The Delta Conjecture

@article{Haglund2015TheDC,
  title={The Delta Conjecture},
  author={James Haglund and Jeffrey B. Remmel and Andrew Timothy Wilson},
  journal={Discrete Mathematics \& Theoretical Computer Science},
  year={2015}
}
International audience 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… 

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References

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Generalized Shuffle Conjectures for the Garsia-Haiman Delta Operator

We conjecture two combinatorial interpretations for the symmetric function Delta/ek en, where [Delta]f is an eigenoperator for the modified Macdonald polynomials defined by Garsia and Haiman. The

A combinatorial formula for the character of the diagonal coinvariants

Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of

A proof of the q,t-Schröder conjecture

We prove a recent conjecture of Egge, Haglund, Killpatrick and Kremer (Elec. J. Combin. 10 (2003), #R16), which gives a combinatorial formula for the coefficients of a hook shape in the Schur

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A proof of the shuffle conjecture

We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial

Ordered set partition statistics and the Delta Conjecture

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Abstract We introduce a $q,\,t$ -enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action

A proof of the 4-variable Catalan polynomial of the Delta conjecture

In The Delta Conjecture (arXiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values

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In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of