Perhaps it is more natural to look at Gog or Magog-triangles, which are special types of GT-patterns, equinumerous with ASMs.

These GT-patterns can then be mapped to SSYTs of triangle shape, and on these, one can do lots of statistics.

I tried a few, but no obvious candidate.

Perhaps it is more natural to look at Gog or Magog-triangles, which are special types of GT-patterns, equinumerous with ASMs.

These GT-patterns can then be mapped to SSYTs of triangle shape, and on these, one can do lots of statistics.

I tried a few, but no obvious candidate.

EDIT:

This questing has been asked in the end of this article (notices of the AMS). There seem to be a cyclic sieving phenomenon on ASM, but they write that they have no idea what the q-statistic is. However, the CSP might give a lead.