@article {ACR21,
    AUTHOR = {Abram, Antoine and Chapelier-Laget, Nathan and Reutenauer, Christophe},
     TITLE = {An order on circular permutations},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {28},
      YEAR = {2021},
    NUMBER = {3},
     PAGES = {Paper No. 3.31, 43},
       DOI = {10.37236/9982},
}

@article {GSS12looping,
    AUTHOR = {Gashi, Q\"{e}ndrim R. and Schedler, Travis and Speyer, David E.},
     TITLE = {Looping of the numbers game and the alcoved hypercube},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {119},
      YEAR = {2012},
    NUMBER = {3},
     PAGES = {713--730},
       DOI = {10.1016/j.jcta.2011.11.012},
}

@article {LP07alcoved1,
    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},
       DOI = {10.1007/s00454-006-1294-3},
}

@incollection {LP18alcoved2,
    AUTHOR = {Lam, Thomas and Postnikov, Alexander},
     TITLE = {Alcoved polytopes {II}},
 BOOKTITLE = {Lie groups, geometry, and representation theory},
    SERIES = {Progr. Math.},
    VOLUME = {326},
     PAGES = {253--272},
 PUBLISHER = {Birkh\"{a}user/Springer, Cham},
      YEAR = {2018},
       DOI = {10.1007/978-3-030-02191-7\_10},
}

@article {Li12,
    AUTHOR = {Li, Nan},
     TITLE = {Ehrhart {$h^*$}-vectors of hypersimplices},
   JOURNAL = {Discrete Comput. Geom.},
  FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
    VOLUME = {48},
      YEAR = {2012},
    NUMBER = {4},
     PAGES = {847--878},
       DOI = {10.1007/s00454-012-9452-2},
}

@article {S80decomp,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Decompositions of rational convex polytopes},
   JOURNAL = {Ann. Discrete Math.},
  FJOURNAL = {Annals of Discrete Mathematics},
    VOLUME = {6},
      YEAR = {1980},
     PAGES = {333--342},
       DOI = {10.1016/S0167-5060(08)70717-9},
}

@article {abr01desmaj,
    AUTHOR = {Adin, Ron M. and Brenti, Francesco and Roichman, Yuval},
     TITLE = {Descent numbers and major indices for the hyperoctahedral group},
      NOTE = {Special issue in honor of Dominique Foata's 65th birthday (Philadelphia, PA, 2000)},
   JOURNAL = {Adv. in Appl. Math.},
  FJOURNAL = {Advances in Applied Mathematics},
    VOLUME = {27},
      YEAR = {2001},
    NUMBER = {2-3},
     PAGES = {210--224},
       DOI = {10.1006/aama.2001.0731},
}

@article {fh09signedV,
    AUTHOR = {Foata, Dominique and Han, Guo-Niu},
     TITLE = {Signed words and permutations. {V}. {A} sextuple distribution},
   JOURNAL = {Ramanujan J.},
  FJOURNAL = {Ramanujan Journal. An International Journal Devoted to th Areas of Mathematics Influenced by Ramanujan},
    VOLUME = {19},
      YEAR = {2009},
    NUMBER = {1},
     PAGES = {29--52},
       DOI = {10.1007/s11139-007-9059-z},
}

@article {Ehrhart62,
    AUTHOR = {Ehrhart, Eug\`ene},
     TITLE = {Sur les poly\`edres rationnels homoth\'etiques \`a{} {$n$} dimensions},
   JOURNAL = {C. R. Acad. Sci. Paris},
  FJOURNAL = {Comptes Rendus Hebdomadaires des S\'eances de l'Acad\'emie des Sciences},
    VOLUME = {254},
      YEAR = {1962},
     PAGES = {616--618},
}

@article {reading05fans,
    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},
}