Descent statistic for periodic-pattern expansions

Determine whether the infinite multi-row periodic patterns associated with C_λ carry a natural descent statistic that combinatorially explains the noncommutative ribbon-basis expansion coefficients σ_{λ,ν}.

Background

The paper realizes C_λ-structures as infinite multi-row periodic patterns shifted simultaneously by one integer and derives a closed algebraic formula for the Kronecker-product coefficients bλ_{α,β}. A related result from the preceding work expands C_λ in the ribbon basis {r_ν} of noncommutative symmetric functions, with coefficients denoted σ_{λ,ν}.

The unresolved question is whether the periodic-pattern model admits a natural descent statistic whose enumeration gives these ribbon-expansion coefficients. Such a statistic would provide a combinatorial interpretation of σ{λ,ν}, analogous in spirit to the way the Möbius-inversion theorem gives an algebraic explanation of the coefficients bλ{α,β}.

References

The paper also expands C_\lambda in the ribbon basis {r_\nu} of non-commutative symmetric functions. Determine whether the periodic patterns of Section~\ref{sec:realization} carry a natural descent statistic explaining the resulting expansion coefficients \sigma_{\lambda,\nu} combinatorially, in the same spirit as Theorem~\ref{thm:mobius} explains b\lambda_{\alpha,\beta} algebraically.

— The Molecular Species $\mathbf{C}_α$: Geometric Realization and a Closed Formula for Kronecker Coefficients  (2609.05010 - Baolahy et al., 4 Sep 2026) in Open Problem 2, Section 4 (Open problems), labeled prob:ribbon