The Delta Conjecture

TL;DR

This paper proposes two combinatorial interpretations for a symmetric function related to the Delta operator, extending the Shuffle Conjecture and connecting to Catalan number generalizations.

Contribution

It introduces new combinatorial interpretations for the Delta conjecture, building on previous work and extending the theory to 4-variable Catalan number generalizations.

Findings

Verified several cases of the conjectures using Tesler matrices and set partitions.

Proved a case using reciprocity identity and LLT polynomials.

Extended work to 4-variable Catalan number generalizations.

Abstract

We conjecture two combinatorial interpretations for the symmetric function $\Delta_{e_k} e_n$, where $\Delta_f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture of Haglund, Haiman, Remmel, Loehr, and Ulyanov, which was proved recently by Carlsson and Mellit. We show how previous work of the third author on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.