Pattern for the promotion order on zig-zag posets

Determine whether the order of promotion on increasing labelings of the zig-zag poset Z_n with labels in [q] follows a pattern as n grows, and, if so, characterize that pattern or derive a general formula.

Background

The paper computes the order of promotion on increasing labelings of the zig-zag posets Z_3, Z_4, and Z_5, obtaining 2q, 15q, and 120q, respectively, for sufficiently large q. Computational evidence suggests that the order for Z_6 is 13090q, but the authors do not establish a general formula or explain the sequence of multiplicative factors.

The unresolved issue is whether these computed values reflect a systematic pattern for arbitrary n and whether the apparent linear dependence on q has a uniform, mathematically describable coefficient.