Combinatorial interpretation of Schur-plethysm coefficients

Determine the coefficients a_{λ,μ}^{ν} in the Schur expansion s_{λ}[s_{μ}]=∑_{ν}a_{λ,μ}^{ν}s_{ν} for arbitrary partitions λ and μ, and construct a combinatorial interpretation of these nonnegative integer coefficients.

Background

Plethysm coefficients describe the decomposition of the Schur functor Sλ(SμV), equivalently the Schur expansion of s_λ[s_μ]. Although these coefficients are known to be nonnegative integers and several special cases and computational algorithms are available, no general combinatorial interpretation is known. The paper uses this unresolved problem as the combinatorial counterpart of the Reidemeister-spectrum problem.

References

One of the major open problems (\cref{obj:PlethysmSchurFunctions}) in the field of symmetric functions is to find a combinatorial interpretation of the coefficients a_{\mu, \lambda}{\nu} in s_{\lambda}[s_{\mu}] = \sum_{\nu} a_{\lambda, \mu}{\nu} s_{\nu}.

— Reidemeister spectra of free nilpotent groups and plethysms of Schur functions  (2501.02591 - Senden, 5 Jan 2025) in Section 1, subsection “Destination: plethysm of Schur functions”; Objective obj:PlethysmSchurFunctions