numbers of zeros and ones. Exhaustive computations up to n=42 suggest that its diameter lies in a narrow quadratic range, leading the authors to formulate a conjectural estimate.

References

Based on that it is tempting to conjecture that the diameters of such graphs are between $n2/6$ and $n2/5$, while growth again can be approximated by the Guimbel distribution.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 5.5, final paragraph on the Schreier coset graph