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  1. Home > Articles & Issues >
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  3. The Delta Conjecture
Conference paper

The Delta Conjecture

James Haglund (1), Jeffrey B. Remmel (2), Andrew Timothy Wilson ORCID (1)
(1) Department of Mathematics [Philadelphia]
(2) Department of Mathematics [Univ California San Diego]
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Imported on
July 4, 2016
Published on
April 22, 2020
Last modified on
March 31, 2025
Volume
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
DOI
10.46298/dmtcs.6384
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https://about.hal.science/hal-authorisation-v1
National Science Foundation
1200296
Combinatorics of Symmetric Functions
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The Delta Conjecture

James Haglund (1), Jeffrey B. Remmel (2), Andrew Timothy Wilson ORCID (1)
(1) Department of Mathematics [Philadelphia]
(2) Department of Mathematics [Univ California San Diego]
Abstract
We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆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, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors 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.
Keywords
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
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