Minimum-length maximal green sequences for the affine Tamari algebra

Determine whether the number of maximal green sequences of minimum length $2n-2$ for the affine Tamari algebra $A_Q$ is equal to $n!$.

Background

The paper identifies maximal green sequences for the affine Tamari algebra AQA_Q with maximal chains in the affine Tamari lattice ATamATam. It proves that the possible lengths of such sequences are precisely the integers in [2n−2,(n+12)−1][2n-2,\binom{n+1}{2}-1], so the shortest sequences have length $2n-2$. The authors leave unresolved the enumeration of these shortest sequences and report numerical evidence suggesting a factorial formula.

References

Numerical evidence suggests that the number of minimum-length maximal green sequences of $A_Q$ (i.e., the number of maximal chains of $ATam$ of length $2n-2$) is $n!$.

— The Affine Tamari Lattice  (2502.07198 - Barkley et al., 11 Feb 2025) in Section 7, “Maximal Green Sequences” / Section 10, “Future Work,” subsection “Maximal Green Sequences”