Journal of Algebraic Combinatorics

, Volume 5, Issue 3, pp 191–244 | Cite as

A Remarkable q, t-Catalan Sequence and q-Lagrange Inversion

  • A.M. Garsia
  • M. Haiman


We introduce a rational function Cn(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number \(\frac{1}{{n + 1}}\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)\). We give supporting evidence by computing the specializations \(D_n \left( q \right) = C_n \left( {q{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \right)q^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)} \) and Cn(q) = Cn(q, 1) = Cn(1,q). We show that, in fact, Dn(q)q -counts Dyck words by the major index and Cn(q) q -counts Dyck paths by area. We also show that Cn(q, t) is the coefficient of the elementary symmetric function en in a symmetric polynomial DHn(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that Cn(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DHn(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {Pμ(x; q, t)}μ which are best dealt with in λ-ring notation. In particular we derive here the λ-ring version of several symmetric function identities.

Catalan number diagonal harmonic Macdonald polynomial Lagrange inversion 


  1. 1.
    E. Allen,”The Kostka-Macdonald coefficients for c-hooks,”Manuscript.Google Scholar
  2. 2.
    G.E. Andrews,”Identities in combinatorics II:A q-analog of the Lagrange inversion theorem,”A.M.S.Pro-ceedings 53 (1975),240–245Google Scholar
  3. 3.
    Y.M. Chen, A.M. Garsia,and J. Remmel,”Algorithms for plethysm,”Combinatorics and Algebra,Curtis Greene,(Ed.), Contemporary Math. 34, Amer. Math. Society, Providence RI (1984), 109–153.Google Scholar
  4. 4.
    J. Furlinger and J. Hofbauer,”q-Catalan numbers,”J.Combin.Theory (A)40 (1985),248–264.Google Scholar
  5. 5.
    A.M. Garsia,”A q-analogue of the Lagrange inversion formula,”Houston J.Math.7 (1981),205–237.Google Scholar
  6. 6.
    A.M. Garsia and C. Procesi,”On certain graded Sn-modules and the q-Kostka polynomials,”Advances in Math.94 (1992),82–138.Google Scholar
  7. 7.
    A.M. Garsia and M. Hairaan,”A graded representation model for Macdonald's polynomials,”Proc,Nat.Acad. U.S.A.90 (1993),3607–3610.Google Scholar
  8. 8.
    A.M. Garsia and M. Haiman,”Orbit harmonics and graded representations,”LA.C.I.M.Research Monograph Series,S. Brlek,(Ed.),U.du Quebec a Montreal,to appear.Google Scholar
  9. 9.
    A.M. Garsia and M. Haiman,”Some natural bigraded Sn-modules and q,t-Kotska coefficients,”to appear.Google Scholar
  10. 10.
    A.M. Garsia and M. Haiman,”Factorizations of Pieri rules for Macdonald polynomials,”Discrete Mathematics,139 (1995),219–256.Google Scholar
  11. 11.
    I. Gessel,”A noncommutative generalization and q-analog of the Lagrange inversion formula,”A.M.S.Transactions 257 (1980),455–482.Google Scholar
  12. 12.
    M. Haiman,”Conjectures on the quotient ring by diagonal invariants,”J.Alg.Combin.3 (1994),17–76.Google Scholar
  13. 13.
    M. Haiman,”(t,q)-Catalan numbers and the Hilbert scheme.”Discrete Mathematics,to appear.Google Scholar
  14. 14.
    D.E. Knuth,The An of Computer Programming, Vol. II, Addison-Wesley, Reading,Mass.,1981.Google Scholar
  15. 15.
    A.G. Konheim and B. Weiss,”An occupancy discipline and applications,”SIAM J.Appl.Math. 14 (1966), 1266–1274.Google Scholar
  16. 16.
    A. Lascoux and M.P. Schutzenberger,”Sur une conjecture de H.O.Foulkes,”C.R.Acad.Sci.Paris 286 (1978), 323–324.Google Scholar
  17. 17.
    A. Lascoux and M.-P. Schutzenberger,”Le monoide plaxique,”Quaderni della Ricerca scientifica 109 (1981), 129–156.Google Scholar
  18. 18.
    M. Lothaire,”Combinatorics on words,”Encyclopedia of Mathematics and its Applications 17,G.-C. Rota (Ed.)Addison-Wesley, Reading,MA,1983.Google Scholar
  19. 19.
    I.G. Macdonald,Symmetric Functions and Hall Polynomials,Second Edition,Oxford Univ.Press,1995.Google Scholar
  20. 20.
    I.G. Macdonald,”A new class of symmetric functions,”Actes du 20 Seminaire Lotharingien, Publ. I.R.M.A. Strasbourg 372/S-20(1988),131–171.Google Scholar
  21. 21.
    E.Reiner,”A proof of the n!conjecture for extended hooks,”Manuscript.See also Some Applications of the Theory of Orbit Harmonics,Doctoral dissertation UCSD (1993).Google Scholar
  22. 22.
    R.P. Stanley,Enumerative Combinatorics, Vol. I, Wadsworth-Brooks/Cole, Monterey,Calif.,1986.Google Scholar
  23. 23.
    R.P. Stanley,”Some combinatorial properties of Jack symmetric functions,”Advances in Math.77 (1989), 76–117.Google Scholar
  24. 24.
    D.Stanton,”Recent Results for the q-Lagrange inversion formula,”Ramanujan Revisited, Acad. Press (1988), 525–536.Google Scholar
  25. 25.
    A. Young,On Quantitative Substitutional Analysis (sixth paper).The collected papers of A. Young, University of Toronto Press 1977,434–435.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • A.M. Garsia
  • M. Haiman
    • 1
  1. 1.Department of MathematicsLa Jolla, University of California

Personalised recommendations