Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)

Extended abstracts listed by author > Haglund James

The Delta Conjecture
James Haglund  1  , Jeffrey B. Remmel  2  
1 : Department of Mathematics [Philadelphia]  -  Website
David Rittenhouse Lab. 209 South 33rd Street Philadelphia, PA 19104-6395 -  États-Unis
2 : Department of Mathematics [San Diego]  -  Website
Mathematics Department, University of California, San Diego, La Jolla, CA, USA -  États-Unis

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. 

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