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A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

  Published:2011-10-22
 Printed: Aug 2012
  • J. Haglund,
    Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
  • J. Morse,
    Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA
  • M. Zabrocki,
    Department of Mathematics and Statistics, York University , Toronto, ON M3J 1P3
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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 of the Macdonald theory $\nabla$ operator applied to a Hall--Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the ``shuffle conjecture" (Duke J. Math. $\mathbf {126}$ ($2005$), 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall--Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,t$-Catalan sequences, and we prove a number of identities involving these functions.
Keywords: Dyck Paths, Parking functions, Hall--Littlewood symmetric functions Dyck Paths, Parking functions, Hall--Littlewood symmetric functions
MSC Classifications: 05E05, 33D52 show english descriptions Symmetric functions and generalizations
Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
05E05 - Symmetric functions and generalizations
33D52 - Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
 

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