@article {OSSZ, AUTHOR = {Orellana, R. and Saliola, F. and Schilling, A. and Zabrocki, M.}, TITLE = {From {Q}uasi-{S}ymmetric to {S}chur {E}xpansions with {A}pplications to {S}ymmetric {C}hain {D}ecompositions and {P}lethysm}, JOURNAL = {Electron. J. Combin.}, FJOURNAL = {Electronic Journal of Combinatorics}, VOLUME = {31}, YEAR = {2024}, NUMBER = {4}, MRCLASS = {99-06}, MRNUMBER = {4816502}, DOI = {10.37236/12950}, URL = {https://doi.org/10.37236/12950}, } @article {Stanley, AUTHOR = {Stanley, R.}, TITLE = {Weyl groups, the hard {L}efschetz theorem, and the {S}perner property}, JOURNAL = {SIAM J. Algebraic Discrete Methods}, FJOURNAL = {Society for Industrial and Applied Mathematics. Journal on Algebraic and Discrete Methods}, VOLUME = {1}, YEAR = {1980}, NUMBER = {2}, PAGES = {168--184}, MRCLASS = {20G05 (05A05 06A10 14M17)}, MRNUMBER = {578321}, DOI = {10.1137/0601021}, URL = {https://doi.org/10.1137/0601021}, } @misc{CDGK, title={A visual approach to symmetric chain decompositions of finite Young lattices}, author={Coggins, T. and {Donley Jr.}, R. and Gondal, A. and Krishna, A.}, year={2024}, eprint={2407.20008}, archivePrefix={arXiv}, primaryClass={math.CO}, } @misc{DST, title={Geometric Symmetric Chain Decompositions}, author={David, S. and Spink, H. and Tiba, M.}, year={2017}, eprint={1703.10954}, archivePrefix={arXiv}, primaryClass={math.CO} } @article {Greene, AUTHOR = {Greene, J.}, TITLE = {Greedy matching in {Y}oung's lattice}, JOURNAL = {Order}, FJOURNAL = {Order. A Journal on the Theory of Ordered Sets and its Applications}, VOLUME = {6}, YEAR = {1990}, NUMBER = {4}, PAGES = {351--366}, MRCLASS = {06A07 (05E25)}, MRNUMBER = {1063818}, MRREVIEWER = {M.\ Aigner}, DOI = {10.1007/BF00346131}, URL = {https://doi.org/10.1007/BF00346131}, } @article {Lindstrom, AUTHOR = {Lindstr\"om, B.}, TITLE = {A partition of {$L(3,\,n)$}\ into saturated symmetric chains}, JOURNAL = {European J. Combin.}, FJOURNAL = {European Journal of Combinatorics}, VOLUME = {1}, YEAR = {1980}, NUMBER = {1}, PAGES = {61--63}, MRCLASS = {06A10}, MRNUMBER = {576767}, MRREVIEWER = {E.\ Rodney\ Canfield}, DOI = {10.1016/S0195-6698(80)80022-9}, URL = {https://doi.org/10.1016/S0195-6698(80)80022-9}, } @misc{Xin-Zhong, title={An explicit order matching for $L(3,n)$ from several approaches and its extension for $L(4,n)$}, author={Xin, G. and Zhong, Y.}, year={2021}, eprint={2104.11003}, archivePrefix={arXiv}, primaryClass={math.CO}, } @article {Wen, AUTHOR = {Wen, X.}, TITLE = {Computer-generated symmetric chain decompositions for {$L(4,n)$} and {$L(3,n)$}}, JOURNAL = {Adv. in Appl. Math.}, FJOURNAL = {Advances in Applied Mathematics}, VOLUME = {33}, YEAR = {2004}, NUMBER = {2}, PAGES = {409--412}, MRCLASS = {05A17 (05E10 06A07)}, MRNUMBER = {2074405}, DOI = {10.1016/j.aam.2003.11.003}, URL = {https://doi.org/10.1016/j.aam.2003.11.003}, } @article {Wen2, AUTHOR = {Wen, X.}, TITLE = {A symmetric chain decomposition of {$L(5,n)$}}, JOURNAL = {Enumer. Comb. Appl.}, FJOURNAL = {Enumerative Combinatorics and Applications}, VOLUME = {4}, YEAR = {2024}, NUMBER = {1}, MRCLASS = {06A07 (06B05)}, MRNUMBER = {4632113}, MRREVIEWER = {Keith\ A.\ Kearnes}, DOI = {10.54550/eca2024v4s1r5}, URL = {https://doi.org/10.54550/eca2024v4s1r5}, } @article {West, AUTHOR = {West, D.}, TITLE = {A symmetric chain decomposition of {$L(4,\,n)$}}, JOURNAL = {European J. Combin.}, FJOURNAL = {European Journal of Combinatorics}, VOLUME = {1}, YEAR = {1980}, NUMBER = {4}, PAGES = {379--383}, MRCLASS = {06A10 (05B99)}, MRNUMBER = {595939}, MRREVIEWER = {R.\ W.\ Quackenbush}, DOI = {10.1016/S0195-6698(80)80040-0}, URL = {https://doi.org/10.1016/S0195-6698(80)80040-0}, } @article {Riess, AUTHOR = {Rie\ss, W.}, TITLE = {Zwei {O}ptimierungsprobleme auf {O}rdnungen}, JOURNAL = {Arbeitsber. Inst. Math. Masch. Datenverarb. (Inform.)}, FJOURNAL = {Institut f\"ur Mathematische Maschinen und Datenverarbeitung (Informatik). Arbeitsberichte}, VOLUME = {11}, YEAR = {1978}, NUMBER = {5}, MRCLASS = {52A45 (52A40)}, MRNUMBER = {580530}, } @book {BS, AUTHOR = {Bump, D. and Schilling, A.}, TITLE = {Crystal bases}, NOTE = {Representations and combinatorics}, PUBLISHER = {World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ}, YEAR = {2017}, MRCLASS = {05-01 (05E10 14T05 17B10)}, MRNUMBER = {3642318}, DOI = {10.1142/9876}, URL = {https://doi.org/10.1142/9876}, } @article{Dhand, author = {Dhand, V.}, title = {Tropical decomposition of {Y}oung’s partition lattice}, journal = {J. Algebr. Comb.}, volume = {39}, year = {2014}, doi = {10.1007/s10801-013-0467-4} } @misc{Treteault, title={Recurrence relation for plethysm}, author={T\'{e}treault, {\'{E}}.}, year={2020}, eprint={2008.13070}, archivePrefix={arXiv}, primaryClass={math.CO}, } @article{StanleySl2, author = {Stanley, R.}, title = {Unimodality and {L}ie Superalgebras}, journal = {Studies in Applied Mathematics}, volume = {72}, number = {3}, pages = {263-281}, doi = {10.1002/sapm1985723263}, year = {1985} } @incollection{AF, AUTHOR = {Almkvist, Gert and Fossum, Robert}, TITLE = {Decomposition of exterior and symmetric powers of indecomposable {${\bf Z}/p{\bf Z}$}-modules in characteristic {$p$}\ and relations to invariants}, BOOKTITLE = {S\'eminaire d'{A}lg\`ebre {P}aul {D}ubreil, 30\`eme ann\'ee}, SERIES = {Lecture Notes in Math.}, VOLUME = {641}, PUBLISHER = {Springer Berlin}, YEAR = {1978}, MRCLASS = {14L30 (13H10 16A26 20C05)}, MRNUMBER = {499459}, MRREVIEWER = {H.\ R\"ohrl}, } @book{Gordan, author = {Gordan, P.}, title = {Vorlesungen ber Invariantentheorie}, volume = {II}, editor = {G. Kerschensteiner}, publisher = {Leipzig, Teubner}, year = {1887} } @incollection{mystery, AUTHOR = {Colmenarejo, Laura and Orellana, Rosa and Saliola, Franco and Schilling, Anne and Zabrocki, Mike}, TITLE = {The mystery of plethysm coefficients}, BOOKTITLE = {Open problems in algebraic combinatorics}, SERIES = {Proc. Sympos. Pure Math.}, VOLUME = {110}, PAGES = {275--292}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {2024}, doi = {10.1090/pspum/110} } @article{Kashiwara, AUTHOR = {Kashiwara, Masaki}, TITLE = {Crystalizing the {$q$}-analogue of universal enveloping algebras}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {133}, YEAR = {1990}, NUMBER = {2}, PAGES = {249--260}, doi = {10.1007/BF02097367} } @incollection{StanleyList, AUTHOR = {Stanley, Richard P.}, TITLE = {Positivity problems and conjectures in algebraic combinatorics}, BOOKTITLE = {Mathematics: frontiers and perspectives}, PAGES = {295--319}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {2000}, } @incollection{Pak, AUTHOR = {Pak, Igor}, TITLE = {What is a combinatorial interpretation?}, BOOKTITLE = {Open problems in algebraic combinatorics}, SERIES = {Proc. Sympos. Pure Math.}, VOLUME = {110}, PAGES = {191--260}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {2024}, doi = {10.1090/pspum/110} } @book {EC2, AUTHOR = {Stanley, R.}, TITLE = {Enumerative combinatorics. {V}ol. 2}, SERIES = {Cambridge Studies in Advanced Mathematics}, VOLUME = {62}, 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} } @manual{sage, Key = {SageMath}, Author = {{Sage Developers}, The}, Title = {{S}ageMath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 9.5)}, url = {https://www.sagemath.org}, Year = {2022}, } @misc{Gut, title={Towards plethystic $\mathfrak{sl}_2$ crystals}, author={Gutiérrez, {\'A}lvaro}, year={2024}, eprint={2412.15006}, archivePrefix={arXiv}, primaryClass={math.CO} }