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#Other combinatorial objects

#Generalizations of Catalan numbers

The \(k\)-Catalan numbers or Fuß–Catalan numbers are defined as \[\catalan^k(n) \coloneqq \frac{1}{kn+1}\binom{kn+1}{n}.\] There are many interpretations of these, see, for example, [Sta15].

The s-binomial coefficients are defined via \[(1+x+x^2+\dotsb+x^s)^n = \sum_{j=0}^{sn} \binom{n}{j}_s x^j.\] The s-Catalan numbers are then defined as \[\catalan^s(n) \coloneqq \binom{2n}{sn}_s - \binom{2n}{sn}_{s+1}.\] There is a relationship between s-Catalan numbers and Littlewood–Richardson coefficients, see [Lin21]. In fact, let \(\delta = (n,n-1,\dotsc,2,1).\) Then \[c^{2 \delta_{2n}}_{2 \delta_{2n-1} , (sn,sn)} = \binom{2n}{sn}_s - \binom{2n}{sn}_{s+1}.\] Note that the left-hand side is a polynomial in \(s,\) due to a result by E. Rassart [Ras04].

F. Bergeron and M. Mazin introduce triangular partitions, whose cells lie below a line segment from \((r,0)\) to \((0,s)\) [BM22]. Partitions contained in a triangular partition generalize Dyck paths and parking functions, and the induced subposet of Young lattice has a planar Hasse diagram with a generalized first-return recurrence.

V. Mazorchuk studies the sequence A393920, counting extension-closed additive idempotent-split subcategories for uniformly oriented type \(A\) quivers [Maz26]. The paper gives a recurrence using subsets of triangular arrays of lattice points, relates a quotient restriction graph to Fibonacci numbers, identifies a Catalan sequence on the boundary of the recurrence, and shows that a companion sequence counts convex topologies on finite chains.

#Set partitions

A. Prasad and S. Ram introduce polynomials indexed by integer partitions which interpolate between several familiar objects [PR25]. At \(1\) they count set partitions of prescribed block sizes, at \(0\) they count standard tableaux of a fixed shape, and at \(-1\) they count standard shifted tableaux of that shape; the same polynomials also count subspace profiles over finite fields at prime powers.

#Skew standard Young tableaux

For formulas for the number of skew SYT, see [MZ20]. I. Pak gives a case-based introduction to asymptotic algebraic combinatorics for skew shapes, centered on thick ribbons [Pak21].

D. Grinberg, N. Korniichuk, K. Molokanov, and S. Khomych give a new proof of a unified generating-function identity combining the Pak–Postnikov and Naruse skew hook-length formulas [GKMK23]. The proof uses recurrences, determinants, and elementary combinatorics. T. Shimazaki relates hook-length products for adjacent staircase partitions to special values of Jacobi polynomials [Shi26]. The same identities give special values of stable Grothendieck polynomials and \(K\)-theoretic Schur \(P\)-functions, with coefficients described by excited Young diagrams.

#Parking functions

See the parking functions page.

#Set-valued tableaux

A. Buch introduced set-valued tableaux in [Buc02], in order to study the K-theoretical Grassmannian.

In [Dru18], it is shown that \(\catalan^k(n)\) is equal to the number of set-valued SYT of shape \((n,n),\) where each box in the first row has one entry, while the boxes in the second row all have exactly \(k-1\) entries. Note that for \(k=2,\) we recover the classical Catalan numbers, which count the number of SYT of shape \((n,n).\) Note that the total number of entries is \(kn.\)

Bibliography

  1. [BM22]François Bergeron and Mikhail Mazin. Combinatorics of triangular partitions. arXiv:2203.15942, 2022.
    .bib
    @article{BergeronMazin2022x,
  author = {François Bergeron and Mikhail Mazin},
  title = {Combinatorics of triangular partitions},
  year = {2022},
  eprint = {2203.15942},
  url = {https://arxiv.org/abs/2203.15942},
  journal = {arXiv e-prints}
}
  2. [Buc02]Anders Skovsted Buch. A Littlewood–Richardson rule for the K-theory of Grassmannians. Acta Mathematica, 189(1):37–78, 2002.
    .bib
    @article{Buch2002,
year = {2002},
issn = {0001-5962},
journal = {Acta Mathematica},
volume = {189},
number = {1},
title = {A {L}ittlewood--{R}ichardson rule for the {K}-theory of {G}rassmannians},
doi = {10.1007/BF02392644},
publisher = {Kluwer Academic Publishers},
author = {Anders Skovsted Buch},
pages = {37--78},
language = {English}
}
  3. [Dru18]Paul Drube. Set-valued tableaux and generalized Catalan numbers. Australas. J Comb., 72:55–69, 2018.
    .bib
    @article{Drube2018,
  title={Set-valued tableaux and generalized {C}atalan numbers},
  author={Paul Drube},
  journal={Australas. J Comb.},
  year={2018},
  volume={72},
  pages={55--69}
}
  4. [GKMK23]Darij Grinberg, Nazar Korniichuk, Kostiantyn Molokanov and Severyn Khomych. The Pak–Postnikov and Naruse skew hook length formulas: A new proof. arXiv:2310.18275, 2023.
    .bib
    @article{GrinbergKorniichukMolokanovKhomych2023x,
  author = {Darij Grinberg and Nazar Korniichuk and Kostiantyn Molokanov and Severyn Khomych},
  title = {The {P}ak--{P}ostnikov and {N}aruse skew hook length formulas: a new proof},
  year = {2023},
  eprint = {2310.18275},
  url = {https://arxiv.org/abs/2310.18275},
  journal = {arXiv e-prints}
}
  5. [Lin21]William Linz. $s$-Catalan numbers and Littlewood–Richardson polynomials. arXiv:2110.12095, 2021.
    .bib
    @article{Linz2021x,
Author = {William Linz},
Title = {$s$-{C}atalan numbers and {L}ittlewood--{R}ichardson polynomials},
Year = {2021},
Eprint = {2110.12095},
  url = {https://arxiv.org/abs/2110.12095},
journal = {arXiv e-prints}
}
  6. [Maz26]Volodymyr Mazorchuk. On the number of extension closed additive subcategories for uniformly oriented ${A}_n$ quivers. arXiv:2607.00651, 2026.
    .bib
    @article{Mazorchuk2026x,
  author = {Volodymyr Mazorchuk},
  title = {On the number of extension closed additive subcategories for
    uniformly oriented ${A}_n$ quivers},
  year = {2026},
  eprint = {2607.00651},
  url = {https://arxiv.org/abs/2607.00651},
  journal = {arXiv e-prints}
}
  7. [MZ20]Alejandro H. Morales and Daniel G. Zhu. On the Okounkov–Olshanski formula for standard tableaux of skew shapes. arXiv:2007.05006, 2020.
    .bib
    @article{MoralesZhu2020x,
Author = {Alejandro H. Morales and Daniel G. Zhu},
Title = {On the {O}kounkov--{O}lshanski formula for standard tableaux of skew shapes},
Year = {2020},
Eprint = {2007.05006},
  url = {https://arxiv.org/abs/2007.05006},
journal = {arXiv e-prints}
}
  8. [Pak21]Igor Pak. Skew shape asymptotics, a case-based introduction. Seminaire Lotharingien de Combinatoire, 84:Article B84a, 2021.
    .bib
    @article{Pak2021SkewShapeAsymptotics,
  author = {Igor Pak},
  title = {Skew Shape Asymptotics, a Case-Based Introduction},
  year = {2021},
  journal = {Seminaire Lotharingien de Combinatoire},
  volume = {84},
  pages = {Article B84a},
  url = {https://www.mat.univie.ac.at/~slc/wpapers/s84pak.html}
}
  9. [PR25]Amritanshu Prasad and Samrith Ram. Set partitions, tableaux, and subspace profiles under regular diagonal matrices. European Journal of Combinatorics, 124:104060, 2025.
    .bib
    @article{PrasadRam2025,
  author = {Amritanshu Prasad and Samrith Ram},
  title = {Set partitions, tableaux, and subspace profiles under regular
    diagonal matrices},
  year = {2025},
  journal = {European Journal of Combinatorics},
  volume = {124},
  pages = {104060},
  doi = {10.1016/j.ejc.2024.104060},
  url = {https://doi.org/10.1016/j.ejc.2024.104060},
  eprint = {2112.00479}
}
  10. [Ras04]Etienne Rassart. A polynomiality property for Littlewood–Richardson coefficients. J. Comb. Theory Ser. A, 107(2):161–179, August 2004.
    .bib
    @article{Rassart2004,
 author = {Etienne Rassart},
 title = {A polynomiality property for {L}ittlewood--{R}ichardson coefficients},
 journal = {J. Comb. Theory Ser. A},
 issue_date = {August 2004},
 volume = {107},
 number = {2},
 month = aug,
 year = {2004},
 issn = {0097-3165},
 pages = {161--179},
 numpages = {19},
 doi = {10.1016/j.jcta.2004.04.003},
 acmid = {1032085},
 publisher = {Academic Press, Inc.},
 address = {Orlando, FL, USA},
}
  11. [Shi26]Tatsushi Shimazaki. Staircase hook-length ratios and special values of Jacobi polynomials. arXiv:2601.05471, 2026.
    .bib
    @article{Shimazaki2026x,
  author = {Tatsushi Shimazaki},
  title = {Staircase hook-length ratios and special values of {J}acobi
    polynomials},
  year = {2026},
  eprint = {2601.05471},
  url = {https://arxiv.org/abs/2601.05471},
  journal = {arXiv e-prints}
}
  12. [Sta15]Richard P. Stanley. Catalan numbers. Cambridge University Press, 2015.
    .bib
    @book{StanleyCatalan,
  Author = {Richard P. Stanley},
  Title = {Catalan Numbers},
  Publisher = {Cambridge University Press},
  Year = {2015},
  ISBN = {1107427746},
  doi = {10.1017/CBO9781139871495}
}

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