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Dyck paths and a bijection for multisets of hook numbers [article]

Ian Goulden, Alexander Yong
2001 arXiv   pre-print
We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams.  ...  The bijection proves a result that is stronger and more symmetric than the original conjecture, by means of a construction involving Dyck paths, a particular type of lattice path.  ...  Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada, through a grant to IG, and a PGSA to AY.  ... 
arXiv:math/0102223v1 fatcat:27scfub4xzbofjffxf72pobfbm

Dyck paths and a bijection for multisets of hook numbers

Ian Goulden, Alexander Yong
2002 Discrete Mathematics  
We give a bijective proof of a result of Regev and Vershik (Electron J. Combin. 4 (1997) R22) on the equality of two multisets of hook numbers of certain skew-Young diagrams.  ...  The bijection is given in terms of Dyck paths, a particular type of lattice path. It is extended to also prove a recent, more reÿned result of Regev (European J.  ...  We thank a referee of an earlier version of this paper for informing us about references [3] [4] [5] , and suggesting that the methods of Sections 1 and 2 could be extended to treat the projective case  ... 
doi:10.1016/s0012-365x(01)00356-9 fatcat:pwc5f5tx5jg3njpnbfxv7hbg4a

Page 7300 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
paths and a bijection for multisets of hook numbers.  ...  (electronic); MR 98g:05153] con- jectured, as a consequence of an asymptotic calculation, surprising identities for multisets of hooks of certain diagrams.  ... 

Rational Shi tableaux and the skew length statistic

Robin Sulzgruber
2020 Discrete Mathematics & Theoretical Computer Science  
Moreover we present a uniform definition of the rational Shi tableau for Weyl groups and conjecture injectivity in the general case.  ...  The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core.  ...  Acknowledgements The author is grateful to Cesar Ceballos, Jim Haglund, Christian Krattenthaler and Marko Thiel for insightful discussions.  ... 
doi:10.46298/dmtcs.6349 fatcat:7nnoapsoifbcnpiqct2l2yb2gy

Generalized Dyck tilings

Matthieu Josuat-Vergès, Jang Soo Kim
2016 European journal of combinatorics (Print)  
The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials.  ...  Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths.  ...  The set of up steps of a Dyck path µ is denoted by UP(µ) and for u ∈ UP(µ), ht(u) is the number of squares between u and the line y = x plus 1.  ... 
doi:10.1016/j.ejc.2015.07.015 fatcat:zfzr67ddcngkroewv44dgodwqi

Generalized Dyck tilings (Extended Abstract)

Matthieu Josuat-Vergès, Jang Soo Kim
2014 Discrete Mathematics & Theoretical Computer Science  
The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials.  ...  Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths.  ...  The set of up steps of a Dyck path µ is denoted by UP(µ) and for u ∈ UP(µ), ht(u) is the number of squares between u and the line y = x plus 1.  ... 
doi:10.46298/dmtcs.2391 fatcat:wrmj53scgzeltizo7xfagx5fi4

Rational Shi tableaux and the skew length statistic [article]

Robin Sulzgruber
2016 arXiv   pre-print
The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core.  ...  Moreover, we provide a uniform generalisation of rational Shi tableaux to Weyl groups, and conjecture injectivity in the general case.  ...  For example consider the Dyck path in Theorem 3.1. [And02, Prop. 1] The map ϕ : D n,p → C n,p is a bijection.  ... 
arXiv:1512.04320v2 fatcat:orj5hfj6v5fgbbaoya336xu4ku

Page 6748 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
Haglund’s model involves two statis- tics on Dyck paths, which are lattice paths consisting of unit north and unit east steps, beginning at (0,0), ending on the line y = x, and staying inside a triangle  ...  with a prescribed number ¢ of pebbles.  ... 

Results and conjectures on simultaneous core partitions [article]

Drew Armstrong, Christopher R. H. Hanusa, Brant C. Jones
2014 arXiv   pre-print
We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b.  ...  of type A and type C.  ...  ACKNOWLEDGEMENTS We would like to thank Monica Vazirani for discussions of related topics and the Institute for Computational and Experimental Research in Mathematics for supporting stimulating collaboration  ... 
arXiv:1308.0572v2 fatcat:zb3aw2raxfbu7p6kkgmmrmqwsu

Pattern Avoidance of Generalized Permutations [article]

Zhousheng Mei, Suijie Wang
2018 arXiv   pre-print
Extending both Dyck path and Riordan path, we introduce the Catalan-Riordan path which turns out to be a combinatorial interpretation of the difference array of Catalan numbers.  ...  In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding π is independent of the choice of π∈ S_3, which extends the classic  ...  permutation matrices can establish a direct bijection between S n (π) and Dyck paths of length 2n for each π ∈ S 3 .  ... 
arXiv:1804.06265v3 fatcat:mz3zszxiajfslkb4djawrsrinq

Numerical Sets, Core Partitions, and Integer Points in Polytopes [article]

Hannah Constantin, Benjamin Houston-Edwards, Nathan Kaplan
2015 arXiv   pre-print
For small values of a, we give formulas for the number of (a,b)-core partitions corresponding to numerical semigroups. We also study the number of partitions with a given hook set.  ...  We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope.  ...  We would like to thank Florencia Orosz-Hunziker and Daniel Corey for their assistance throughout this project.  ... 
arXiv:1509.06077v1 fatcat:7yskdhpi45ekjho36dxnug5g4y

Nested quantum Dyck paths and nabla(s_lambda) [article]

Nicholas A. Loehr, Gregory S. Warrington
2007 arXiv   pre-print
The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal inversion" statistic.  ...  We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator.  ...  Then area(g) is the number of squares between the path and the line y = x.  ... 
arXiv:0705.4608v1 fatcat:zwvecfoxnrd75olw5etmpuq7eq

The Delta Conjecture [article]

James Haglund, Jeffrey Remmel, Andrew Timothy Wilson
2017 arXiv   pre-print
Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.  ...  Furthermore, we use a reciprocity identity and LLT polynomials to prove another case.  ...  The number of these peaks for any Dyck path is equal to the number of valleys of the path.  ... 
arXiv:1509.07058v4 fatcat:kprwpinnmnaypovmk7amsou5ie

The Delta Conjecture

James Haglund, Jeffrey B. Remmel, Andrew Timothy Wilson
2020 Discrete Mathematics & Theoretical Computer Science  
Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.  ...  Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit  ...  In particular, we give bijections between contributing labeled Dyck paths and ordered set partitions, and these bijections lead to each of the four statistics.  ... 
doi:10.46298/dmtcs.6384 fatcat:rpqjtppsnnhdfglldorgrc3unu

The Delta Conjecture

J. Haglund, J. B. Remmel, A. T. Wilson
2018 Transactions of the American Mathematical Society  
Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author. Résumé.  ...  Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit  ...  In particular, we give bijections between contributing labeled Dyck paths and ordered set partitions, and these bijections lead to each of the four statistics.  ... 
doi:10.1090/tran/7096 fatcat:yuujzmos25cepomhhvhf2m7lsu
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