Derivation of the fixed-shape major-index formula via noncrossing involutions

Derive the q-counting formula for the major-index statistic of Richardson tableaux of any fixed shape using the correspondence with noncrossing involutions.

Background

The paper cites an existing formula for q-counting the major-index statistic of Richardson tableaux of a fixed shape. Its main construction identifies Richardson tableaux with insertion tableaux of noncrossing involutions, and the preceding q-enumeration results use this correspondence successfully for comajor-index statistics in particular families.

The authors explicitly state that they do not know how to obtain the fixed-shape major-index formula from noncrossing involutions, leaving this as an unresolved methodological problem.