Nested Quantum Dyck Paths and ∇ ( s λ )

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron–Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q = 1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.