The symmetric functions catalog

#Introduction

For an introduction to the topic of \(qt\)-Catalan numbers and \(qt\)-Kostka numbers, see J. Haglund’s book, [Hag07] and F. Bergeron [Ber09]. F. Bergeron’s lecture notes on Macdonald polynomials and rectangular Catalan combinatorics give a more recent gateway to the symmetric-function operators used in this area [Ber21, Ber21].

Let \(DP(n)\) be the set of Dyck paths of size \(n.\) Then the \(qt\)-Catalan numbers can then be defined as \[\catalan_n(q,t) \coloneqq \sum_{P \in DP(n)} q^{\area(P)} t^{\bounce(P)}.\] These are symmetric in \((q,t),\) but a combinatorial proof of this fact is not known.

There are various other ways to define these polynomials, for example \[\catalan_n(q,t) = \sum_{P \in DP(n)} q^{\dinv(P)} t^{\area(P)},\] or via diagonal harmonics as \(\langle DH_n(q,t), \elementaryE_{n} \rangle,\) see [Thm. I.2, GH02].

#Generalizations

In [QZ25], a \(qt\)-Catalan symmetry is proved involving the two statistics \((area,depth)\) as well as \((dinv,ddinv).\) They also provide proofs in the \(\vec{k}\)-Dyck path setting.

Bibliography

1. [Ber09]François Bergeron. Algebraic combinatorics and coinvariant spaces. A K Peters/CRC Press, 2009.
   .bib

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   @book{Bergeron2009,
     Author = {Fran{\c{c}}ois Bergeron},
     Title = {Algebraic Combinatorics and Coinvariant Spaces},
     Publisher = {A K Peters/CRC Press},
     Year = {2009},
     ISBN = {1568813244}
   }
   

2. [Ber21]François Bergeron. Macdonald polynomials and operators and Catalan combinatorics. arXiv:2112.09800, 2021.
   .bib

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   @article{Bergeron2021MacdonaldCatalan,
     author = {Fran{\c c}ois Bergeron},
     title = {Macdonald polynomials and operators and {C}atalan combinatorics},
     year = {2021},
     eprint = {2112.09800},
     url = {https://arxiv.org/abs/2112.09800},
     journal = {arXiv e-prints}
   }
   

3. [Ber21]François Bergeron. Symmetric functions and rectangular Catalan combinatorics. arXiv:2112.09799, 2021.
   .bib

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   @article{Bergeron2021RectangularCatalan,
     author = {Fran{\c c}ois Bergeron},
     title = {Symmetric functions and rectangular {C}atalan combinatorics},
     year = {2021},
     eprint = {2112.09799},
     url = {https://arxiv.org/abs/2112.09799},
     journal = {arXiv e-prints}
   }
   

4. [GH02]Adriano M. Garsia and James Haglund. A proof of the $q,t$-Catalan positivity conjecture. Discrete Mathematics, 256(3):677–717, October 2002.
   .bib

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   @article{GarsiaHaglund2000,
     doi = {10.1016/s0012-365x(02)00343-6},
     url2 = {https://doi.org/10.1016/s0012-365x(02)00343-6},
     year = {2002},
     month = oct,
     publisher = {Elsevier {BV}},
     volume = {256},
     number = {3},
     pages = {677--717},
      author = {Adriano M. Garsia and James Haglund},
     title ={A Proof of the $q,t$-{C}atalan Positivity Conjecture},
     journal = {Discrete Mathematics}
   }
   

5. [QZ25]Menghao Qu and Yingrui Zhang. Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths. arXiv:2510.08196v1, 2025.
   .bib

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   @article{QuZhang2025x,
     Author        = {Menghao Qu and Yingrui Zhang},
     Title         = {Symmetry of the Refined {$q,t$}-{C}atalan Polynomials for {$\vec{k}$}-{D}yck Paths},
     Year          = {2025},
     Eprint        = {2510.08196v1},
     url = {https://arxiv.org/abs/2510.08196v1},
     journal       = {arXiv e-prints}
   }
   

6. [Hag07]James Haglund. The $q,t$-Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007.
   .bib

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   @book{qtCatalanBook,
     Author = {James Haglund},
     Title = {The $q,t$-{C}atalan numbers and the space of diagonal harmonics ({U}niversity lecture series)},
     Publisher = {American Mathematical Society},
     Year = {2007},
     ISBN = {0821844113},
     url = {https://www.math.upenn.edu/~jhaglund/books/qtcat.pdf}
   }