% BibTeX Entries


@article{fukuda2000energy,
  title={Energy functions in box ball systems},
  author={Fukuda, K. and Yamada, Y. and Okado, M.},
  journal={International Journal of Modern Physics A},
  volume={15},
  number={09},
  pages={1379--1392},
  year={2000},
  publisher={World Scientific}
}

@article{fukuda04,
  title={Box-ball systems and Robinson-Schensted-Knuth correspondence},
  author={Fukuda, K.},
  journal={Journal of Algebraic Combinatorics},
  volume={19},
  number={1},
  pages={67--89},
  year={2004},
  publisher={Springer}
}

@article {Gre74,
    AUTHOR = {Greene, C.},
     TITLE = {An extension of {S}chensted's theorem},
   JOURNAL = {Advances in Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {14},
      YEAR = {1974},
     PAGES = {254--265},
   MRCLASS = {05A17},
  MRNUMBER = {354395},
MRREVIEWER = {A. O. Morris}
}

@article {GK76,
    AUTHOR = {Greene, Curtis and Kleitman, Daniel J.},
     TITLE = {The structure of {S}perner {$k$}-families},
   JOURNAL = {J. Combinatorial Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {20},
      YEAR = {1976},
    NUMBER = {1},
     PAGES = {41--68},
      ISSN = {0097-3165},
   MRCLASS = {05A05 (06A20)},
  MRNUMBER = {398844},
MRREVIEWER = {Ralph Freese},
       DOI = {10.1016/0097-3165(76)90077-7},
       URL = {https://doi-org.ezproxy.lib.uconn.edu/10.1016/0097-3165(76)90077-7},
}



@article{ILP16,
    AUTHOR = {Inoue, Rei and Lam, Thomas and Pylyavskyy, Pavlo},
     TITLE = {On the cluster nature and quantization of geometric {$R$}-matrices},
   JOURNAL = {Publ. Res. Inst. Math. Sci.},
  FJOURNAL = {Publications of the Research Institute for Mathematical Sciences},
    VOLUME = {55},
      YEAR = {2019},
    NUMBER = {1},
     PAGES = {25--78},
      ISSN = {0034-5318},
   MRCLASS = {13F60 (81R12)},
  MRNUMBER = {3898323},
       DOI = {10.4171/PRIMS/55-1-2},
}

@book {JK81,
    AUTHOR = {James, Gordon and Kerber, Adalbert},
     TITLE = {The representation theory of the symmetric group},
    SERIES = {Encyclopedia of Mathematics and its Applications},
    VOLUME = {16},
      NOTE = {With a foreword by P. M. Cohn,
              With an introduction by Gilbert de B. Robinson},
 PUBLISHER = {Addison-Wesley Publishing Co., Reading, Mass.},
      YEAR = {1981},
     PAGES = {xxviii+510},
      ISBN = {0-201-13515-9},
   MRCLASS = {20-02 (20C30)},
  MRNUMBER = {644144},
MRREVIEWER = {A. O. Morris},
}


@article {Knuth70,
    AUTHOR = {Knuth, D. E.},
     TITLE = {Permutations, matrices, and generalized {Y}oung tableaux},
   JOURNAL = {Pacific J. Math.},
  FJOURNAL = {Pacific Journal of Mathematics},
    VOLUME = {34},
      YEAR = {1970},
     PAGES = {709--727},
   MRCLASS = {05.30},
  MRNUMBER = {272654},
MRREVIEWER = {M. Doob}
}


 @misc{Lang02OEIS, 
 title={{The On-Line Encyclopedia of Integer Sequences}},
 howpublished = "\url{https://oeis.org/A077415}", journal={OEIS}, publisher={OEIS}, author={}, year={}, month={}, note = "Number of standard tableaux of shape (n-1,2,1) [Online; accessed 14-April-2021]"} 


@misc{Lewis20Blog,
    author = {Lewis, J.},
    title = {A localized version of Greene’s theorem},
    url = {https://realopacblog.wordpress.com/2019/11/24/a-localized-version-of-greenes-theorem/},
    note="[Online; accessed 14-April-2021]"
  }


@misc{LLPS19,
  title={Scaling limit of soliton lengths in a multicolor box-ball system},
  author={Lewis, J. and Lyu, H. and Pylyavskyy, P. and Sen, A.},
  EPRINTTYPE = {arxiv},
  Eprint = {1911.04458},
  year={2019}
}

@book{marshall2007number,
  title={Number Theory Through Inquiry},
  author={Marshall, D.C. and Odell, E. and Starbird, M. and MAA, The Mathematical Association of America },
  isbn={9780883857519},
  lccn={2007938223},
  series={MAA textbooks},
  url={https://books.google.com/books?id=9\_x81PuCs0MC},
  year={2007},
  publisher={Mathematical Association of America}
}


@book {Sag01,
    AUTHOR = {Sagan, Bruce E.},
     TITLE = {The symmetric group},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {203},
   EDITION = {Second},
      NOTE = {Representations, combinatorial algorithms, and symmetric
              functions},
 PUBLISHER = {Springer-Verlag, New York},
      YEAR = {2001},
     PAGES = {xvi+238},
      ISBN = {0-387-95067-2},
   MRCLASS = {05E10 (05E05 20C30)},
  MRNUMBER = {1824028},
       DOI = {10.1007/978-1-4757-6804-6},
       URL = {https://doi-org.ezproxy.lib.uconn.edu/10.1007/978-1-4757-6804-6},
}
	

@manual{sage,
 author               = {The Sage Developers},
 key                  = {Sage},
 url                  = {https://www.sagemath.org},
 organization         = {The Sage Development Team},
 title                = {{S}age {M}athematics {S}oftware ({V}ersion 9.1)},
 year                 = {2020},
 }


@article {Sch61,
    AUTHOR = {Schensted, C.},
     TITLE = {Longest increasing and decreasing subsequences},
   JOURNAL = {Canadian J. Math.},
  FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de
              Math\'{e}matiques},
    VOLUME = {13},
      YEAR = {1961},
     PAGES = {179--191},
   MRCLASS = {05.00},
  MRNUMBER = {121305},
MRREVIEWER = {D. E. Rutherford}
}


@book {Sta18AlgebraicCombinatorics2ed,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Algebraic combinatorics},
    SERIES = {Undergraduate Texts in Mathematics},
      NOTE = {Walks, trees, tableaux, and more,
              Second edition of [ MR3097651]},
 PUBLISHER = {Springer, Cham},
      YEAR = {2018},
     PAGES = {xvi+263},
      ISBN = {978-3-319-77172-4; 978-3-319-77173-1},
   MRCLASS = {05-01 (05Axx 05Exx)},
  MRNUMBER = {3823204},
       DOI = {10.1007/978-3-319-77173-1},
       URL = {https://doi-org.ezproxy.lib.uconn.edu/10.1007/978-3-319-77173-1},
}
		
@book {Sta15Catalan,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Catalan numbers},
 PUBLISHER = {Cambridge University Press, New York},
      YEAR = {2015},
     PAGES = {viii+215},
      ISBN = {978-1-107-42774-7; 978-1-107-07509-2},
   MRCLASS = {05-01 (01A05 11B75 11B83)},
  MRNUMBER = {3467982},
MRREVIEWER = {David Callan},
       DOI = {10.1017/CBO9781139871495},
       URL = {https://doi-org.ezproxy.lib.uconn.edu/10.1017/CBO9781139871495},
}
		
@book {Sta12EC1seconded,
    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},
}
		
@book {Sta99EC2,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Enumerative combinatorics. {V}ol. 2},
    SERIES = {Cambridge Studies in Advanced Mathematics},
    VOLUME = {62},
      NOTE = {With a foreword by Gian-Carlo Rota and appendix 1 by Sergey
              Fomin},
 PUBLISHER = {Cambridge University Press, Cambridge},
      YEAR = {1999},
     PAGES = {xii+581},
      ISBN = {0-521-56069-1; 0-521-78987-7},
   MRCLASS = {05A15 (05-02 05E05 05E10 68R05)},
  MRNUMBER = {1676282},
MRREVIEWER = {Ira Gessel},
       DOI = {10.1017/CBO9780511609589},
       URL = {https://doi-org.ezproxy.lib.uconn.edu/10.1017/CBO9780511609589},
}

@inproceedings{Takahashi93,
  title={On some soliton systems defined by using boxes and balls},
  author={Takahashi, D.},
  booktitle={Proceedings of the international symposium on nonlinear theory and its applications (NOLTA’93)},
  pages={555--558},
  year={1993}
}

@article{TM97carrier,
  title={Box and ball system with a carrier and ultradiscrete modified KdV equation},
  author={Takahashi, D. and Matsukidaira, J.},
  journal={Journal of Physics A: Mathematical and General},
  volume={30},
  number={21},
  pages={L733},
  year={1997},
  publisher={IOP Publishing}
}

@article {TS90,
    AUTHOR = {Takahashi, D. and Satsuma, J.},
     TITLE = {A soliton cellular automaton},
   JOURNAL = {J. Phys. Soc. Japan},
  FJOURNAL = {Journal of the Physical Society of Japan},
    VOLUME = {59},
      YEAR = {1990},
    NUMBER = {10},
     PAGES = {3514--3519},
      ISSN = {0031-9015},
     CODEN = {JUPSAU},
   MRCLASS = {58F08 (68Q80)},
  MRNUMBER = {1082435},
       DOI = {10.1143/JPSJ.59.3514},
}