Dens, nests and the Loehr-Warrington conjecture

Abstract: In a companion paper, we introduced raising operator series called Catalanimals. Among them are Schur Catalanimals which represent Schur functions \(s_{\mu }(X^{m,n})\) in alphabets corresponding to copies of the algebra of symmetric functions inside the elliptic Hall algebra of Burban and Schiffmann.

Here we obtain a combinatorial formula for symmetric functions given by a class of Catalanimals that includes the Schur Catalanimals. Our formula is expressed as a weighted sum of LLT polynomials, with terms indexed by configurations of nested lattice paths called nests, having endpoints and bounding constraints controlled by data called a den.

Applied to Schur Catalanimals for the alphabets \(X^{m,1}\) with \(n=1\), our `nests in a den’ formula proves the combinatorial formula conjectured by Loehr and Warrington for \(\nabla^m s_{\mu }\) as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. When \(n\) is arbitrary, our formula establishes an \((m,n)\) version of the Loehr-Warrington conjecture.

In the case where each nest consists of a single lattice path, the nests in a den formula reduces to our previous shuffle theorem for paths under any line. Both this and the \((m,n)\) Loehr-Warrington formula generalize the \((km,kn)\) shuffle theorem proven by Carlsson and Mellit (for \(n=1\)) and Mellit. Our formula here unifies these two generalizations.

Date published: Wednesday, December 15, 2021