Combinatorial proof of plethysm-coefficient nonnegativity

Derive a combinatorial proof that plethysm coefficients are nonnegative.

Background

Plethysm coefficients are nonnegative because they represent multiplicities of irreducible general linear group representations. The paper explicitly observes that a proof of this nonnegativity based purely on combinatorial properties is not known, leaving the construction of such a proof unresolved.

References

The connection to representation theory also shows that the plethysm coefficients $a_{\mu,\lambda}\pi$ for $\lambda,\mu,\pi$ partitions with ${\pi}={\mu}\cdot{\lambda}$ are non-negative, of which no combinatorial proof is known p. 499.

— Asymptotics of Plethysm  (2509.06424 - Kuppel, 8 Sep 2025) in Section 2, subsection “What is plethysm?”