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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Square $\boldsymbol {q,t}$-lattice paths and $\boldsymbol {\nabla (p_n)}$
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by Nicholas A. Loehr and Gregory S. Warrington PDF
Trans. Amer. Math. Soc. 359 (2007), 649-669 Request permission

Abstract:

The combinatorial $q,t$-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The $q,t$-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the $n$’th $q,t$-Catalan number is the Hilbert series for the module of diagonal harmonic alternants in $2n$ variables; it is also the coefficient of $s_{1^n}$ in the Schur expansion of $\nabla (e_n)$. Using $q,t$-analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of $\nabla (e_n)$ and the Hilbert series of the diagonal harmonics modules. This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several $q,t$-analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the $q,t$-Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of $\nabla (p_n)$, the “Hilbert series” $\langle \nabla (p_n),h_{1^n}\rangle$, and the sign character $\langle \nabla (p_n),s_{1^n}\rangle$.
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Additional Information
  • Nicholas A. Loehr
  • Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187
  • Email: nick@math.wm.edu
  • Gregory S. Warrington
  • Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
  • MR Author ID: 677560
  • Email: warrings@wfu.edu
  • Received by editor(s): November 19, 2004
  • Published electronically: August 16, 2006
  • Additional Notes: Both authors’ research was supported by NSF Postdoctoral Research Fellowships.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 649-669
  • MSC (2000): Primary 05E10; Secondary 05A30, 20C30
  • DOI: https://doi.org/10.1090/S0002-9947-06-04044-X
  • MathSciNet review: 2255191