Square 𝑞,𝑡-lattice paths and ∇(𝑝_{𝑛})

Loehr, Nicholas; Warrington, Gregory

doi:10.1090/S0002-9947-06-04044-X

Square $\boldsymbol {q,t}$-lattice paths and $\boldsymbol {\nabla (p_n)}$
HTML articles powered by AMS MathViewer

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 $\langle \nabla (p_n),s_{1^n}\rangle$.

References

F. Bergeron and A. M. Garsia, Science fiction and Macdonald’s polynomials, Algebraic methods and $q$-special functions (Montréal, QC, 1996) CRM Proc. Lecture Notes, vol. 22, Amer. Math. Soc., Providence, RI, 1999, pp. 1–52. MR 1726826, DOI 10.1090/crmp/022/01
François Bergeron, Nantel Bergeron, Adriano M. Garsia, Mark Haiman, and Glenn Tesler, Lattice diagram polynomials and extended Pieri rules, Adv. Math. 142 (1999), no. 2, 244–334. MR 1680202, DOI 10.1006/aima.1998.1791
F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999), no. 3, 363–420. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III. MR 1803316, DOI 10.4310/MAA.1999.v6.n3.a7
E. S. Egge, J. Haglund, K. Killpatrick, and D. Kremer, A Schröder generalization of Haglund’s statistic on Catalan paths, Electron. J. Combin. 10 (2003), Research Paper 16, 21. MR 1975766
A. M. Garsia and J. Haglund, A proof of the $q,t$-Catalan positivity conjecture, Discrete Math. 256 (2002), no. 3, 677–717. LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal, QC). MR 1935784, DOI 10.1016/S0012-365X(02)00343-6
A. M. Garsia and J. Haglund, A positivity result in the theory of Macdonald polynomials, Proc. Natl. Acad. Sci. USA 98 (2001), no. 8, 4313–4316. MR 1819133, DOI 10.1073/pnas.071043398
A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191–244. MR 1394305, DOI 10.1023/A:1022476211638
J. Haglund, Conjectured statistics for the $q,t$-Catalan numbers, Adv. Math. 175 (2003), no. 2, 319–334. MR 1972636, DOI 10.1016/S0001-8708(02)00061-0
J. Haglund, A proof of the $q,t$-Schröder conjecture, Int. Math. Res. Not. 11 (2004), 525–560. MR 2038776, DOI 10.1155/S1073792804132509
J. Haglund, A combinatorial model for the Macdonald polynomials, Proc. Natl. Acad. Sci. USA 101 (2004), no. 46, 16127–16131. MR 2114585, DOI 10.1073/pnas.0405567101
J. Haglund, The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics, AMS University Lecture Series, to appear.
J. Haglund, M. Haiman, and N. Loehr, A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc. 18 (2005), no. 3, 735–761. MR 2138143, DOI 10.1090/S0894-0347-05-00485-6
J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005), no. 2, 195–232. MR 2115257, DOI 10.1215/S0012-7094-04-12621-1
J. Haglund and N. Loehr, A conjectured combinatorial formula for the Hilbert series for diagonal harmonics, Discrete Math. 298 (2005), no. 1-3, 189–204. MR 2163448, DOI 10.1016/j.disc.2004.01.022
Mark Haiman, Notes on Macdonald polynomials and the geometry of Hilbert schemes, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 1–64. MR 2059359, DOI 10.1007/978-94-010-0524-1_{1}
Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39–111. MR 2051783
Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919, DOI 10.1090/S0894-0347-01-00373-3
Mark Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), no. 2, 371–407. MR 1918676, DOI 10.1007/s002220200219
N. Loehr, Multivariate Analogues of Catalan Numbers, Parking Functions, and their Extensions. Ph.D. thesis, University of California at San Diego, June 2003.
N. Loehr, Trapezoidal lattice paths and multivariate analogues, Adv. in Appl. Math. 31 (2003), no. 4, 597–629. MR 2008039, DOI 10.1016/S0196-8858(03)00028-9
Nicholas A. Loehr, Combinatorics of $q$, $t$-parking functions, Adv. in Appl. Math. 34 (2005), no. 2, 408–425. MR 2110560, DOI 10.1016/j.aam.2004.08.002
Nicholas A. Loehr, Conjectured statistics for the higher $q,t$-Catalan sequences, Electron. J. Combin. 12 (2005), Research Paper 9, 54. MR 2134172
Nicholas A. Loehr and Jeffrey B. Remmel, Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules, Electron. J. Combin. 11 (2004), no. 1, Research Paper 68, 64. MR 2097334
N. Loehr, The major index specialization of the $q,t$-Catalan, to appear in Ars Combinatoria.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Bruce E. Sagan, The symmetric group, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Representations, combinatorial algorithms, and symmetric functions. MR 1093239