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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Square $ \boldsymbol{q,t}$-lattice paths and $ \boldsymbol{\nabla(p_n)}$


Authors: Nicholas A. Loehr and Gregory S. Warrington
Journal: Trans. Amer. Math. Soc. 359 (2007), 649-669
MSC (2000): Primary 05E10; Secondary 05A30, 20C30
Published electronically: August 16, 2006
MathSciNet review: 2255191
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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
Email: warrings@wfu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04044-X
PII: S 0002-9947(06)04044-X
Keywords: Lattice paths, Catalan numbers, Dyck paths, diagonal harmonics, nabla operator, Macdonald polynomials
Received by editor(s): November 19, 2004
Published electronically: August 16, 2006
Additional Notes: Both authors’ research was supported by NSF Postdoctoral Research Fellowships.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.