Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • 1. F. Bergeron and A. Garsia, Science Fiction and Macdonald Polynomials, CRM Proceedings and Lecture Notes AMS VI 3 (1999), 363--429. MR 1726826 (2002d:20013)
  • 2. F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Lattice diagram polynomials and extended Pieri rules, Adv. in Math. 2 (1999), 244--334. MR 1680202 (2000h:05231)
  • 3. F. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods and Applications of Analysis VII 3 (1999), 363--420. MR 1803316 (2002g:05184)
  • 4. E. Egge, J. Haglund, D. Kremer, and K. Killpatrick, A Schröder generalization of Haglund's statistic on Catalan paths, Electronic Journal of Combinatorics 10 (2003), R16; 21 pages. MR 1975766 (2004c:05008)
  • 5. A. Garsia and J. Haglund, A proof of the $ q,t$-Catalan positivity conjecture, LACIM 2000 Conference on Combinatorics, Computer Science, and Applications (Montreal), Discrete Math. 256 (2002), 677--717. MR 1935784 (2004c:05207)
  • 6. A. Garsia and J. Haglund, A positivity result in the theory of Macdonald polynomials, Proc. Nat. Acad. Sci. 98 (2001), 4313--4316. MR 1819133 (2002c:05159)
  • 7. A. Garsia and M. Haiman, A remarkable $ q,t$-Catalan sequence and $ q$-Lagrange Inversion, J. Algebraic Combinatorics 5 (1996), 191--244. MR 1394305 (97k:05208)
  • 8. J. Haglund, Conjectured statistics for the $ q,t$-Catalan numbers, Advances in Mathematics 175 (2003), 319--334. MR 1972636 (2004c:05021)
  • 9. J. Haglund, A proof of the $ q,t$-Schröder conjecture, Intl. Math. Res. Notices 11 (2004), 525--560. MR 2038776 (2005d:05147)
  • 10. J. Haglund, A combinatorial model for the Macdonald polynomials, Proc. Nat. Acad. Sci. USA, 101, 46 (2004), 16127-16131. MR 2114585 (2006e:05178)
  • 11. J. Haglund, The $ q,t$-Catalan Numbers and the Space of Diagonal Harmonics, AMS University Lecture Series, to appear.
  • 12. J. Haglund, M. Haiman, and N. Loehr, A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc. 18 (2005), 735-736. MR 2138143
  • 13. J. Haglund, M. Haiman, N. Loehr, J. Remmel, and A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005), 195--232. MR 2115257 (2006f:05186)
  • 14. J. Haglund and N. Loehr, A conjectured combinatorial formula for the Hilbert series for diagonal harmonics, Proceedings of FPSAC 2002 Conference (Melbourne, Australia), Discrete Mathematics 298 (2005), no. 1-3, 189-204. MR 2163448
  • 15. M. Haiman, Notes on Macdonald polynomials and the geometry of Hilbert schemes, Symmetric Functions 2001: Surveys of Developments and Perspectives, Proceedings of the NATO Advaced Study Institute. Sergey Fomin, ed. Kluwer, Dordrecht (2002), 1--64. MR 2059359 (2005f:14010)
  • 16. M. Haiman, Combinatorics, symmetric functions, and Hilbert schemes, CDM 2002: Current Developments in Mathematics, Intl. Press Books (2003), 39--112. MR 2051783 (2006e:05179)
  • 17. M. Haiman, Hilbert schemes, polygraphs, and the Macdonald positivity conjecture. J. Amer. Math. Soc. 14 (2001), 941--1006. MR 1839919 (2002c:14008)
  • 18. M. Haiman. Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371--407. MR 1918676 (2003f:14006)
  • 19. N. Loehr, Multivariate Analogues of Catalan Numbers, Parking Functions, and their Extensions. Ph.D. thesis, University of California at San Diego, June 2003.
  • 20. N. Loehr, Trapezoidal lattice paths and multivariate analogues, Adv. in Appl. Math. 31 (2003), 597--629. MR 2008039 (2004i:05155)
  • 21. N. Loehr, Combinatorics of $ q,t$-parking functions, Adv. in Appl. Math. 34 (2005), 408--425. MR 2110560 (2005j:05007)
  • 22. N. Loehr, Conjectured statistics for the higher $ q,t$-Catalan sequences, Electron. J. Combin. 12 (2005), R9; 54 pages. MR 2134172 (2006e:05010)
  • 23. N. Loehr and J. Remmel, Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules, Electron. J. Combin. 11 (2004), R68; 64 pages. MR 2097334 (2005h:05202)
  • 24. N. Loehr, The major index specialization of the $ q,t$-Catalan, to appear in Ars Combinatoria.
  • 25. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, 1995. MR 1354144 (96h:05207)
  • 26. B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth and Brooks/Cole, 1991. MR 1093239 (93f:05102)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E10, 05A30, 20C30

Retrieve articles in all journals with MSC (2000): 05E10, 05A30, 20C30

Additional Information

Nicholas A. Loehr
Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187

Gregory S. Warrington
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109

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.

American Mathematical Society