% s-weak order

@article {CeballosPons-sWeakOrderI,
    AUTHOR = {Ceballos, Cesar and Pons, Viviane},
     TITLE = {The {$s$}-weak order and {$s$}-permutahedra {I}:
              {C}ombinatorics and lattice structure},
   JOURNAL = {SIAM J. Discrete Math.},
  FJOURNAL = {SIAM Journal on Discrete Mathematics},
    VOLUME = {38},
      YEAR = {2024},
    NUMBER = {4},
     PAGES = {2855--2895}, 
       DOI = {10.1137/23M1605818},
       URL = {https://doi.org/10.1137/23M1605818},
}

@article {CeballosPons-sWeakOrderII,
    AUTHOR = {Ceballos, Cesar and Pons, Viviane},
     TITLE = {The {$s$}-weak order and {$s$}-permutahedra {II}: {T}he
              combinatorial complex of pure intervals},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {31},
      YEAR = {2024},
    NUMBER = {3},
     PAGES = {Paper No. 3.12, 59 pp.}, 
       DOI = {10.37236/12438},
       URL = {https://doi.org/10.37236/12438},
}

@article {Lacina,
	AUTHOR = {Lacina, Stephen},
	TITLE = {Poset topology of {$s$}-weak order via {SB}-labelings},
	JOURNAL = {J. Comb.},
	FJOURNAL = {Journal of Combinatorics},
	VOLUME = {13},
	YEAR = {2022},
	NUMBER = {3},
	PAGES = {357--395},
}

@unpublished {DLeonMoralesPhilippeTamayoYip,
	AUTHOR = {Rafael S. González D'León and Alejandro H. Morales and Eva Philippe and Daniel Tamayo Jiménez and Martha Yip},
	TITLE = {Realizing the $s$-permutahedron via flow polytopes},
	eprinttype = {arXiv},
	eprint = {2307.03474},
 	NOTE = {To appear in \emph{Trans. Amer. Math. Soc.}}, 
	YEAR = {2023},
}

@misc {Pons-sagedemo,
	AUTHOR = {Viviane Pons},
	TITLE = {Sagemath code and demo for s-weak order and s-permutahedra},
	NOTE = {\href{https://doi.org/10.5281/zenodo.8380308}{\texttt{doi:10.5281/zenodo.8380308}}},
	YEAR = {2022}
}

@misc {Pons-personnalCommunication,
	AUTHOR = {Viviane Pons},
	TITLE = {Personnal communications},
	YEAR = {2024}
}

@unpublished {PhilippePilaud,
	AUTHOR = {Eva Philippe and Vincent Pilaud},
	TITLE = {Geometric realizations of the $\boldsymbol{s}$-weak order and its quotients},
	eprinttype = {arXiv},
	eprint = {2405.02092},
	YEAR = {2024},
}


% lattice theory weak order

@article {GuilbaudRosenstiehl,
	AUTHOR = {Guilbaud, Georges Th. and Rosenstiehl, Pierre},
	TITLE = {Analyse algébrique d’un scrutin},
	JOURNAL = {Math. Inform. Sci. Humaines},
	FJOURNAL = {Math\'{e}matiques Informatique et Sciences Humaines},
	NUMBER = {4},
	YEAR = {1963},
	PAGES = {9--33},
}

@article {DuquenneCherfouh,
	AUTHOR = {Duquenne, Vincent and Cherfouh, Ameziane},
	TITLE = {On permutation lattices},
	JOURNAL = {Math. Social Sci.},
	FJOURNAL = {Mathematical Social Sciences},
	VOLUME = {27},
	YEAR = {1994},
	NUMBER = {1},
	PAGES = {73--89},
}

@article {ContePolyBarbut,
	AUTHOR = {Le Conte de Poly-Barbut, Claude},
	TITLE = {Sur les treillis de {C}oxeter finis},
	JOURNAL = {Math. Inform. Sci. Humaines},
	FJOURNAL = {Math\'{e}matiques Informatique et Sciences Humaines},
	NUMBER = {125},
	YEAR = {1994},
	PAGES = {41--57},
}

@article {CaspardContePolyBarbutMorvan,
	AUTHOR = {Caspard, Nathalie and Le Conte de Poly-Barbut, Claude and Morvan, Michel},
	TITLE = {Cayley lattices of finite {C}oxeter groups are bounded},
	JOURNAL = {Adv. in Appl. Math.},
	FJOURNAL = {Advances in Applied Mathematics},
	VOLUME = {33},
	YEAR = {2004},
	NUMBER = {1},
	PAGES = {71--94},
}

@article {Reading-posetRegions,
	AUTHOR = {Reading, Nathan},
	TITLE = {Lattice and order properties of the poset of regions in a hyperplane arrangement},
	JOURNAL = {Algebra Universalis},
	FJOURNAL = {Algebra Universalis},
	VOLUME = {50},
	YEAR = {2003},
	NUMBER = {2},
	PAGES = {179--205},
       DOI = {10.1007/s00012-003-1834-0},
}

@incollection {Reading-posetRegionsChapter,
	AUTHOR = {Reading, Nathan},
	TITLE = {Lattice theory of the poset of regions},
	BOOKTITLE = {Lattice theory: special topics and applications. {V}ol. 2},
	PAGES = {399--487},
	PUBLISHER = {Birkh\"auser/Springer, Cham},
	YEAR = {2016},
}

@article {Reading-latticeCongruences,
	AUTHOR = {Nathan~Reading},
	TITLE = {Lattice congruences of the weak order},
	JOURNAL = {Order},
	FJOURNAL = {Order. A Journal on the Theory of Ordered Sets and its Applications},
	VOLUME = {21},
	YEAR = {2004},
	NUMBER = {4},
	PAGES = {315--344},
       DOI = {10.1007/s11083-005-4803-8},
}

@article {Day,
	AUTHOR = {Day, Alan},
	TITLE = {Congruence normality: the characterization of the doubling class of convex sets},
	JOURNAL = {Algebra Universalis},
	FJOURNAL = {Algebra Universalis},
	VOLUME = {31},
	YEAR = {1994},
	NUMBER = {3},
	PAGES = {397--406},
       URL = {https://doi.org/10.1007/BF01221793},
}

@article {PilaudSantosZiegler,
	AUTHOR = {Pilaud, Vincent and Santos, Francisco and Ziegler, G\"{u}nter M.},
	TITLE = {Celebrating {L}oday's associahedron},
	JOURNAL = {Arch. Math.},
	FJOURNAL = {Archiv der Mathematik},
	VOLUME = {121},
	YEAR = {2023},
	NUMBER = {5-6},
	PAGES = {559--601},
       DOI = {10.1007/s00013-023-01895-6},
}

% special congruences

@inproceedings {Tonks,
	AUTHOR = {Tonks, Andy},
	TITLE = {Relating the associahedron and the permutohedron},
	BOOKTITLE = {Operads: {P}roceedings of {R}enaissance {C}onferences ({H}artford, {CT}/{L}uminy, 1995)},
	SERIES = {Contemp. Math.},
	VOLUME = {202},
	PAGES = {33--36},
	PUBLISHER = {Amer. Math. Soc., Providence, RI},
	YEAR = {1997},
}

@article {LodayRonco,
	AUTHOR = {Loday, Jean-Louis and Ronco, Mar{\'{\i}}a O.},
	TITLE = {Hopf algebra of the planar binary trees},
	JOURNAL = {Adv. Math.},
	FJOURNAL = {Advances in Mathematics},
	VOLUME = {139},
	YEAR = {1998},
	NUMBER = {2},
	PAGES = {293--309},
}

@article {HivertNovelliThibon-algebraBinarySearchTrees,
	AUTHOR = {Hivert, Florent and Novelli, Jean-Christophe and Thibon, Jean-Yves},
	TITLE = {The algebra of binary search trees},
	JOURNAL = {Theoret. Comput. Sci.},
	FJOURNAL = {Theoretical Computer Science},
	VOLUME = {339},
	YEAR = {2005},
	NUMBER = {1},
	PAGES = {129--165},
}

@phdthesis{Tamari,
	AUTHOR = {Dov Tamari},
	TITLE = {Monoides pr\'eordonn\'es et cha\^ines de Malcev},
	SCHOOL = {Universit\'e Paris Sorbonne},
	YEAR = {1951},
}

@article {Reading-CambrianLattices,
	AUTHOR = {Nathan~Reading},
	TITLE = {Cambrian lattices},
	JOURNAL = {Adv.~Math.},
	FJOURNAL = {Advances in Mathematics},
	VOLUME = {205},
	YEAR = {2006},
	NUMBER = {2},
	PAGES = {313--353},
       DOI = {10.1016/j.aim.2005.07.010},
}

@article {LangePilaud,
	AUTHOR = {Lange, Carsten and Pilaud, Vincent},
	TITLE = {Associahedra via spines},
	JOURNAL = {Combinatorica},
	FJOURNAL = {Combinatorica. An International Journal on Combinatorics and the Theory of Computing},
	VOLUME = {38},
	YEAR = {2018},
	NUMBER = {2},
	PAGES = {443--486},
}

@article {ChatelPilaud,
	AUTHOR = {Gr\'egory Chatel and Vincent Pilaud},
	TITLE = {{C}ambrian {H}opf {Algebras}},
	JOURNAL = {Adv. Math.},
	FJOURNAL = {Advances in Mathematics},
	VOLUME = {311},
	YEAR = {2017},
	PAGES = {598--633},
}

@article {PilaudPons-permutrees,
    AUTHOR = {Pilaud, Vincent and Pons, Viviane},
     TITLE = {Permutrees},
   JOURNAL = {Algebr. Comb.},
  FJOURNAL = {Algebraic Combinatorics},
    VOLUME = {1},
      YEAR = {2018},
    NUMBER = {2},
     PAGES = {173--224}, 
       DOI = {10.5802/alco},
       URL = {https://doi.org/10.5802/alco},
}

@article {Pilaud-brickAlgebra,
	AUTHOR = {Vincent Pilaud},
	TITLE = {Brick polytopes, lattice quotients, and {H}opf algebras},
	JOURNAL = {J. Combin. Theory Ser. A},
	FJOURNAL = {Journal of Combinatorial Theory. Series A},
	VOLUME = {155},
	YEAR = {2018},
	PAGES = {418--457},
}

@article {HoangMutze,
	AUTHOR = {Hoang, Hung P. and M\"{u}tze, Torsten},
	TITLE = {Combinatorial generation via permutation languages. {II}. {L}attice congruences},
	JOURNAL = {Israel J. Math.},
	FJOURNAL = {Israel Journal of Mathematics},
	VOLUME = {244},
	YEAR = {2021},
	NUMBER = {1},
	PAGES = {359--417},
       DOI = {10.1007/s11856-021-2186-1},
}

@article {DemonetIyamaReadingReitenThomas,
	AUTHOR = {Demonet, Laurent and Iyama, Osamu and Reading, Nathan and Reiten, Idun and Thomas, Hugh},
	TITLE = {Lattice theory of torsion classes: {B}eyond {$\tau$}-tilting theory},
	JOURNAL = {Trans. Amer. Math. Soc. Ser. B},
	FJOURNAL = {Transactions of the American Mathematical Society. Series B},
	VOLUME = {10},
	YEAR = {2023},
	PAGES = {542--612},
       DOI = {10.1090/btran/100},
}

@unpublished {BarnardNovelliPilaud,
	AUTHOR = {Emily Barnard and Jean-Christophe Novelli and Vincent Pilaud},
	TITLE = {On simple congruences of the weak order},
	NOTE = {In preparation},
	YEAR = {2023},
}

@article {LawReading,
	AUTHOR = {Law, Shirley and Reading, Nathan},
	TITLE = {The {H}opf algebra of diagonal rectangulations},
	JOURNAL = {J. Combin. Theory Ser. A},
	FJOURNAL = {Journal of Combinatorial Theory. Series A},
	VOLUME = {119},
	YEAR = {2012},
	NUMBER = {3},
	PAGES = {788--824},
}

@article {Giraudo,
	AUTHOR = {Giraudo, Samuele},
	TITLE = {Algebraic and combinatorial structures on pairs of twin binary trees},
	JOURNAL = {J. Algebra},
	FJOURNAL = {Journal of Algebra},
	VOLUME = {360},
	YEAR = {2012},
	PAGES = {115--157},
}

@article {Reading-rectangulations,
	AUTHOR = {Reading, Nathan},
	TITLE = {Generic rectangulations},
	JOURNAL = {European J. Combin.},
	FJOURNAL = {European Journal of Combinatorics},
	VOLUME = {33},
	YEAR = {2012},
	NUMBER = {4},
	PAGES = {610--623},
}

@unpublished {AsinowskiCardinalFelsnerFusy,
	AUTHOR = {Andrei Asinowski and Jean Cardinal and Stefan Felsner and {\'{E}}ric Fusy},
	TITLE = {Combinatorics of rectangulations: Old and new bijections},
	eprinttype = {},
	eprint = {},
 	NOTE = {Preprint, \href{http://arxiv.org/abs/2402.01483}{\texttt{arXiv:2402.01483}}}, 
	YEAR = {2024},
}

@unpublished {CardinalPilaud,
	AUTHOR = {Cardinal, Jean and Pilaud, Vincent},
	TITLE = {Rectangulotopes},
	eprinttype = {},
	eprint = {},
 	NOTE = {Preprint, \href{http://arxiv.org/abs/2404.17349}{\texttt{arXiv:2404.17349}}}, 
	YEAR = {2024},
}

% canonical representations, canonical complexes

@article {Reading-arcDiagrams,
	AUTHOR = {Reading, Nathan},
	TITLE = {Noncrossing arc diagrams and canonical join representations},
	JOURNAL = {SIAM J. Discrete Math.},
	FJOURNAL = {SIAM Journal on Discrete Mathematics},
	VOLUME = {29},
	YEAR = {2015},
	NUMBER = {2},
	PAGES = {736--750},
       DOI = {10.1137/140972391},
}

@article {Barnard,
	AUTHOR = {Barnard, Emily},
	TITLE = {The canonical join complex},
	JOURNAL = {Electron. J. Combin.},
	FJOURNAL = {Electronic Journal of Combinatorics},
	VOLUME = {26},
	YEAR = {2019},
	NUMBER = {1},
	PAGES = {Paper No. 1.24, 25},
}

@article {AlbertinPilaud,
	AUTHOR = {Albertin, Doriann and Pilaud, Vincent},
	TITLE = {The canonical complex of the weak order},
	JOURNAL = {Order},
	FJOURNAL = {Order. A Journal on the Theory of Ordered Sets and its Applications},
	VOLUME = {40},
	YEAR = {2023},
	NUMBER = {2},
	PAGES = {349--370},
}

% quotientopes

@article {Reading-HopfAlgebras,
	AUTHOR = {Reading, Nathan},
	TITLE = {Lattice congruences, fans and {H}opf algebras},
	JOURNAL = {J. Combin. Theory Ser. A},
	FJOURNAL = {Journal of Combinatorial Theory. Series A},
	VOLUME = {110},
	YEAR = {2005},
	NUMBER = {2},
	PAGES = {237--273},
       DOI = {10.1016/j.jcta.2004.11.001},
}

@article {PilaudSantos-quotientopes,
	AUTHOR = {Vincent Pilaud and Francisco Santos},
	TITLE = {Quotientopes},
	JOURNAL = {Bull. Lond. Math. Soc.},
	FJOURNAL = {Bulletin of the London Mathematical Society},
	YEAR = {2019},
	VOLUME = {51},
	NUMBER = {3},
	PAGES = {406--420},
       DOI = {10.1112/blms.12231},
}

@article {PadrolPilaudRitter,
	AUTHOR = {Arnau Padrol and Vincent Pilaud and Julian Ritter},
	TITLE = {Shard polytopes},
	JOURNAL = {Int. Math. Res. Not. IMRN},
	FJOURNAL = {International Mathematics Research Notices. IMRN},
	YEAR = {2023},
	NUMBER = {9},
	PAGES = {7686--7796},
       URL = {https://doi.org/10.1093/imrn/rnac042},
}

@unpublished {DanaHansonThomas,
	AUTHOR = {Will Dana and Eric Hanson and Hugh Thomas},
	TITLE = {Shard polytopes via representation theory},
	NOTE = {In preparation},
	YEAR = {2023},
}

@article {AlbertinPilaudRitter,
	AUTHOR = {Albertin, Doriann and Pilaud, Vincent and Ritter, Julian},
	TITLE = {Removahedral congruences versus permutree congruences},
	JOURNAL = {Electron. J. Combin.},
	FJOURNAL = {Electronic Journal of Combinatorics},
	VOLUME = {28},
	YEAR = {2021},
	NUMBER = {4},
	PAGES = {Paper No. 4.8, 38},
}

@unpublished {Pilaud-acyclicReorientationLattices,
	AUTHOR = {Vincent Pilaud},
	TITLE = {Acyclic reorientation lattices and their lattice quotients},
	eprinttype = {},
	eprint = {},
 	NOTE = {Preprint, \href{http://arxiv.org/abs/2111.12387}{\texttt{arXiv:2111.12387}}}, 
	YEAR = {2021},
}

% deformed permutahedra

@inproceedings {Edmonds,
	AUTHOR = {Edmonds, Jack},
	TITLE = {Submodular functions, matroids, and certain polyhedra},
	BOOKTITLE = {Combinatorial {S}tructures and their {A}pplications ({P}roc. {C}algary {I}nternat. {C}onf., {C}algary, {A}lta., 1969)},
	PAGES = {69--87},
	PUBLISHER = {Gordon and Breach, New York},
	YEAR = {1970},
}

@article {Postnikov,
	AUTHOR = {Alexander~Postnikov},
	TITLE = {Permutohedra, associahedra, and beyond},
	JOURNAL = {Int. Math. Res. Not. IMRN},
	FJOURNAL = {International Mathematics Research Notices. IMRN},
	YEAR = {2009},
	NUMBER = {6},
	PAGES = {1026--1106},
}

@article {PostnikovReinerWilliams,
    AUTHOR = {Postnikov, Alex and Reiner, Victor and Williams, Lauren},
     TITLE = {Faces of generalized permutohedra},
   JOURNAL = {Doc. Math.},
  FJOURNAL = {Documenta Mathematica},
    VOLUME = {13},
      YEAR = {2008},
     PAGES = {207--273},
     DOI = {10.4171/DM/248}
}

@article {Reading-coarseningPolyhedralComplexes,
	AUTHOR = {Reading, Nathan},
	TITLE = {Coarsening polyhedral complexes},
	JOURNAL = {Proc. Amer. Math. Soc.},
	FJOURNAL = {Proceedings of the American Mathematical Society},
	VOLUME = {140},
	YEAR = {2012},
	NUMBER = {10},
	PAGES = {3593--3605},
}

% facial weak order

@unpublished {KrobLatapyNovelliPhanSchwer,
	AUTHOR = {Daniel Krob and Matthieu Latapy and Jean-Christophe Novelli and Ha-Duong Phan and Sylviane Schwer},
	TITLE = {Pseudo-{P}ermutations {I}: {F}irst {C}ombinatorial and {L}attice {P}roperties},
	NOTE = {13th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2001)},
	YEAR = {2001},
}

@incollection {BoulierHivertKrobNovelli,
	AUTHOR = {Boulier, Fran\ced{c}ois and Hivert, Florent and Krob, Daniel and Novelli, Jean-Christophe},
	TITLE = {Pseudo-permutations. {II}. {G}eometry and representation theory},
	BOOKTITLE = {Discrete models: combinatorics, computation, and geometry ({P}aris, 2001)},
	SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AA},
	PAGES = {123--132 (electronic)},
	PUBLISHER = {Maison Inform. Math. Discr\`et. (MIMD), Paris},
	YEAR = {2001},
}

@article {PalaciosRonco,
	AUTHOR = {Palacios, Patricia and Ronco, Mar{\'{\i}}a O.},
	TITLE = {Weak {B}ruhat order on the set of faces of the permutohedron and the associahedron},
	JOURNAL = {J. Algebra},
	FJOURNAL = {Journal of Algebra},
	VOLUME = {299},
	YEAR = {2006},
	NUMBER = {2},
	PAGES = {648--678},
}

@article {DermenjianHohlwegPilaud,
	AUTHOR = {Aram Dermenjian and Christophe Hohlweg and Vincent Pilaud},
	TITLE = {The facial weak order and its lattice quotients},
	JOURNAL = {Trans. Amer. Math. Soc.},
	FJOURNAL = {Transactions of the American Mathematical Society},
	VOLUME = {370},
	YEAR = {2018},
	NUMBER = {2},
	PAGES = {1469--1507},
}

@article {DermenjianHohlwegMcConvillePilaud,
	AUTHOR = {Aram Dermenjian and Christophe Hohlweg and Thomas McConville and Vincent Pilaud},
	TITLE = {The facial weak order on hyperplane arrangements},
	JOURNAL = {Discrete Comput. Geom.},
	FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
	VOLUME = {67},
	YEAR = {2022},
	NUMBER = {1},
	PAGES = {166--202},
}

@unpublished {Hanson,
	AUTHOR = {Eric Hanson},
	TITLE = {A facial order for torsion classes},
	eprinttype = {},
	eprint = {},
	NOTE = {Preprint, \href{http://arxiv.org/abs/2305.06031}{\texttt{arXiv:2305.06031}}}, 
	YEAR = {2023},
}

% tropical geometry

@incollection {Joswig16,
	AUTHOR = {Joswig, Michael},
	TITLE = {The {C}ayley trick for tropical hypersurfaces with a view toward {R}icardian economics},
	BOOKTITLE = {Homological and computational methods in commutative algebra},
	SERIES = {Springer INdAM Ser.},
	VOLUME = {20},
	PAGES = {107--128},
	PUBLISHER = {Springer, Cham},
	YEAR = {2017},
}

@book{Joswig21,
	AUTHOR = {Joswig, Michael},
	TITLE = {Essentials of tropical combinatorics},
	SERIES = {Graduate Studies in Mathematics},
	VOLUME = {219},
	PUBLISHER = {American Mathematical Society, Providence, RI},
	YEAR = {[2021] \copyright 2021},
	PAGES = {xx+398},
 }
 
 @article {McMullen,
	AUTHOR = {McMullen, Peter},
	TITLE = {Fibre tilings},
	JOURNAL = {Mathematika},
	FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
	VOLUME = {50},
	YEAR = {2003},
	NUMBER = {1-2},
	PAGES = {1--33 (2005)},
}
 
 % nu-Tamari
 
 @article {BergeronPrevilleRatelle,
	AUTHOR = {Bergeron, Fran\ced{c}ois and Pr\'eville-Ratelle, Louis-Fran\ced{c}ois},
	TITLE = {Higher trivariate diagonal harmonics via generalized {Tamari} posets},
	JOURNAL = {Journal of Combinatorics},
	VOLUME = {3},
	NUMBER = {3},
	YEAR = {2012},
	PAGES = {317--341},
}

@article {PrevilleRatelleViennot,
	AUTHOR = {Pr\'eville-Ratelle, Louis-Fran\ced{c}ois and Viennot, Xavier},
	TITLE = {The enumeration of generalized {T}amari intervals},
	JOURNAL = {Trans. Amer. Math. Soc.},
	FJOURNAL = {Transactions of the American Mathematical Society},
	VOLUME = {369},
	YEAR = {2017},
	NUMBER = {7},
	PAGES = {5219--5239},
       DOI = {10.1090/tran/7004},
}

 @article {CeballosPadrolSarmiento-geometryNuTamari,
	AUTHOR = {Ceballos, Cesar and Padrol, Arnau and Sarmiento, Camilo},
	TITLE = {Geometry of {$\nu$}-{T}amari lattices in types~{$A$} and~{$B$}},
	JOURNAL = {Trans. Amer. Math. Soc.},
	FJOURNAL = {Transactions of the American Mathematical Society},
	VOLUME = {371},
	YEAR = {2019},
	NUMBER = {4},
	PAGES = {2575--2622},
       DOI = {10.1090/tran/7405},
}