Chain Decompositions of qt-Catalan Numbers via Local Chains


The qt-Catalan number \({{\,\mathrm{Cat}\,}}_n(q,t)\) enumerates integer partitions contained in an \(n\times n\) triangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018) proposed a new approach to understanding the symmetry property \({{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)\) based on decomposing the set of all integer partitions into infinite chains. Each such global chain \(\mathcal {C}_{\mu }\) has an opposite chain \(\mathcal {C}_{\mu ^*}\); these combine to give a new small slice of \({{\,\mathrm{Cat}\,}}_n(q,t)\) that is symmetric in q and t. Here, we advance the agenda of Lee et al. (SIAM J Discr Math 32:191–232, 2018) by developing a new general method for building the global chains \(\mathcal {C}_{\mu }\) from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most \(11\). This proves that for all n, the terms in \({{\,\mathrm{Cat}\,}}_n(q,t)\) of degree at least \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -11\) are symmetric in q and t.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    D. Armstrong, N. Loehr, and G. Warrington, “Rational parking functions and Catalan numbers,” Ann. Comb. 20 (2016), 21–58.

    MathSciNet  Article  Google Scholar 

  2. 2.

    E. Carlsson and A. Mellit, “A proof of the shuffle conjecture,” J. Amer. Math. Soc. 31 (2018), 661–697.

    MathSciNet  Article  Google Scholar 

  3. 3.

    A. Garsia and J. Haglund, “A positivity result in the theory of Macdonald polynomials,” Proc. Natl. Acad. Sci. USA 98 (2001), 4313–4316.

    MathSciNet  Article  Google Scholar 

  4. 4.

    A. Garsia and J. Haglund, “A proof of the \(q,t\)-Catalan positivity conjecture,” Discrete Math. 256 (2002), 677–717.

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. Garsia and M. Haiman, “A remarkable \(q,t\)-Catalan sequence and \(q\)-Lagrange inversion,” J. Algebraic Combin. 5 (1996), 191–244.

    MathSciNet  Article  Google Scholar 

  6. 6.

    J. Haglund, “Conjectured statistics for the \(q,t\)-Catalan numbers,” Adv. in Math. 175 (2003), 319–334.

    MathSciNet  Article  Google Scholar 

  7. 7.

    J. Haglund, The \(q,t\)-Catalan Numbers and the Space of Diagonal Harmonics, with an Appendix on the Combinatorics of Macdonald Polynomials, AMS University Lecture Series (2008).

  8. 8.

    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.

    MathSciNet  Article  Google Scholar 

  9. 9.

    M. Haiman, “Vanishing theorems and character formulas for the Hilbert scheme of points in the plane,” Invent. Math. 149 (2002), 371–407.

    MathSciNet  Article  Google Scholar 

  10. 10.

    S. Han, K. Lee, L. Li, and N. Loehr, Extended Appendix for “Chain Decompositions of \(q,t\)-Catalan Numbers via Local Chains,” available online at

  11. 11.

    K. Lee, L. Li, and N. Loehr, “Limits of modified higher \(q,t\)-Catalan numbers, Electron. J. Combin. 20(3) (2013), research paper P4, 23 pages (electronic).

  12. 12.

    K. Lee, L. Li, and N. Loehr, “Combinatorics of certain higher \(q,t\)-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula,” J. Algebraic Combin. 39 (2014), 749–781.

    MathSciNet  Article  Google Scholar 

  13. 13.

    K. Lee, L. Li, and N. Loehr, “A combinatorial approach to the symmetry of \(q,t\)-Catalan numbers,” SIAM J. Discrete Math. 32 (2018), 191–232.

    MathSciNet  Article  Google Scholar 

  14. 14.

    N. Loehr and G. Warrington, “A continuous family of partition statistics equidistributed with length,” J. Combin. Theory Ser. A 116 (2009), 379–403.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Anton Mellit, “Toric braids and \((m,n)\)-parking functions,” arXiv:1604.07456 (2016).

Download references


The authors are grateful to the anonymous referees for their valuable comments.

Author information



Corresponding author

Correspondence to Li Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Kyungyong Lee was supported by NSF Grant DMS 1800207, the Korea Institute for Advanced Study (KIAS), and the University of Alabama. This work was supported by a grant from the Simons Foundation/SFARI (Grant #633564 to Nicholas A. Loehr).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Han, S., Lee, K., Li, L. et al. Chain Decompositions of qt-Catalan Numbers via Local Chains. Ann. Comb. 24, 739–765 (2020).

Download citation


  • q
  •  t-Catalan numbers
  • Dyck paths
  • Dinv statistic
  • Joint symmetry
  • Integer partitions
  • Chain decompositions

Mathematics Subject Classification

  • 05A19
  • 05A17
  • 05E05