Combinatorial formula for the Schur expansion of the diagonal coinvariant Frobenius series

Find a combinatorial formula for the Schur expansion of the symmetric function \(\nabla e_n\), equivalently for the graded Frobenius image of the diagonal coinvariant algebra \(DR_n\).

Background

The paper studies the diagonal coinvariant algebra $DR_n$ through its bigraded Frobenius image, which is $\nabla e_n$. Although the Schur coefficients encode the multiplicities of Specht modules and are central to representation stability, the paper uses a combinatorial formula for the monomial expansion instead, obtained from labeled Dyck paths and the shuffle theorem of Carlsson and Mellit. Establishing a direct combinatorial description of the Schur expansion would provide a more explicit understanding of these representation multiplicities.