@book {kemeny,
    AUTHOR = {Kemeny, John G. and Snell, J. Laurie},
     TITLE = {Finite {M}arkov chains},
    SERIES = {Undergraduate Texts in Mathematics},
      NOTE = {Reprinting of the 1960 original},
 PUBLISHER = {Springer-Verlag, New York-Heidelberg},
      YEAR = {1976},
     PAGES = {ix+210},
   MRCLASS = {60J10 (60J20)},
  MRNUMBER = {410929},
}

@misc{ash2023introducing,
      title={Introducing DASEP: the doubly asymmetric simple exclusion process}, 
      author={David W. Ash},
      year={2023},
      eprint={2201.00040},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}

@article{Derrida_1993,
doi = {10.1088/0305-4470/26/7/011},
year = {1993},
month = {apr},
publisher = {},
volume = {26},
number = {7},
pages = {1493},
author = {B Derrida and  M R Evans and  V Hakim and  V Pasquier},
title = {Exact solution of a 1D asymmetric exclusion model using a matrix formulation},
journal = {Journal of Physics A: Mathematical and General},
abstract = {Several recent works have shown that the one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, can be solved exactly in the case of open boundaries. Here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices. With this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules. They obtain several explicit forms for these non-commuting matrices which are, in the general case, infinite-dimensional. Their approach allows exact expressions to be derived for the current and density profiles. Finally they discuss briefly two possible generalizations of their results: the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles.}
}

@article {Martin2020,
    AUTHOR = {Martin, James B.},
     TITLE = {Stationary distributions of the multi-type {ASEP}},
   JOURNAL = {Electron. J. Probab.},
  FJOURNAL = {Electronic Journal of Probability},
    VOLUME = {25},
      YEAR = {2020},
     PAGES = {Paper No. 43, 41},
      ISSN = {1083-6489},
   MRCLASS = {60K35 (60K25 82C22)},
  MRNUMBER = {4089793},
MRREVIEWER = {Christian\ Tutschka},
       DOI = {10.1214/20-ejp421},
}

@article{MGP, 
    author = {MacDonald, Carolyn T. and Gibbs, Julian H. and Pipkin, Allen C.},
title = {Kinetics of biopolymerization on nucleic acid templates},
journal = {Biopolymers},
volume = {6},
number = {1},
pages = {1-25},
doi = {https://doi.org/10.1002/bip.1968.360060102},
year = {1968}
}


@book{Liggett1985InteractingParticleSystems,
    AUTHOR = {Liggett, Thomas M.},
     TITLE = {Interacting particle systems},
    SERIES = {Grundlehren der mathematischen Wissenschaften [Fundamental
              Principles of Mathematical Sciences]},
    VOLUME = {276},
 PUBLISHER = {Springer-Verlag, New York},
      YEAR = {1985},
     PAGES = {xv+488},
      ISBN = {0-387-96069-4},
   MRCLASS = {60K35},
  MRNUMBER = {776231},
MRREVIEWER = {F.\ L.\ Spitzer},
       DOI = {10.1007/978-1-4613-8542-4},
}

@article{Liggett1975ErgodicTF,
    AUTHOR = {Liggett, Thomas M.},
     TITLE = {Ergodic theorems for the asymmetric simple exclusion process},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {213},
      YEAR = {1975},
     PAGES = {237--261},
      ISSN = {0002-9947,1088-6850},
   MRCLASS = {60K35},
  MRNUMBER = {410986},
MRREVIEWER = {Richard\ Holley},
       DOI = {10.2307/1998046},
}

@article{Johansson2000,
  title={Shape Fluctuations and Random Matrices},
  author={Kurt Johansson},
  journal={Communications in Mathematical Physics},
abstract = {We study a certain random growth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy–Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).},
  year={2000},
  volume={209},
  issue = {2},
  pages={437-476},
}

@article{Tracy2008AFD,
    AUTHOR = {Tracy, Craig A. and Widom, Harold},
     TITLE = {A {F}redholm determinant representation in {ASEP}},
   JOURNAL = {J. Stat. Phys.},
  FJOURNAL = {Journal of Statistical Physics},
    VOLUME = {132},
      YEAR = {2008},
    NUMBER = {2},
     PAGES = {291--300},
      ISSN = {0022-4715,1572-9613},
   MRCLASS = {82C22 (47B35 47N30 60K35)},
  MRNUMBER = {2415104},
MRREVIEWER = {M\'{a}rton\ Bal\'{a}zs},
       DOI = {10.1007/s10955-008-9562-7},
}

@article{Tracy2008AsymptoticsIA,
    AUTHOR = {Tracy, Craig A. and Widom, Harold},
     TITLE = {Asymptotics in {ASEP} with step initial condition},
   JOURNAL = {Comm. Math. Phys.},
  FJOURNAL = {Communications in Mathematical Physics},
    VOLUME = {290},
      YEAR = {2009},
    NUMBER = {1},
     PAGES = {129--154},
      ISSN = {0010-3616,1432-0916},
   MRCLASS = {60K35 (47N30 82C22)},
  MRNUMBER = {2520510},
MRREVIEWER = {Timo\ Sepp\"{a}l\"{a}inen},
       DOI = {10.1007/s00220-009-0761-0},
}

@article{Spitzer1970InteractionOM,
    AUTHOR = {Spitzer, Frank},
     TITLE = {Interaction of {M}arkov processes},
   JOURNAL = {Advances in Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {5},
      YEAR = {1970},
     PAGES = {246--290},
      ISSN = {0001-8708},
   MRCLASS = {60.60},
  MRNUMBER = {268959},
MRREVIEWER = {B.\ Weiss},
       DOI = {10.1016/0001-8708(70)90034-4},
}

@article{Corteel2006AMC,
  title={A Markov Chain on Permutations which Projects to the PASEP},
  author={Sylvie Corteel and Lauren K. Williams},
  journal={International Mathematics Research Notices},
  year={2006},
  volume={2007},
}

@article{Corteel2006TableauxCF,
    AUTHOR = {Corteel, Sylvie and Williams, Lauren K.},
     TITLE = {A {M}arkov chain on permutations which projects to the
              {PASEP}},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2007},
    NUMBER = {17},
     PAGES = {Art. ID rnm055, 27},
      ISSN = {1073-7928,1687-0247},
   MRCLASS = {60J10 (05A05 05A30)},
  MRNUMBER = {2354800},
MRREVIEWER = {Stuart\ G.\ Whittington},
       DOI = {10.1093/imrn/rnm055},
}

@article{Corteel2009StaircaseTT,
    AUTHOR = {Corteel, Sylvie and Williams, Lauren K.},
     TITLE = {Staircase tableaux, the asymmetric exclusion process, and
              {A}skey-{W}ilson polynomials},
   JOURNAL = {Proc. Natl. Acad. Sci. USA},
  FJOURNAL = {Proceedings of the National Academy of Sciences of the United
              States of America},
    VOLUME = {107},
      YEAR = {2010},
    NUMBER = {15},
     PAGES = {6726--6730},
      ISSN = {0027-8424,1091-6490},
   MRCLASS = {05E10 (33C45 33D45 60K35)},
  MRNUMBER = {2630104},
MRREVIEWER = {Tomohiro\ Sasamoto},
       DOI = {10.1073/pnas.0909915107},
}

@article{Corteel2009TableauxCF,
    AUTHOR = {Corteel, Sylvie and Williams, Lauren K.},
     TITLE = {Tableaux combinatorics for the asymmetric exclusion process
              and {A}skey-{W}ilson polynomials},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {159},
      YEAR = {2011},
    NUMBER = {3},
     PAGES = {385--415},
      ISSN = {0012-7094,1547-7398},
   MRCLASS = {05E10 (33C45 60K35 82C22)},
  MRNUMBER = {2831874},
MRREVIEWER = {Timo\ Sepp\"{a}l\"{a}inen},
       DOI = {10.1215/00127094-1433385},
}

@article{Corteel2018FromMQ,
    AUTHOR = {Corteel, Sylvie and Mandelshtam, Olya and Williams, Lauren},
     TITLE = {From multiline queues to {M}acdonald polynomials via the
              exclusion process},
   JOURNAL = {Amer. J. Math.},
  FJOURNAL = {American Journal of Mathematics},
    VOLUME = {144},
      YEAR = {2022},
    NUMBER = {2},
     PAGES = {395--436},
      ISSN = {0002-9327,1080-6377},
   MRCLASS = {60K35 (05E05 33C52)},
  MRNUMBER = {4401508},
       DOI = {10.1353/ajm.2022.0007},
}

@article{Pang2015LumpingsOA,
    AUTHOR = {Pang, C. Y. Amy},
     TITLE = {Lumpings of algebraic {M}arkov chains arise from subquotients},
   JOURNAL = {J. Theoret. Probab.},
  FJOURNAL = {Journal of Theoretical Probability},
    VOLUME = {32},
      YEAR = {2019},
    NUMBER = {4},
     PAGES = {1804--1844},
      ISSN = {0894-9840,1572-9230},
   MRCLASS = {60J10 (05E05 16T30)},
  MRNUMBER = {4020688},
       DOI = {10.1007/s10959-018-0834-0},
}

@article{Bertini1997StochasticBA,
    AUTHOR = {Bertini, Lorenzo and Giacomin, Giambattista},
     TITLE = {Stochastic {B}urgers and {KPZ} equations from particle
              systems},
   JOURNAL = {Comm. Math. Phys.},
  FJOURNAL = {Communications in Mathematical Physics},
    VOLUME = {183},
      YEAR = {1997},
    NUMBER = {3},
     PAGES = {571--607},
      ISSN = {0010-3616,1432-0916},
   MRCLASS = {60K35 (60H15 82C22 82C24)},
  MRNUMBER = {1462228},
MRREVIEWER = {Ellen\ Saada},
       DOI = {10.1007/s002200050044},
}

@article{Corwin2016OpenAI,
    AUTHOR = {Corwin, Ivan and Shen, Hao},
     TITLE = {Open {ASEP} in the weakly asymmetric regime},
   JOURNAL = {Comm. Pure Appl. Math.},
  FJOURNAL = {Communications on Pure and Applied Mathematics},
    VOLUME = {71},
      YEAR = {2018},
    NUMBER = {10},
     PAGES = {2065--2128},
      ISSN = {0010-3640,1097-0312},
   MRCLASS = {82C70},
  MRNUMBER = {3861074},
       DOI = {10.1002/cpa.21744},
}

@article{Spohn1991LargeSD,
    AUTHOR = {Spohn, Herbert},
     TITLE = {Exact solutions for {KPZ}-type growth processes, random
              matrices, and equilibrium shapes of crystals},
   JOURNAL = {Phys. A},
  FJOURNAL = {Physica A. Statistical Mechanics and its Applications},
    VOLUME = {369},
      YEAR = {2006},
    NUMBER = {1},
     PAGES = {71--99},
      ISSN = {0378-4371},
   MRCLASS = {82C31 (82-02 82B31 82D25)},
  MRNUMBER = {2246567},
MRREVIEWER = {Rapha\"{e}l\ Cerf},
       DOI = {10.1016/j.physa.2006.04.006},
}

@article{Blythe2004DyckPM,
    AUTHOR = {Blythe, R. A. and Janke, W. and Johnston, D. A. and Kenna, R.},
     TITLE = {The grand-canonical asymmetric exclusion process and the
              one-transit walk},
   JOURNAL = {J. Stat. Mech. Theory Exp.},
  FJOURNAL = {Journal of Statistical Mechanics: Theory and Experiment},
      YEAR = {2004},
    NUMBER = {6},
     PAGES = {001, 10},
      ISSN = {1742-5468},
   MRCLASS = {82C22},
  MRNUMBER = {2115122},
MRREVIEWER = {Herv\'{e}\ Jean F. Guiol},
}

@article{Duchi2004ACA,
    AUTHOR = {Duchi, Enrica and Schaeffer, Gilles},
     TITLE = {A combinatorial approach to jumping particles: the parallel
              {TASEP}},
   JOURNAL = {Random Structures Algorithms},
  FJOURNAL = {Random Structures \& Algorithms},
    VOLUME = {33},
      YEAR = {2008},
    NUMBER = {4},
     PAGES = {434--451},
      ISSN = {1042-9832,1098-2418},
   MRCLASS = {05A15 (60C05 60J10)},
  MRNUMBER = {2462250},
MRREVIEWER = {E.\ J.\ Janse van Rensburg},
       DOI = {10.1002/rsa.20229},
}

@article{Brak2012SimpleAE,
    AUTHOR = {Brak, R. and Essam, J. W.},
     TITLE = {Simple asymmetric exclusion model and lattice paths:
              bijections and involutions},
   JOURNAL = {J. Phys. A},
  FJOURNAL = {Journal of Physics. A. Mathematical and Theoretical},
    VOLUME = {45},
      YEAR = {2012},
    NUMBER = {49},
     PAGES = {494007, 22},
      ISSN = {1751-8113,1751-8121},
   MRCLASS = {82B41 (60K35)},
  MRNUMBER = {2999725},
MRREVIEWER = {Stuart\ G.\ Whittington},
       DOI = {10.1088/1751-8113/45/49/494007},
}

@article{Ayyer2021CombinatoricsOA,
    AUTHOR = {Ayyer, Arvind and Nadeau, Philippe},
     TITLE = {Combinatorics of a disordered two-species {ASEP} on a torus},
   JOURNAL = {European J. Combin.},
  FJOURNAL = {European Journal of Combinatorics},
    VOLUME = {103},
      YEAR = {2022},
     PAGES = {Paper No. 103511, 20},
      ISSN = {0195-6698,1095-9971},
   MRCLASS = {82C44 (05A15 05A19)},
  MRNUMBER = {4379818},
       DOI = {10.1016/j.ejc.2022.103511},
}

@article{Kardar1986DynamicSO,
  title = {Dynamic Scaling of Growing Interfaces},
  author = {Kardar, Mehran and Parisi, Giorgio and Zhang, Yi-Cheng},
  journal = {Phys. Rev. Lett.},
  volume = {56},
  issue = {9},
  pages = {889--892},
  numpages = {0},
  year = {1986},
  month = {Mar},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevLett.56.889},
}


@book{stanley_1999,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Enumerative combinatorics. {V}ol. 2},
    SERIES = {Cambridge Studies in Advanced Mathematics},
    VOLUME = {208},
   EDITION = {Second},
      NOTE = {With an appendix by Sergey Fomin},
 PUBLISHER = {Cambridge University Press, Cambridge},
      YEAR = {[2024] \copyright 2024},
     PAGES = {xvi+783},
      ISBN = {978-1-009-26249-1; 978-1-009-26248-4},
   MRCLASS = {05-02 (05A15 05E05 05E10)},
  MRNUMBER = {4621625},
    ANNOTE = {For {V}olume {I} see [2868112\endremfr]},
}

@incollection{Ferrari2010RandomGM,
    AUTHOR = {Ferrari, P. L. and Spohn, H.},
     TITLE = {Random growth models},
 BOOKTITLE = {The {O}xford handbook of random matrix theory},
     PAGES = {782--801},
 PUBLISHER = {Oxford Univ. Press, Oxford},
      YEAR = {2011},
      ISBN = {978-0-19-957400-1},
   MRCLASS = {60K35 (60B20)},
  MRNUMBER = {2932658},
MRREVIEWER = {Tomohiro\ Sasamoto},
}

@misc{oeis,
  author={Sloane, Neil J. A. and The OEIS Foundation Inc.},
  title={The on-line encyclopedia of integer sequences},
  year={2020},
  url={http://oeis.org/?language=english},
}

@article {ProppRoby,
    AUTHOR = {Propp, James and Roby, Tom},
     TITLE = {Homomesy in products of two chains},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {22},
      YEAR = {2015},
    NUMBER = {3},
     PAGES = {Paper 3.4, 29},
      ISSN = {1077-8926},
   MRCLASS = {05E18 (06A07 06A11)},
  MRNUMBER = {3367853},
MRREVIEWER = {Konrad\ P.\ Pi\'{o}ro},
       DOI = {10.37236/3579},
}

@article {striker2015,
    AUTHOR = {Striker, Jessica},
     TITLE = {The toggle group, homomesy, and the {R}azumov-{S}troganov
              correspondence},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {22},
      YEAR = {2015},
    NUMBER = {2},
     PAGES = {Paper 2.57, 17},
      ISSN = {1077-8926},
   MRCLASS = {05E18 (06A06 82B05)},
  MRNUMBER = {3367300},
       DOI = {10.37236/5158},
}

@article {noncrossingtoggle,
    AUTHOR = {Einstein, David and Farber, Miriam and Gunawan, Emily and
              Joseph, Michael and Macauley, Matthew and Propp, James and
              Rubinstein-Salzedo, Simon},
     TITLE = {Noncrossing partitions, toggles, and homomesies},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {23},
      YEAR = {2016},
    NUMBER = {3},
     PAGES = {Paper 3.52, 26},
      ISSN = {1077-8926},
   MRCLASS = {05E18 (05A17 06A07 20F55)},
  MRNUMBER = {3576300},
MRREVIEWER = {Jan\ \'{S}liwa},
       DOI = {10.37236/5648},
}

@incollection {roby2016,
    AUTHOR = {Roby, Tom},
     TITLE = {Dynamical algebraic combinatorics and the homomesy phenomenon},
 BOOKTITLE = {Recent trends in combinatorics},
    SERIES = {IMA Vol. Math. Appl.},
    VOLUME = {159},
     PAGES = {619--652},
 PUBLISHER = {Springer, [Cham]},
      YEAR = {2016},
      ISBN = {978-3-319-24296-5; 978-3-319-24298-9},
   MRCLASS = {05-02 (05E18 06A07)},
  MRNUMBER = {3526426},
       DOI = {10.1007/978-3-319-24298-9\_25},
}

@article {dmitri,
    AUTHOR = {Panyushev, Dmitri I.},
     TITLE = {On orbits of antichains of positive roots},
   JOURNAL = {European J. Combin.},
  FJOURNAL = {European Journal of Combinatorics},
    VOLUME = {30},
      YEAR = {2009},
    NUMBER = {2},
     PAGES = {586--594},
      ISSN = {0195-6698,1095-9971},
   MRCLASS = {05E18 (17B20)},
  MRNUMBER = {2489252},
       DOI = {10.1016/j.ejc.2008.03.009},
}

@article {nestcross,
    AUTHOR = {Armstrong, Drew and Stump, Christian and Thomas, Hugh},
     TITLE = {A uniform bijection between nonnesting and noncrossing
              partitions},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {365},
      YEAR = {2013},
    NUMBER = {8},
     PAGES = {4121--4151},
      ISSN = {0002-9947,1088-6850},
   MRCLASS = {05A05 (05A19 20F55)},
  MRNUMBER = {3055691},
MRREVIEWER = {Alessandro\ Conflitti},
       DOI = {10.1090/S0002-9947-2013-05729-7},
}

@article {salez2023,
    AUTHOR = {Salez, Justin},
     TITLE = {Universality of cutoff for exclusion with reservoirs},
   JOURNAL = {Ann. Probab.},
  FJOURNAL = {The Annals of Probability},
    VOLUME = {51},
      YEAR = {2023},
    NUMBER = {2},
     PAGES = {478--494},
      ISSN = {0091-1798,2168-894X},
   MRCLASS = {60J27 (60K35 82C22)},
  MRNUMBER = {4546624},
MRREVIEWER = {David\ J.\ Aldous},
       DOI = {10.1214/22-aop1600},
       URL = {https://doi.org/10.1214/22-aop1600},
}

@article {FerrariMartin,
    AUTHOR = {Ferrari, Pablo A. and Martin, James B.},
     TITLE = {Stationary distributions of multi-type totally asymmetric
              exclusion processes},
   JOURNAL = {Ann. Probab.},
  FJOURNAL = {The Annals of Probability},
    VOLUME = {35},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {807--832},
      ISSN = {0091-1798,2168-894X},
   MRCLASS = {60K35 (82C22 90B22)},
  MRNUMBER = {2319708},
MRREVIEWER = {C\'{e}dric\ Bernardin},
       DOI = {10.1214/009117906000000944},
       URL = {https://doi.org/10.1214/009117906000000944},
}