#Border strip tableaux and the Littlewood map
The notion of border-strip tableaux, also known as rim-hook tableaux, show up in several areas, most notably the Murnaghan–Nakayama rule and the original definition of LLT polynomials. Holmes gives an efficient recursion for irreducible symmetric-group character values [Hol17]. A. R. Miller uses a reformulation of the Murnaghan–Nakayama rule to study congruences in symmetric-group character tables [Mil19]. T. Y. Chow and J. Paulhus study algorithms for distinguishing irreducible characters of the symmetric group [CP21].
#Border strip tableaux
A border strip tableau (also known as ribbon tableau or rim-hook tableau) of shape \(\lambda\) and weight \(\mu\) is a filling of the diagram \(\lambda\) such that exactly \(\mu_i\) boxes are labeled \(i,\) and rows and columns are weakly increasing. Moreover, for each \(i,\) the boxes labeled \(i\) must form a border strip. We let \(\BST(\lambda,\mu)\) denote this set of border strip tableaux.
Example (Border-strip tableaux for \(\lambda=44321,\) \(\mu=5333\)).
Here are the 9 tableaux in \(\BST(44321, 5333).\)
The height of such a tableau is the sum of the heights of the border strips, where the height of a border strip is the number of rows it touches, minus one. We let \(\ht(B)\) denote the height of \(B.\)
When all strips have the same size, several additional properties hold, and we let \(\BST(\lambda,d)\) denote the set of border-strip tableaux of shape \(\lambda\) and all strips have size \(d.\)
Y. Chen and V. Kargin study ribbon tilings of large rectangles and fixed-width strips [CK23]. For order-\(n\) ribbons they prove the existence of a rectangle growth rate, bound it by \((n-1)\log 2,\) and derive asymptotic bounds for strip growth rates. They also give recursive systems that enumerate order-\(n\) ribbon tilings of strips for \(n\leq 8\) and produce the corresponding generating functions.
#\(k\)-ribbon tableaux
There is a semi-standard version of border strip tableaux, called \(k\)-ribbon tableaux. Let \(k \geq 1\) be an integer. A \(k\)-horizontal strip is a skew shape formed by a disjoint union of \(k\)-ribbons such that all their heads are in different columns. A \(k\)-ribbon tableau of shape \(\lambda/\mu\) and weight \(\nu,\) is a sequence of shapes \[\mu = \mu^0 \subseteq \mu^1 \subseteq \dotsb \subseteq \mu^{\ell} = \lambda\] such that each \(\mu^{i+1}/\mu^{i}\) is a horizontal \(k\)-ribbon consisting of \(\nu_i\) ribbons labeled \(i.\) This definition is from [Og20]; see [Des08] for an alternative way to phrase this.
We let \(\SSYT^{(k)}(\lambda/\mu,\nu)\) denote the set of \(k\)-ribbon tableaux of shape \(\lambda/\mu\) and weight \(\nu.\) Moreover, we let \(K^{(k)}_{\lambda/\mu,\nu}\) be the cardinality of this set. Note that for \(k=1\) we recover the set of semistandard Young tableaux, and the Kostka coefficients.
F. Maas-Gariépy and R. Patrias introduce set-valued domino tableaux and shifted set-valued domino tableaux [MP20]. These are \(K\)-theoretic analogues of domino tableaux and shifted domino tableaux. D. Abbasian, L. Difulvio, R. Muth, G. Pasternak, I. Sholtes, and F. Sinclair study cuspidal ribbon tableaux in affine type \(A\) [ADMP+23]. E. A. Vassilieva studies products of generating functions for domino tableaux and bi-tableaux [Vas19].
Example (k-ribbon tableaux).
The following figure shows the elements in \(\SSYT^{(3)}(4422,1111)\) where we have put the label of each \(3\)-ribbon on its head.
The following figure shows the elements in \(\SSYT^{(3)}(4422,211).\)
The notion of \(k\)-ribbon tableaux appears frequently when studying plethysm coefficients, evaluations at roots of unity, and in the study of LLT polynomials.
We have [p. 29, DLT94] that \[\schurS_\nu[ \powerSum_k \circ \completeH_\mu ] = \sum_{\lambda} \epsilon_k(\lambda/\mu) K^{(k)}_{\lambda/\nu,\mu} \schurS_\lambda.\]
Theorem (See [p.29, DLT94]).
Let \(\lambda/\mu\) be a skew shape and \(\nu\) a weak composition. Let \(\xi\) be a primitive \(j^\thsup\) root of unity. Then \[K^{(j)}_{\lambda/\mu,\nu} = (-1)^{|\nu|(j-1)}\varepsilon_k(\lambda/\mu) K_{\lambda/\mu,\nu^j}(\xi),\] where \(K_{\lambda/\mu,\nu^j}(q)\) is a Kostka–Foulkes polynomial.
For example, \(K_{4422,2^31^31^3}(q)\) is given by \[\begin{aligned} &q^{25}+q^{24}+4 q^{23}+5 q^{22}+10 q^{21}+13 q^{20}+21 q^{19}+ \\ &24 q^{18}+33 q^{17}+34 q^{16}+39 q^{15}+36 q^{14}+36 q^{13}+ \\ &27 q^{12}+23 q^{11}+14 q^{10}+9 q^9+4 q^8+2 q^7 \end{aligned}\] and it evaluates to \(-3\) at \(q=e^{2\pi i/3},\) so \(K^{(3)}(4422,211)=3.\)
#Partition quotients
Background on this subsection can be found in [p.12, Mac95], and also [Chapter 2.7, JK84]. See also the lecture notes by G. Seelinger (taught by J. Morse).
Let \(\ell \geq 2\) be a fixed integer. The \(\ell\)-core of a partition \(\lambda,\) denoted \(\widetilde{\lambda},\) is the shape which remains after removing all possible \(\ell\)-ribbons from \(\lambda.\) One can show that the \(\ell\)-core is independent of the choice of ribbons. A partition is an \(\ell\)-core if and only if no box in \(\ell\) has hook-length divisible by \(\ell.\)
The \(\ell\)-quotient of a partition \(\lambda\) is an \(\ell\)-tuple \((\lambda^0,\lambda^1,\dotsc,\lambda^{\ell-1})\) with several nice properties:
The \(\ell\)-quotient map \[\lambda \to \left( \widetilde{\lambda}; \; \lambda^0,\lambda^1,\dotsc,\lambda^{\ell-1} \right)\] is injective, and \(|\lambda| = |\widetilde{\lambda}| + \ell |\lambda^0| + \dotsb + \ell |\lambda^{\ell-1}|.\) This explains the usage of the word quotient.
Conjugating \(\lambda\) conjugates the core and all partitions in the quotient.
Warning! Other definitions of the quotient might give another cyclic rotation of the entries in the \(\ell\)-tuple. See the discussion in Macdonald’s book.
There are several ways to compute the quotient, see [p.12, Mac95]. The following example shows one possible method.
Example (Computing the quotient).
Let \(\lambda = 8766641\) and \(\ell=3.\) We label the path outlining the partition as follows. Each east step is labeled with the content (row index minus column index) of the box above it, while north steps are labeled with the content of the box to its right. We only consider the remainder of the content mod \(\ell,\) which gives the labeling as in the figure.
To find \(\lambda^j,\) simply consider all boxes in \(\lambda,\) such that the label of the horizontal step in the same row, and the vertical step in the same column are both labeled \(j.\) After removing empty columns and rows, these boxes form the shape \(\lambda^j.\)
The \(3\)-quotient of \(\lambda\) is therefore \((33,\;2,\; 211)\) in this case. For example, \(\lambda^0\) has shape \(33\) as the boxes marked \(\blacksquare\) form this shape.
The partition \(\lambda\) defines a binary word, by considering the outline, starting from the first column. A vertical step is represented by a 0, and a horizontal one by a \(1.\) In this case, we have the word \(\dotsc0010111011000101011\dotsc.\) Taking every third entry in this word gives the outline associated to one of the \(\lambda^j\) — this can easily be seen from the above definition of quotient.
The quotient map extends to the so-called Littlewood quotient map described below. The name Littlewood map appears in [p.92, Hag07] so I stick with that.
See also the article [BN19], which explores the quotient map from a Frobenius coordinate perspective.
#Littlewood’s quotient map
The Littlewood quotient map sends an element in \(\BST(\lambda,\ell)\) to an \(\ell\)-tuple \((T_0,\dotsc,T_{\ell-1})\) of standard Young tableaux, but where the entries \(1,2\dotsc,\) are distributed among the tableaux. The shape of \(T_j\) is \(\lambda^j,\) the \(j^\thsup\) element in the \(\ell\)-quotient of \(\lambda.\)
It is easiest to explain the map with an example.
Example
Let \(\lambda = 8766641\) and \(\ell=3.\) The \(3\)-core of \(\lambda\) is \((2),\) and the \(3\)-quotient is \((33,\;2,\; 211).\)
The figure shows a border-strip tableau \(B,\) where the label has been written in each strip. Recall that the content of a box \((r,c)\) is given by \(c-r.\) In each strip, there is a unique box with minimal content — this is where the label has been written. Each label has a content whose remainder is in \(\{0,1,\dotsc,\ell-1\},\) mod \(\ell.\) Labels whose content-remainder is \(j\) are in correspondence with boxes in \(T_j.\) Furthermore, labels with the same content in \(B\) also have the same content in \(T_j.\)
For example, the strips with labels in \(\{1,2,3,8,11,12\}\) have content-remainder \(0,\) and are therefore the labels in \(T_0.\)
(* This code follows the description in [Macdonald1995] *)
PartitionCore[lam_List, d_Integer] :=
Module[{m = Length@lam, xi, xis, mr, xiTildes},
xi = PadRight[lam, m] + Range[m - 1, 0, -1];
xiTildes = Sort[
Join @@ Table[
xis = Select[xi, Mod[# - r, d] == 0 &];
mr = Length[xis];
Table[d s + r, {s, 0, mr - 1}]
, {r, 0, d - 1}], Greater];
DeleteCases[MapIndexed[#1 - m + #2[[1]] &, xiTildes], 0]
];
PartitionQuotient[lam_List, d_Integer] :=
Module[{m = Length@lam, xi, xis, mr},
xi = PadRight[lam, m] + Range[m - 1, 0, -1];
Table[
xis = Select[xi, Mod[# - r, d] == 0 &];
mr = Length[xis];
MapIndexed[
#1 - mr + #2[[1]] &,
Sort[(xis - r)/d, Greater]
], {r, 0, d - 1}]
];
(* Skew shapes are given by skewing the quotients. *)
PartitionQuotient[{lam_List, mu_List}, d_Integer] :=
MapThread[Join, {
List /@ PartitionQuotient[lam, d],
List /@ PartitionQuotient[mu, d]}, 1];
(* Example, from J.Haglund's qt-Catalan book *)
PartitionQuotient[{5, 5, 5, 5, 5, 5}, 3] == {{2, 2}, {2, 2}, {1, 1}}Example
The following examples appear in the literature, in [p.13, LLT97], [p.94, Hag07] and [Fig. 2.6, Pak00], respectively.
$\lambda$ $\textbf{3-core}$ $\textbf{3-quotient}$ $87^241^5$ $211$ $21,\; 22,\; 2$ $5^6$ $\emptyset$ $22,\; 22,\; 11$ $987^34$ $\emptyset$ $33,\; 21,\; 32$Note that I. Pak has a different labeling convention, so one needs to cyclically rotate the entries in the quotient for the calculations to agree.
It is straightforward to generalize this map to skew shapes, sending elements in \(\BST(\lambda/\mu,\ell)\) to \(\ell\)-tuples of skew tableaux, see [Wil18]. This is possible as long as \(\lambda\) and \(\mu\) have the same \(\ell\)-core, otherwise \(\BST(\lambda/\mu,\ell)\) is empty. The Littlewood quotient map can be extended to semistandard \(k\)-ribbon tableaux. This shows why the original definition of LLT polynomials (using ribbon tableaux) agree with the Bylund–Haiman combinatorial model for LLT polynomials.
By using the Littlewood map, one can give an alternative proof of Thurston’s theorem [Thu90], which shows that the flip graph of domino tilings of rectangles is connected.
#The abacus model
Partition quotients and border-strip tableaux can also be described using abaci, see [Wil18].
#Hook formulas for border strip tableaux
Using Littlewood’s quotient map the following result follows.
Theorem (Hook formula for border-strip tableaux, see [JK84, FL97]).
Let \(\mu \vdash d\ell.\) Then \[|\BST(\mu,\ell)| = \binom{\ell}{|\mu^0|,|\mu^1|,\dotsc,|\mu^d|} f^{\mu^0} f^{\mu^1}\dotsm f^{\mu^{d-1}},\] where the \(\mu^j\) are the partitions in the \(d\)-quotient of \(\mu.\)
There are analogs of this formula for shifted shapes as well as for trees.
The above formula can also be phrased as follows.
Theorem (The modular hook formula, see [FL97]).
Let \(\mu \vdash d\ell.\) Then \[|\BST(\mu,\ell)| = \frac{d!}{\prod_{\substack{s \in \mu \\ \ell | h(s) } } h(s)/\ell }\] where the product is over all boxes in \(\mu\) such that the hook value is divisible by \(\ell.\)
A \(q\)-analog of this formula is given in [ML19].
Theorem (See [Corollary 9, Whi83] and [Thm. 2.7.27, JK84]).
Let \(\lambda \vdash d\ell.\) Then \(|\BST(\lambda,d)| = |\chi^{\lambda}(d^\ell)|.\) In other words, the Murnaghan–Nakayama rule is cancellation-free.
In fact, [Thm. 2.7.27, JK84] is the following statement. Let \(\lambda \vdash d\ell + m\) where the \(d\)-core \(\widetilde{\lambda}\) has size \(m.\) Suppose \(\sigma \in \symS_n\) such that the type of \(\sigma\) is \(d^\ell,\mu\) where \(\mu \vdash m.\) Then \[\chi^{\lambda}(\sigma) = \sign(\sigma) |\BST(\lambda/\tilde{\lambda},d)| \chi^{\tilde{\lambda}}(\mu) .\]
Bibliography
- [ADMP+23]Dina Abbasian, Lena Difulvio, Robert Muth, Gabrielle Pasternak, Isabella Sholtes and Frances Sinclair. Cuspidal ribbon tableaux in affine type A. Algebraic Combinatorics, 6(2):285–319, 2023.
.bib
@article{AbbasianDifulvioMuthPasternakSholtesSinclair2023, author = {Dina Abbasian and Lena Difulvio and Robert Muth and Gabrielle Pasternak and Isabella Sholtes and Frances Sinclair}, title = {Cuspidal ribbon tableaux in affine type {A}}, year = {2023}, journal = {Algebraic Combinatorics}, volume = {6}, number = {2}, pages = {285--319}, doi = {10.5802/alco.260}, eprint = {2009.07344} } - [BN19]Olivier Brunat and Rishi Nath. Cores and quotients of partitions through the Frobenius symbol. arXiv:1911.12098, 2019.
.bib
@article{BrunatNath2019, Author = {Olivier Brunat and Rishi Nath}, Title = {Cores and quotients of partitions through the {F}robenius symbol}, Year = {2019}, Eprint = {1911.12098}, url = {https://arxiv.org/abs/1911.12098}, journal = {arXiv e-prints} } - [CK23]Yinsong Chen and Vladislav Kargin. The Number of Ribbon Tilings for Strips. arXiv:2307.00767, 2023.
.bib
@article{ChenKargin2023x, author = {Yinsong Chen and Vladislav Kargin}, title = {The {N}umber of {R}ibbon {T}ilings for {S}trips}, year = {2023}, eprint = {2307.00767}, url = {https://arxiv.org/abs/2307.00767}, journal = {arXiv e-prints} } - [CP21]Timothy Y. Chow and Jennifer Paulhus. Algorithmically distinguishing irreducible characters of the symmetric group. The Electronic Journal of Combinatorics, 28(2), 2021.
.bib
@article{ChowPaulhus2021, author = {Timothy Y. Chow and Jennifer Paulhus}, title = {Algorithmically distinguishing irreducible characters of the symmetric group}, year = {2021}, journal = {The Electronic Journal of Combinatorics}, volume = {28}, number = {2}, doi = {10.37236/9753}, url = {https://doi.org/10.37236/9753} } - [DLT94]Jacques Désarménien, Bernard Leclerc and Jean-Yves Thibon. Hall-Littlewood functions and Kostka–Foulkes polynomials in representation theory. Séminaire Lotharingien de Combinatoire [electronic only], 32:38, 1994.
.bib
@article{DesarmenienLeclercThibon1994, author = {Jacques D{\'{e}}sarm{\'{e}}nien and Bernard Leclerc and Jean-Yves Thibon}, journal = {S{\'{e}}minaire Lotharingien de Combinatoire [electronic only]}, language = {eng}, pages = {38}, publisher = {Universit{\"{a}}t Wien, Fakult{\"{a}}t f{\"{u}}r Mathematik}, title = {Hall-{L}ittlewood functions and {K}ostka--{F}oulkes polynomials in representation theory}, url = {http://eudml.org/doc/119019}, volume = {32}, year = {1994}, } - [Des08]Francois Descouens. Ribbon tableaux, ribbon rigged configurations and Hall–Littlewood functions at roots of unity. Journal of Combinatorial Theory, Series A, 115(3):361–375, April 2008.
.bib
@article{Descouens2008, doi = {10.1016/j.jcta.2007.06.005}, url2 = {https://doi.org/10.1016/j.jcta.2007.06.005}, year = {2008}, month = apr, publisher = {Elsevier {BV}}, volume = {115}, number = {3}, pages = {361--375}, author = {Francois Descouens}, title = {Ribbon tableaux, ribbon rigged configurations and {H}all--{L}ittlewood functions at roots of unity}, journal = {Journal of Combinatorial Theory, Series A} } - [FL97]Sergey V. Fomin and Nathan Lulov. On the number of rim hook tableaux. Journal of Mathematical Sciences (New York), 87(6):4118–4123, December 1997.
.bib
@article{FominLulov1997, doi = {10.1007/bf02355806}, url2 = {https://doi.org/10.1007/bf02355806}, year = {1997}, month = dec, publisher = {Springer Nature}, volume = {87}, number = {6}, pages = {4118--4123}, author = {Sergey V. Fomin and Nathan Lulov}, title = {On the number of rim hook tableaux}, journal = {Journal of Mathematical Sciences (New York)} } - [Hol17]Randall R. Holmes. A recursion formula for the irreducible characters of the symmetric group. arXiv:1712.08023, 2017.
.bib
@article{Holmes2017x, Author = {Randall R. Holmes}, Title = {A recursion formula for the irreducible characters of the symmetric group}, Year = {2017}, Eprint = {1712.08023}, url = {https://arxiv.org/abs/1712.08023}, journal = {arXiv e-prints} } - [JK84]Gordon James and Adalbert Kerber. The representation theory of the symmetric group. Cambridge University Press, 1984.
.bib
@book{JamesKerber1984, doi = {10.1017/cbo9781107340732}, url2 = {https://doi.org/10.1017/cbo9781107340732}, year = {1984}, publisher = {Cambridge University Press}, author = {Gordon James and Adalbert Kerber}, title = {The Representation Theory of the Symmetric Group} } - [LLT97]Alain Lascoux, Bernard Leclerc and Jean-Yves Thibon. Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras and unipotent varieties. J. Math. Phys, 38:1041–1068, 1997.
.bib
@article{Lascoux97ribbontableaux, author = {Lascoux, Alain and Leclerc, Bernard and Thibon, Jean-Yves}, title = {Ribbon tableaux, {H}all--{L}ittlewood Functions, Quantum Affine algebras and Unipotent Varieties}, journal = {J. Math. Phys}, year = {1997}, volume = {38}, pages = {1041--1068} } - [MP20]Florence Maas-Gariépy and Rebecca Patrias. Set-valued domino tableaux and shifted set-valued domino tableaux. Involve, a Journal of Mathematics, 13(5):721–746, 2020.
.bib
@article{MaasGariepyPatrias2020, author = {Florence Maas-Gari{\'e}py and Rebecca Patrias}, title = {Set-valued domino tableaux and shifted set-valued domino tableaux}, year = {2020}, journal = {Involve, a Journal of Mathematics}, volume = {13}, number = {5}, pages = {721--746}, doi = {10.2140/involve.2020.13.721}, eprint = {2011.12493} } - [Mac95]Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
.bib
@book{Macdonald1995, address = {New York}, author = {Ian G. Macdonald}, edition = {Second}, isbn = {0-19-853489-2}, mrclass = {05E05 (05-02 20C30 20C33 20K01 33C80 33D80)}, note = {With contributions by A. Zelevinsky, Oxford Science Publications}, pages = {x+475}, publisher = {The Clarendon Press, Oxford University Press}, series = {Oxford Mathematical Monographs}, title = {Symmetric functions and {H}all polynomials}, year = {1995} } - [ML19]Anthony Mendes and Sam Lindbloom-Airey. Position sequences and a $q$-analogue for the modular hook length formula. The Electronic Journal of Combinatorics, 26(4), October 2019.
.bib
@article{MendesLindbloomAirey2019, doi = {10.37236/8685}, url2 = {https://doi.org/10.37236/8685}, year = {2019}, month = oct, publisher = {The Electronic Journal of Combinatorics}, volume = {26}, number = {4}, author = {Anthony Mendes and Sam Lindbloom-Airey}, title = {Position Sequences and a $q$-Analogue for the Modular Hook Length Formula}, journal = {The Electronic Journal of Combinatorics} } - [Mil19]Alexander R. Miller. Congruences in character tables of symmetric groups. arXiv:1908.03741, 2019.
.bib
@article{Miller2019x, author = {Alexander R. Miller}, title = {Congruences in character tables of symmetric groups}, year = {2019}, eprint = {1908.03741}, url = {https://arxiv.org/abs/1908.03741}, journal = {arXiv e-prints} } - [Og20]Ezgi Kantarcı Oğuz. A shifted analogue to ribbon tableaux. Journal of Combinatorics, 11(1):169–202, 2020.
.bib
@article{Oguz2020, doi = {10.4310/joc.2020.v11.n1.a8}, url2 = {https://doi.org/10.4310/joc.2020.v11.n1.a8}, year = {2020}, publisher = {International Press of Boston}, volume = {11}, number = {1}, pages = {169--202}, author = {Ezgi Kantarc{\i} O{\u{g}}uz}, title = {A shifted analogue to ribbon tableaux}, journal = {Journal of Combinatorics} } - [Pak00]Igor Pak. Ribbon tile invariants. Transactions of the American Mathematical Society, 352(12):5525–5562, December 2000.
.bib
@article{Pak2000Ribbon, doi = {10.1090/s0002-9947-00-02666-0}, year = {2000}, month = dec, publisher = {American Mathematical Society ({AMS})}, volume = {352}, number = {12}, pages = {5525--5562}, author = {Igor Pak}, title = {Ribbon tile invariants}, journal = {Transactions of the American Mathematical Society} } - [Thu90]William P. Thurston. Conways tiling groups. The American Mathematical Monthly, 97(8):757, October 1990.
.bib
@article{Thurston1990, doi = {10.2307/2324578}, url2 = {https://doi.org/10.2307/2324578}, year = {1990}, month = oct, publisher = {{JSTOR}}, volume = {97}, number = {8}, pages = {757}, author = {William P. Thurston}, title = {Conway{\textquotesingle}s Tiling Groups}, journal = {The American Mathematical Monthly} } - [Vas19]Ekaterina A. Vassilieva. On the product of generating functions for domino and bi-tableaux. arXiv:1911.10430, 2019.
.bib
@article{Vassilieva2019x, author = {Ekaterina A. Vassilieva}, title = {On the product of generating functions for domino and bi-tableaux}, year = {2019}, eprint = {1911.10430}, archivePrefix = {arXiv}, primaryClass = {math.CO} } - [Whi83]Dennis E. White. A bijection proving orthogonality of the characters of $S_n$. Advances in Mathematics, 50(2):160–186, November 1983.
.bib
@article{White1983, doi = {10.1016/0001-8708(83)90038-5}, year = {1983}, month = nov, publisher = {Elsevier {BV}}, volume = {50}, number = {2}, pages = {160--186}, author = {Dennis E. White}, title = {A bijection proving orthogonality of the characters of {$S_n$}}, journal = {Advances in Mathematics} } - [Wil18]Mark Wildon. A generalized SXP rule proved by bijections and involutions. Annals of Combinatorics, 22(4):885–905, November 2018.
.bib
@article{Wildon2018, doi = {10.1007/s00026-018-0409-x}, url2 = {https://doi.org/10.1007/s00026-018-0409-x}, year = {2018}, month = nov, publisher = {Springer Nature}, volume = {22}, number = {4}, pages = {885--905}, author = {Mark Wildon}, title = {A Generalized {SXP} Rule Proved by Bijections and Involutions}, journal = {Annals of Combinatorics} } - [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} }