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@article {Stembridge,
    AUTHOR = {Stembridge, John R.},
     TITLE = {On the fully commutative elements of {C}oxeter groups},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {5},
      YEAR = {1996},
    NUMBER = {4},
     PAGES = {353--385},
      ISSN = {0925-9899},
   MRCLASS = {20F55 (05E15)},
  MRNUMBER = {1406459},
MRREVIEWER = {Kimmo Eriksson},
       DOI = {10.1023/A:1022452717148},
       URL = {https://doi.org/10.1023/A:1022452717148},
}

@article {Kahn-Linial,
    AUTHOR = {Kahn, Jeff and Linial, Nathan},
     TITLE = {Balancing extensions via {B}runn-{M}inkowski},
   JOURNAL = {Combinatorica},
  FJOURNAL = {Combinatorica. An International Journal on Combinatorics and
              the Theory of Computing},
    VOLUME = {11},
      YEAR = {1991},
    NUMBER = {4},
     PAGES = {363--368},
      ISSN = {0209-9683},
   MRCLASS = {52A39 (06A06 52B11)},
  MRNUMBER = {1137768},
MRREVIEWER = {A. Florian},
       DOI = {10.1007/BF01275670},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BF01275670},
}

@article {Kahn-Saks,
    AUTHOR = {Kahn, Jeff and Saks, Michael},
     TITLE = {Balancing poset extensions},
   JOURNAL = {Order},
  FJOURNAL = {Order},
    VOLUME = {1},
      YEAR = {1984},
    NUMBER = {2},
     PAGES = {113--126},
      ISSN = {0167-8094},
   MRCLASS = {06A10 (68P99 68R99)},
  MRNUMBER = {764319},
MRREVIEWER = {Dean Sturtevant},
       DOI = {10.1007/BF00565647},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BF00565647},
}

@article {Brightwell,
    AUTHOR = {Brightwell, Graham},
     TITLE = {Balanced pairs in partial orders},
   JOURNAL = {Discrete Math.},
  FJOURNAL = {Discrete Mathematics},
    VOLUME = {201},
      YEAR = {1999},
    NUMBER = {1-3},
     PAGES = {25--52},
      ISSN = {0012-365X},
   MRCLASS = {06A07},
  MRNUMBER = {1687862},
MRREVIEWER = {Gerhard Behrendt},
       DOI = {10.1016/S0012-365X(98)00311-2},
       URL = {https://doi-org.libproxy.mit.edu/10.1016/S0012-365X(98)00311-2},
}

@book {Bjorner-Brenti,
    AUTHOR = {Bj\"{o}rner, Anders and Brenti, Francesco},
     TITLE = {Combinatorics of {C}oxeter groups},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {231},
 PUBLISHER = {Springer, New York},
      YEAR = {2005},
     PAGES = {xiv+363},
      ISBN = {978-3540-442387; 3-540-44238-3},
   MRCLASS = {05-01 (05E15 20F55)},
  MRNUMBER = {2133266},
MRREVIEWER = {Jian-yi Shi},
}

@article {Kislicyn,
    AUTHOR = {Kislicyn, S. S.},
     TITLE = {Finite partially ordered sets and their corresponding
              permutation sets},
   JOURNAL = {Mat. Zametki},
  FJOURNAL = {Akademiya Nauk SSSR. Matematicheskie Zametki},
    VOLUME = {4},
      YEAR = {1968},
     PAGES = {511--518},
      ISSN = {0025-567X},
   MRCLASS = {06.20},
  MRNUMBER = {244113},
MRREVIEWER = {J. Jakub\'{\i}k},
}

@article {Linial,
    AUTHOR = {Linial, Nathan},
     TITLE = {The information-theoretic bound is good for merging},
   JOURNAL = {SIAM J. Comput.},
  FJOURNAL = {SIAM Journal on Computing},
    VOLUME = {13},
      YEAR = {1984},
    NUMBER = {4},
     PAGES = {795--801},
      ISSN = {0097-5397},
   MRCLASS = {68P10},
  MRNUMBER = {764179},
       DOI = {10.1137/0213049},
       URL = {https://doi-org.libproxy.mit.edu/10.1137/0213049},
}

@article {Brightwell-Winkler,
    AUTHOR = {Brightwell, Graham and Winkler, Peter},
     TITLE = {Counting linear extensions},
   JOURNAL = {Order},
  FJOURNAL = {Order. A Journal on the Theory of Ordered Sets and its
              Applications},
    VOLUME = {8},
      YEAR = {1991},
    NUMBER = {3},
     PAGES = {225--242},
      ISSN = {0167-8094},
   MRCLASS = {06A06 (68Q25)},
  MRNUMBER = {1154926},
MRREVIEWER = {Christoph Meinel},
       DOI = {10.1007/BF00383444},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BF00383444},
}

@misc{Dittmer-Pak,
       author = {{Dittmer}, Samuel and {Pak}, Igor},
        title = "{Counting linear extensions of restricted posets}",
         year = 2018,
eprinttype = {arXiv},
       eprint = {1802.06312},
}
	
@incollection {Viennot,
    AUTHOR = {Viennot, G\'{e}rard Xavier},
     TITLE = {Heaps of pieces. {I}. {B}asic definitions and combinatorial
              lemmas},
 BOOKTITLE = {Combinatoire \'{e}num\'{e}rative ({M}ontreal, {Q}ue., 1985/{Q}uebec,
              {Q}ue., 1985)},
    SERIES = {Lecture Notes in Math.},
    VOLUME = {1234},
     PAGES = {321--350},
 PUBLISHER = {Springer, Berlin},
      YEAR = {1986},
   MRCLASS = {05A99 (20M10)},
  MRNUMBER = {927773},
MRREVIEWER = {Dominique Perrin},
       DOI = {10.1007/BFb0072524},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BFb0072524},
}	

@article {Bjorner-Wachs-lin-ext,
    AUTHOR = {Bj\"{o}rner, Anders and Wachs, Michelle L.},
     TITLE = {Permutation statistics and linear extensions of posets},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {58},
      YEAR = {1991},
    NUMBER = {1},
     PAGES = {85--114},
      ISSN = {0097-3165},
   MRCLASS = {06A07 (05E99 20B35)},
  MRNUMBER = {1119703},
MRREVIEWER = {Sergey V. Fomin},
       DOI = {10.1016/0097-3165(91)90075-R},
       URL = {https://doi-org.libproxy.mit.edu/10.1016/0097-3165(91)90075-R},
}

@article {Bjorner-Wachs-gen-quo,
    AUTHOR = {Bj\"{o}rner, Anders and Wachs, Michelle L.},
     TITLE = {Generalized quotients in {C}oxeter groups},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {308},
      YEAR = {1988},
    NUMBER = {1},
     PAGES = {1--37},
      ISSN = {0002-9947},
   MRCLASS = {05A99 (06F99 20B30 20F99)},
  MRNUMBER = {946427},
MRREVIEWER = {Joseph Kung},
       DOI = {10.2307/2000946},
       URL = {https://doi-org.libproxy.mit.edu/10.2307/2000946},
}

@article {Dilworth,
    AUTHOR = {Dilworth, R. P.},
     TITLE = {A decomposition theorem for partially ordered sets},
   JOURNAL = {Ann. of Math. (2)},
  FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {51},
      YEAR = {1950},
     PAGES = {161--166},
      ISSN = {0003-486X},
   MRCLASS = {09.1X},
  MRNUMBER = {32578},
MRREVIEWER = {P. M. Whitman},
       DOI = {10.2307/1969503},
       URL = {https://doi-org.libproxy.mit.edu/10.2307/1969503},
}

@book {Tits,
    AUTHOR = {Tits, Jacques},
     TITLE = {Buildings of spherical type and finite {BN}-pairs},
    SERIES = {Lecture Notes in Mathematics, Vol. 386},
 PUBLISHER = {Springer-Verlag, Berlin-New York},
      YEAR = {1974},
     PAGES = {x+299},
   MRCLASS = {20G15 (50A20)},
  MRNUMBER = {0470099},
MRREVIEWER = {Bruce Cooperstein},
}

@incollection {Tits-words,
    AUTHOR = {Tits, Jacques},
     TITLE = {Le probl\`eme des mots dans les groupes de {C}oxeter},
 BOOKTITLE = {Symposia {M}athematica ({INDAM}, {R}ome, 1967/68), {V}ol. 1},
     PAGES = {175--185},
 PUBLISHER = {Academic Press, London},
      YEAR = {1969},
   MRCLASS = {20.10},
  MRNUMBER = {0254129},
MRREVIEWER = {J. E. Humphreys},
}

@book {Humphreys,
    AUTHOR = {Humphreys, James E.},
     TITLE = {Introduction to {L}ie algebras and representation theory},
      NOTE = {Graduate Texts in Mathematics, Vol. 9},
 PUBLISHER = {Springer-Verlag, New York-Berlin},
      YEAR = {1972},
     PAGES = {xii+169},
   MRCLASS = {17BXX},
  MRNUMBER = {0323842},
MRREVIEWER = {F. W. Lemire},
}

@book {ec1,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Enumerative combinatorics. {V}olume 1},
    SERIES = {Cambridge Studies in Advanced Mathematics},
    VOLUME = {49},
   EDITION = {Second},
 PUBLISHER = {Cambridge University Press, Cambridge},
      YEAR = {2012},
     PAGES = {xiv+626},
      ISBN = {978-1-107-60262-5},
   MRCLASS = {05-02 (05A15 06-02)},
  MRNUMBER = {2868112},
}

@article {Aigner,
    AUTHOR = {Aigner, Martin},
     TITLE = {A note on merging},
   JOURNAL = {Order},
  FJOURNAL = {Order},
    VOLUME = {2},
      YEAR = {1985},
    NUMBER = {3},
     PAGES = {257--264},
      ISSN = {0167-8094},
   MRCLASS = {06A10 (68P10)},
  MRNUMBER = {824699},
       DOI = {10.1007/BF00333131},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BF00333131},
}

@article {Brightwell-semiorder,
    AUTHOR = {Brightwell, Graham R.},
     TITLE = {Semiorders and the {$\frac 13$}--{$\frac 23$} conjecture},
   JOURNAL = {Order},
  FJOURNAL = {Order. A Journal on the Theory of Ordered Sets and its
              Applications},
    VOLUME = {5},
      YEAR = {1989},
    NUMBER = {4},
     PAGES = {369--380},
      ISSN = {0167-8094},
   MRCLASS = {06A10},
  MRNUMBER = {1010386},
MRREVIEWER = {Michael Saks},
       DOI = {10.1007/BF00353656},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BF00353656},
}

@article {Lam-Postnikov,
    AUTHOR = {Lam, Thomas and Postnikov, Alexander},
     TITLE = {Alcoved polytopes. {I}},
   JOURNAL = {Discrete Comput. Geom.},
  FJOURNAL = {Discrete \& Computational Geometry. An International Journal
              of Mathematics and Computer Science},
    VOLUME = {38},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {453--478},
      ISSN = {0179-5376},
   MRCLASS = {52B12 (52B40)},
  MRNUMBER = {2352704},
MRREVIEWER = {Jes\'{u}s A. De Loera},
       DOI = {10.1007/s00454-006-1294-3},
       URL = {https://doi.org/10.1007/s00454-006-1294-3},
}

@article {Stanley-poset-polytopes,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Two poset polytopes},
   JOURNAL = {Discrete Comput. Geom.},
  FJOURNAL = {Discrete \& Computational Geometry. An International Journal
              of Mathematics and Computer Science},
    VOLUME = {1},
      YEAR = {1986},
    NUMBER = {1},
     PAGES = {9--23},
      ISSN = {0179-5376},
   MRCLASS = {52A25 (52A40)},
  MRNUMBER = {824105},
MRREVIEWER = {P. McMullen},
       DOI = {10.1007/BF02187680},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/BF02187680},
}

@misc{arxiv-version,
       author = {{Gaetz}, Christian and {Gao}, Yibo},
        title = "{Balance constants for Coxeter groups}",
         year = 2020,
    eprinttype = {arXiv},
       eprint = {2005.09719},
}