@article{Greene, AUTHOR = {Greene, Curtis}, TITLE = {A rational-function identity related to the {M}urnaghan-{N}akayama formula for the characters of {$S_n$}}, JOURNAL = {J. Algebraic Combin.}, FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal}, VOLUME = {1}, YEAR = {1992}, NUMBER = {3}, PAGES = {235--255}, DOI = {10.1023/A:1022435901373}, URL = {https://doi.org/10.1023/A:1022435901373}, } @article{toric, AUTHOR = {Develin, Mike and Macauley, Matthew and Reiner, Victor}, TITLE = {Toric partial orders}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {368}, YEAR = {2016}, NUMBER = {4}, PAGES = {2263--2287}, DOI = {10.1090/tran/6356}, URL = {https://doi.org/10.1090/tran/6356}, } @article{Cones, AUTHOR = {Boussicault, Adrien and F\'eray, Valentin and Lascoux, Alain and Reiner, Victor}, TITLE = {Linear extension sums as valuations on cones}, JOURNAL = {J. Algebraic Combin.}, FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal}, VOLUME = {35}, YEAR = {2012}, NUMBER = {4}, PAGES = {573--610}, DOI = {10.1007/s10801-011-0316-2}, URL = {https://doi.org/10.1007/s10801-011-0316-2}, } @article{linear, title={Counting linear extensions}, author={Brightwell, G. and Winkler, P.}, journal={Order}, volume={8}, number={3}, pages={225--242}, year={1991}, publisher={Springer} } @article{Parke, author = {Parke, S. and Taylor, T. R.}, title = {Amplitude for $n$-Gluon Scattering}, journal = {Phys. Rev. Lett.}, volume = {56}, number = {23}, pages = {2459--2460}, year = {1986}, month = {Jun}, publisher = {American Physical Society}, doi = {10.1103/PhysRevLett.56.2459}, } @misc{Propp, author={Propp, J.}, eprint ={math/0209005}, eprinttype ={arxiv}, title={Lattice structure for orientations of graphs}, year={2002}} @misc{galashin2021poset, author={Galashin, P.}, eprint ={2110.07257}, eprinttype = {arxiv}, title={Poset associahedra}, year={2021}} @misc{early2018honeycomb, author={Early, N.}, epint = {1810.03246}, eprinttype = {arxiv}, title={Honeycomb tessellations and canonical bases for permutohedral blades}, year={2018}} @book{commalg, author={Miller, E. and Sturmfels, B.}, title={Combinatorial commutative algebra}, volume={227}, year={2005}, publisher={Springer}} @inproceedings{Barvinok, author={Barvinok, A. I.}, title={Computing the volume, counting integral points, and exponential sums}, booktitle={Proceedings of the eighth annual symposium on Computational geometry}, pages={161--170}, year={1992}} @article{Berline, author={Berline, N. and Vergne, M.}, title={Local Euler-Maclaurin formula for polytopes}, journal={arXiv preprint math/0507256}, year={2005}} @article{stanleyacyclic, AUTHOR = {Stanley, Richard P.}, TITLE = {Acyclic orientations of graphs}, JOURNAL = {Discrete Math.}, FJOURNAL = {Discrete Mathematics}, VOLUME = {5}, YEAR = {1973}, PAGES = {171--178}, DOI = {10.1016/0012-365X(73)90108-8}, URL = {https://doi.org/10.1016/0012-365X(73)90108-8}, } @article{Zaslavsky, AUTHOR = {Greene, Curtis and Zaslavsky, Thomas}, TITLE = {On the interpretation of {W}hitney numbers through arrangements of hyperplanes, zonotopes, non-{R}adon partitions, and orientations of graphs}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {280}, YEAR = {1983}, NUMBER = {1}, PAGES = {97--126}, DOI = {10.2307/1999604}, URL = {https://doi.org/10.2307/1999604}, } @article{postnikovfaces, AUTHOR = {Postnikov, Alex and Reiner, Victor and Williams, Lauren}, TITLE = {Faces of generalized permutohedra}, JOURNAL = {Doc. Math.}, FJOURNAL = {Documenta Mathematica}, VOLUME = {13}, YEAR = {2008}, PAGES = {207--273}, DOI = {10.4171/DM/248} } @article{pretzel, AUTHOR = {Pretzel, Oliver}, TITLE = {On reorienting graphs by pushing down maximal vertices}, JOURNAL = {Order}, FJOURNAL = {Order}, VOLUME = {3}, YEAR = {1986}, NUMBER = {2}, PAGES = {135--153}, DOI = {10.1007/BF00390104}, URL = {https://doi.org/10.1007/BF00390104}, } @article{mosesian, AUTHOR = {Mosesian, K. M.}, TITLE = {Strongly basable graphs}, JOURNAL = {Akad. Nauk Armjan. SSR Dokl.}, FJOURNAL = {Akademija Nauk Armjansko\u{\i} SSR. Doklady}, VOLUME = {54}, YEAR = {1972}, PAGES = {134--138}, MRCLASS = {05C20}, MRNUMBER = {327560}, MRREVIEWER = {J.\ Bos\'{a}k},} @article{chen, AUTHOR = {Chen, Beifang}, TITLE = {Orientations, lattice polytopes, and group arrangements. {I}. {C}hromatic and tension polynomials of graphs}, JOURNAL = {Ann. Comb.}, FJOURNAL = {Annals of Combinatorics}, VOLUME = {13}, YEAR = {2010}, NUMBER = {4}, PAGES = {425--452}, DOI = {10.1007/s00026-009-0037-6}, URL = {https://doi.org/10.1007/s00026-009-0037-6}, } @article{eriksson, AUTHOR = {Eriksson, Henrik and Eriksson, Kimmo}, TITLE = {Conjugacy of {C}oxeter elements}, JOURNAL = {Electron. J. Combin.}, FJOURNAL = {Electronic Journal of Combinatorics}, VOLUME = {16}, YEAR = {2009}, NUMBER = {2}, PAGES = {Research Paper 4, 7 pp.}, DOI = {10.37236/70}, URL = {https://doi.org/10.37236/70}, } @article{macauley2009posets, AUTHOR = {Macauley, Matthew and Mortveit, Henning S.}, TITLE = {Posets from admissible {C}oxeter sequences}, JOURNAL = {Electron. J. Combin.}, FJOURNAL = {Electronic Journal of Combinatorics}, VOLUME = {18}, YEAR = {2011}, NUMBER = {1}, PAGES = {Paper 197, 18 pp.}, DOI = {10.37236/684}, URL = {https://doi.org/10.37236/684}, } @article{speyer, AUTHOR = {Speyer, David E.}, TITLE = {Powers of {C}oxeter elements in infinite groups are reduced}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {137}, YEAR = {2009}, NUMBER = {4}, PAGES = {1295--1302}, DOI = {10.1090/S0002-9939-08-09638-X}, URL = {https://doi.org/10.1090/S0002-9939-08-09638-X}, } @article{clusters, author={Fomin, S. and Shapiro, M. and Thurston, D.}, title={Cluster algebras and triangulated surfaces. Part I: Cluster complexes}, journal ={Acta. math}, year={2008}} @article{robins, author={Robins, S.}, title={The integer point transform as a complete invariant}, journal={Communications in Mathematics}, volume={13}, year={2023}} @article{meszaros, author={Meszaros, K.}, title={Calculating Greene’s function via root polytopes and subdivision algebras}, journal={Pacific Journal of Mathematics}, volume={286}, number={2}, pages={385--400}, year={2017}, publisher={Mathematical Sciences Publishers}} @article{bernstein, author={Bernstein, I.N. and Gel{\textquotesingle}fand, I. and Ponomarev, V.A.}, title={Coxeter functors and Gabriel's theorem}, journal={Russian Mathematical Surveys}, volume={28}, year={1973}, number={2(170)}, pages = {17--32}, doi = {10.1070/RM1973v028n02ABEH001526}, } @book{kirillov, author={Kirillov Jr, A.}, title={Quiver representations and quiver varieties}, volume={174}, year={2016}, publisher={American Mathematical Soc.}} @book{gelfandmanin, title={Methods of homological algebra}, author={Gelfand, S. I. and Manin, Y. I.}, year={2013}, publisher={Springer Science \& Business Media}} @article{Kleiss, author = {Kleiss, R. and Kuijf, H.}, title = {Multigluon cross sections and 5-jet production at hadron colliders}, journal = {Nuclear Physics B}, volume = {312}, number = {3}, pages = {616--644}, year = {1989}, doi = {10.1016/0550-3213(89)90574-9}, } @article{Shuffle, author = {Eilenberg, S. and Mac Lane, S.}, journal = {Annals of Mathematics}, number = {1}, pages = {55--106}, title = {On the Groups H($\Pi$, n), I}, volume = {58}, year = {1953}} @article{parketaylor, author={Parke, S. J. and Taylor, T. R.}, title={Amplitude for $n$-gluon scattering}, journal = {Phys. Rev. Lett.}, volume={56}, number={23}, year={1986}, publisher={APS}, doi = {10.1103/PhysRevLett.56.2459}, } @article{fomin2002cluster, AUTHOR = {Fomin, Sergey and Zelevinsky, Andrei}, TITLE = {Cluster algebras. {I}. {F}oundations}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {15}, YEAR = {2002}, NUMBER = {2}, PAGES = {497--529}, DOI = {10.1090/S0894-0347-01-00385-X}, } @article{robinsbook, title={A friendly introduction to Fourier analysis on polytopes}, author={Robins, Sinai}, journal={arXiv preprint arXiv:2104.06407}, year={2021}} @article{macauley2016morphisms, author={Macauley, M.}, title={Morphisms and order ideals of toric posets}, journal={Mathematics}, volume={4}, number={2}, note={39}, year={2016}, doi = {10.3390/math4020039}, } @article{megiddo1976partial, author={Megiddo, N.}, title={Partial and complete cyclic orders}, journal ={Bull. Amer. Math. Soc.}, volume = {82}, number = {3}, pages = {274-276}, year={1976}} @misc{Williams, author={Parisi, M. and Sherman-Bennett, M. and Tessler, R. and Williams, L.}, eprint = {2404.03026}, eprinttype = {arxiv}, title={The Magic Number Conjecture for the $m = 2$ amplituhedron and Parke-Taylor identities}, year={2024}} @article{fomin2, AUTHOR = {Fomin, Sergey and Zelevinsky, Andrei}, TITLE = {Cluster algebras. {II}. {F}inite type classification}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {154}, YEAR = {2003}, NUMBER = {1}, PAGES = {63--121}, DOI = {10.1007/s00222-003-0302-y}, URL = {https://doi.org/10.1007/s00222-003-0302-y}, } @article{greenekleitman, author={Greene, C. and Kleitman, D. J.}, title={The structure of Sperner k-families}, journal={Journal of Combinatorial Theory, Series A}, volume={20}, year={1976}, number={1}, pages={41--68}} @article{brightwell, author = {Brightwell, G. and Winkler, P.}, booktitle = {STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of computing}, pages = {175--181}, title = {Counting linear extensions is {\#}P-complete}, year = {1991}, doi = {10.1145/103418.103441}, } @article{pretzel2, author={Pretzel, O.}, title={On reorienting graphs by pushing down maximal vertices—II}, journal={Discrete mathematics}, volume={270}, year={2003}, number={3}, pages={227--240}} @article{defant, AUTHOR = {Defant, Colin and Kravitz, Noah}, TITLE = {Friends and strangers walking on graphs}, JOURNAL = {Comb. Theory}, FJOURNAL = {Combinatorial Theory}, VOLUME = {1}, YEAR = {2021}, PAGES = {Paper No. 6, 34 pp.}, DOI = {10.5070/C61055363}, URL = {https://doi.org/10.5070/C61055363}, } @inproceedings{caldero, title={From triangulated categories to cluster algebras II}, author={Caldero, Philippe and Keller, Bernhard}, booktitle={Annales scientifiques de l'Ecole normale sup{\'e}rieure}, volume={39}, number={6}, pages={983--1009}, year={2006}} @article{coleman, author={Coleman, A.J.}, title={Killing and the Coxeter transformation of Kac-Moody algebras}, journal={Inventiones mathematicae}, volume={95}, year={1989}, number={3}, pages={447--477}} @misc{catania2024toric, author = {Catania, E.}, Eprint ={2409.04907}, Eprinttype = {arxiv}, title={A Toric Analogue for Greene's Rational Function of a Poset}, year={2024}} @article{goldreich2008computational, title={Computational complexity: a conceptual perspective}, author={Goldreich, Oded}, journal={ACM Sigact News}, volume={39}, number={3}, pages={35--39}, year={2008}, publisher={ACM New York, NY, USA} } @article{bibby, AUTHOR = {Bibby, Christin and Delucchi, Emanuele}, TITLE = {Supersolvable posets and fiber-type abelian arrangements}, JOURNAL = {Selecta Math. (N.S.)}, FJOURNAL = {Selecta Mathematica. New Series}, VOLUME = {30}, YEAR = {2024}, NUMBER = {5}, PAGES = {Paper No. 89, 39 pp.}, DOI = {10.1007/s00029-024-00976-w}, URL = {https://doi.org/10.1007/s00029-024-00976-w}, } @article{aguiar, AUTHOR = {Aguiar, Marcelo and Chan, Swee Hong}, TITLE = {Toric arrangements associated to graphs}, JOURNAL = {S\'em. Lothar. Combin.}, FJOURNAL = {S\'eminaire Lotharingien de Combinatoire}, VOLUME = {78B}, YEAR = {2017}, PAGES = {Art. 84, 12 pp.}, } @article{williams2014cluster, title={Cluster algebras: an introduction}, author={Williams, Lauren}, journal={Bulletin of the American Mathematical Society}, volume={51}, number={1}, pages={1--26}, year={2014} } @misc{fomin2017introduction, title={Introduction to cluster algebras. chapters 4-5}, author={Fomin, Sergey and Williams, Lauren and Zelevinsky, Andrei}, eprint={1707.07190}, eprinttype = {arxiv}, year={2017} } @book{arora, title={Computational complexity: a modern approach}, author={Arora, Sanjeev and Barak, Boaz}, year={2009}, publisher={Cambridge University Press} }