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A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Published online by Cambridge University Press:  20 November 2018

J. Haglund
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA email: jhaglund@math.upenn.edu
J. Morse
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA email: morsej@math.drexel.edu
M. Zabrocki
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3 email: zabrocki@mathstat.yorku.ca
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Abstract

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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. 126 (2005), pp. 195 − 232) for $\nabla {{e}_{n}}\left[ X \right]$. 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

References

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