Ribbon-Schur analogues of the asymptotic results

Determine whether the asymptotic regularity results established for inner products with irreducible characters and Schur functions remain valid when those test functions are replaced by ribbon Schur functions indexed by compositions.

Background

The paper’s regularity notion is formulated using multiplicities against irreducible characters, equivalently coefficients of Schur functions. Ribbon Schur functions provide a refinement indexed by compositions and encode descent-set information.

The corresponding coefficients enumerate permutations with prescribed descent sets and cycle structures. The problem asks whether the paper’s asymptotic methods and conclusions extend to this finer statistic.