qt-Catalan numbers

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#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 [GH02, Thm. I.2].

#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.

A. Oblomkov studies a type-$B_n$ $q,t$-Fuss–Catalan polynomial from generalized diagonal coinvariants. Let $W=W(B_n)$ act diagonally on $\mathfrak h\oplus\mathfrak h^*,$ let $S=\setC[\mathfrak h\oplus\mathfrak h^*],$ and let $J\subseteq S$ be the ideal generated by the $W$-alternating polynomials. If $\mathfrak m_S$ is the maximal ideal at the origin, then \[\operatorname{Cat}^{(m)}(B_n;q,t) \coloneqq \operatorname{Hilb}_{q,t} \left(J^m/\mathfrak m_SJ^m\right)_{\det\text{-part}}.\] For fixed $n$ and all sufficiently large $m,$ this series equals an explicit localization polynomial indexed by partitions of $2n$ with empty $2$-core [Obl26, Thm. 9.4]. Its principal specialization is \[q^{mn^2}\operatorname{Cat}^{(m)}(B_n;q,q^{-1}) =\qbinom{n(m+1)}{n}_{q^2}, \qquad \operatorname{Cat}^{(m)}(B_n;1,1)=\binom{n(m+1)}{n}\] in this asymptotic range [Obl26, Thm. 7.2]. The paper also proposes type-$B$ shuffle and labelled-path models for this series; its labelled path function is symmetric and Schur-positive and is conjectured to give the corresponding Frobenius character.

Bibliography

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

.bib

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

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

.bib

@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}
}

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

.bib

@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}
}

[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

@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}
}

[Obl26]Alexei Oblomkov. Asymptotic $q,t$-Fuss–Catalan numbers for type ${B}$. arXiv:2609.11691v3, 2026.

.bib

@article{Oblomkov2026x,
  author = {Alexei Oblomkov},
  title = {Asymptotic $q,t$-{F}uss--{C}atalan numbers for type ${B}$},
  year = {2026},
  eprint = {2609.11691v3},
  url = {https://arxiv.org/abs/2609.11691v3},
  journal = {arXiv e-prints}
}

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

.bib

@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}
}

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

.bib

@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}
}