Proof of the 2-part compositional shuffle conjecture

  • Adriano M. GarsiaEmail author
  • Gouce Xin
  • Mike Zabrocki
Part of the Progress in Mathematics book series (PM, volume 257)


In a recent paper [9] J. Haglund, J. Morse and M. Zabrocki advanced a refinement of the Shuffle Conjecture of Haglund et. al. [8]. They introduce the notion of “touch composition” of a Dyck path, whose parts yield the positions where the path touches the diagonal. They conjectured that the polynomial \(\big\langle \nabla \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}\,1\,\ h_{\mu _{1}}h_{\mu _{2}}\cdots h_{\mu _{l}}\big\rangle\), where \(\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}\,1\) is essentially a rescaled Hall–Littlewood polynomial and \(\nabla \) is the Macdonald eigen-operator introduced in [1], enumerates by \(t^{\mathrm{area}}q^{\mathrm{dinv}}\) the parking functions whose Dyck paths hit the diagonal by (p 1, p 2, , p k ) and whose diagonal word is a shuffle of l increasing words of lengths μ 1, μ 2, , μ l . In this paper we prove the case l = 2 of this conjecture.


Symmetric functions Macdonald polynomials Parking functions 

Mathematics Subject Classification



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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.UC San DiegoLa JollaUSA
  2. 2.Capital Normal UniversityBeijingPeople’s Republic of China
  3. 3.York UniversityTorontoCanada

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