Combinatorics of the Diagonal Harmonics

  • Angela HicksEmail author
Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)


The Shuffle Theorem, recently proven by Carlsson and Mellit, states that the bigraded Frobenius characteristic of the diagonal harmonics is equal to a weighted sum of parking functions. In this introduction to the topic, we discuss the theorem and connections between it and the well-known Macdonald polynomials. Furthermore, we describe important combinatorial bijections which imply various restatements of the theorem and play an important role in its proof. Finally, we briefly discuss the proof and describe various generalizations of the theorem.



The author would like to express her gratitude for the many helpful remarks of the anonymous reviewer.


  1. 1.
    D. Armstrong, Hyperplane arrangements and diagonal harmonics, in 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), pp. 39–50. Discrete Mathematics and Theoretical Computer Science (2011)Google Scholar
  2. 2.
    D. Armstrong, B. Rhoades, The Shi arrangement and the Ish arrangement. Trans. Am. Math. Soc. 364(3), 1509–1528 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Armstrong, N.A. Loehr, G.S. Warrington, Sweep maps: a continuous family of sorting algorithms. Adv. Math. 284, 159–185 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Assaf, Toward the Schur expansion of Macdonald polynomials. Electron. J. Combin. 25(2), Paper 2.44 (2018)Google Scholar
  5. 5.
    C.A. Athanasiadis, S. Linusson, A simple bijection for the regions of the Shi arrangement of hyperplanes. Discret. Math. 204(1–3), 27–39 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Bergeron, Multivariate diagonal coinvariant spaces for complex reflection groups. Adv. Math. 239, 97–108 (2013). ISSN 0001-8708MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Bergeron, Algebraic Combinatorics and Coinvariant Spaces (CRC Press, Boca Raton, 2009)CrossRefGoogle Scholar
  8. 8.
    F. Bergeron, L.-F. Préville-Ratelle, Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb. 3(3), 317–341 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions. Methods Appl. Anal. 6(3), 363–420 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    F. Bergeron, A. Garsia, E.S. Leven, G. Xin, Compositional (km, kn)-shuffle conjectures. Int. Math. Res. Not. 2016(14), 4229–4270 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Bergeron, A. Garsia, E.S. Leven, G. Xin, Some remarkable new plethystic operators in the theory of Macdonald polynomials. J. Comb. 7(4), 671–714 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    E. Carlsson, A. Mellit, A proof of the shuffle conjecture. J. Am. Math. Soc. 31(3), 661–697 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Foata, J. Riordan, Mappings of acyclic and parking functions. Aequ. Math. 10(1), 10–22 (1974)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A.M. Garsia, M. Haiman, Some natural bigraded \(S_n\)-modules and q, t-Kostka coefficients. Electron. J. Combin. 3 (1996)Google Scholar
  15. 15.
    A. Garsia, J. Haglund, A positivity result in the theory of Macdonald polynomials. Proc. Natl. Acad. Sci. 98(8), 4313–4316 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A.M. Garsia, J. Haglund, A proof of the q, t-Catalan positivity conjecture. Discret. Math. 256(3), 677–717 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A.M. Garsia, M. Haiman, A graded representation model for Macdonald’s polynomials. Proc. Natl. Acad. Sci. 90(8), 3607–3610 (1993)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A.M. Garsia, G. Xin, M. Zabrocki, Hall-Littlewood operators in the theory of parking functions and diagonal harmonics. Int. Math. Res. Not. 2012(6), 1264–1299 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A.M. Garsia, G. Xin, M. Zabrocki, A three shuffle case of the compositional parking function conjecture. J. Comb. Theory Ser. A 123(1), 202–238 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    E. Gorsky, A. Negut, Refined knot invariants and Hilbert schemes. J. Math. Pures Appl. (9) 104(3), 403–435 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    E. Gorsky, A. Oblomkov, J. Rasmussen, V. Shende, Torus knots and the rational DAHA. Duke Math. J. 163(14), 2709–2794 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    E. Gorsky, M. Mazin, M. Vazirani, Affine permutations and rational slope parking functions. Trans. Am. Math. Soc. 368(12), 8403–8445 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    J. Haglund, The combinatorics of knot invariants arising from the study of Macdonald polynomials. Recent Trends in Combinatorics (Springer, Berlin, 2016), pp. 579–600CrossRefGoogle Scholar
  24. 24.
    J. Haglund, J. Morse, M. Zabrocki, A compositional shuffle conjecture specifying touch points of the Dyck path. Canad. J. Math. 64(4), 822–844 (2012). ISSN 0008-414XMathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Haglund, G. Xin, Lecture notes on the Carlsson-Mellit proof of the shuffle conjecture (2017), arXiv:1705.11064
  26. 26.
    J. Haglund, A proof of the q, t-Schröder conjecture. Int. Math. Res. Not. 2004(11), 525–560 (2004)CrossRefGoogle Scholar
  27. 27.
    J. Haglund, The genesis of the Macdonald polynomial statistics. Séminaire Lotharingien de Combinatoire 54, B54Ao (2006)MathSciNetzbMATHGoogle Scholar
  28. 28.
    J. Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics, vol. 41 (American Mathematical Society, Providence, 2008)zbMATHGoogle Scholar
  29. 29.
    J. Haglund, N. Loehr, A conjectured combinatorial formula for the Hilbert series for diagonal harmonics. Discret. Math. 298(1), 189–204 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Haglund, M. Haiman, N. Loehr, A combinatorial formula for Macdonald polynomials. J. Am. Math. Soc. 18(3), 735–761 (2005)MathSciNetCrossRefGoogle Scholar
  31. 31.
    J. Haglund, J. Remmel, A.T. Wilson, The delta conjecture. Trans. Am. Math. Soc. 370(6), 4029–4057 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane. Inventiones Math. 149(2), 371–407 (2002)MathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Hicks, E. Leven, A simpler formula for the number of diagonal inversions of an \((m, n)\)-parking function and a returning fermionic formula. Discrete Math. 338(3), 48–65 (2015)Google Scholar
  35. 35.
    A.S. Hicks, Two parking function bijections: a sharpening of the q, t-Catalan and Shröder theorems. Int. Math. Res. Not. 2012(13), 3064–3088 (2011)CrossRefGoogle Scholar
  36. 36.
    T. Hikita, Affine Springer fibers of type \(A\) and combinatorics of diagonal coinvariants. Adv. Math. 263, 88–122 (2014)Google Scholar
  37. 37.
    D.E. Knuth, Linear probing and graphs. Algorithmica 22(4), 561–568 (1998)MathSciNetCrossRefGoogle Scholar
  38. 38.
    A.G. Konheim, B. Weiss, An occupancy discipline and applications. SIAM J. Appl. Math. 14(6), 1266–1274 (1966)CrossRefGoogle Scholar
  39. 39.
    A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties. J. Math. Phys. 38(2), 1041–1068 (1997)MathSciNetCrossRefGoogle Scholar
  40. 40.
    B. Leclerc, J.-Y. Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials. Combinatorial Methods in Representation Theory, Advanced Studies in Pure Mathematics, vol. 28 (Citeseer, 1998)Google Scholar
  41. 41.
    N.A. Loehr, G.S. Warrington, Nested quantum Dyck paths and \(\nabla (s_\lambda )\). Int. Math. Res. Not. 5 (2008)Google Scholar
  42. 42.
    N.A. Loehr, Combinatorics of q, t-parking functions. Adv. Appl. Math. 34(2), 408–425 (2005)MathSciNetCrossRefGoogle Scholar
  43. 43.
    N.A. Loehr, J.B. Remmel, A computational and combinatorial exposé of plethystic calculus. J. Algebr. Comb. 33(2), 163–198 (2011)CrossRefGoogle Scholar
  44. 44.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford University Press, Oxford, 1998)zbMATHGoogle Scholar
  45. 45.
    A. Mellit, Toric braids and \((m, n) \)-parking functions (2016), arXiv:1604.07456
  46. 46.
    A. Oblomkov, V. Shende, The Hilbert scheme of a plane curve singularity and the homfly polynomial of its link. Duke Math. J. 161(7), 1277–1303 (2012)MathSciNetCrossRefGoogle Scholar
  47. 47.
    R. Pyke, The supremum and infimum of the poisson process. Ann. Math. Stat. 30(2), 568–576 (1959)MathSciNetCrossRefGoogle Scholar
  48. 48.
    B. Rhoades, A.T. Wilson, Tail positive words and generalized coinvariant algebras. Electron. J. Combin. 24(3), Paper 3.21, 29 (2017)Google Scholar
  49. 49.
    B. Rhoades, Ordered set partition statistics and the delta conjecture. J. Comb. Theory Ser. A 154, 172–217 (2018)MathSciNetCrossRefGoogle Scholar
  50. 50.
    M. Romero, The delta conjecture at \(q=1\). Trans. Am. Math. Soc. 369(10), 7509–7530 (2017)Google Scholar
  51. 51.
    B.E. Sagan, The Symmetric Group. Volume 203 of Graduate Texts in Mathematics (Springer, New York, 2001), 2nd edn. Representations, combinatorial algorithms, and symmetric functionsGoogle Scholar
  52. 52.
    A. Schilling, M. Shimozono, D. White, Branching formula for q-Littlewood-Richardson coefficients. Adv. Appl. Math. 30(1–2), 258–272 (2003)MathSciNetCrossRefGoogle Scholar
  53. 53.
    E. Sergel, A proof of the square paths conjecture. J. Comb. Theory Ser. A 152, 363–379 (2017)MathSciNetCrossRefGoogle Scholar
  54. 54.
    N.J.A. Sloane, The on-line encyclopedia of integer sequences, Sequence A000272
  55. 55.
    R.P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999)Google Scholar
  56. 56.
    R.P. Stanley, Hyperplane arrangements, interval orders, and trees. Proc. Natl. Acad. Sci. 93(6), 2620–2625 (1996)MathSciNetCrossRefGoogle Scholar
  57. 57.
    H. Thomas, N. Williams, Sweeping up zeta. Sel. Math. (2018)Google Scholar
  58. 58.
    A.T. Wilson, Torus link homology and the nabla operator. J. Comb. Theory Ser. A 154, 129–144 (2018)MathSciNetCrossRefGoogle Scholar
  59. 59.
    C.H. Yan, Parking functions, Handbook of Enumerative Combinatorics, Discrete Mathematics and Applications (CRC Press, Boca Raton, FL, 2015), pp. 835–893CrossRefGoogle Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  1. 1.Lehigh UniversityBethlehemUSA

Personalised recommendations