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@article{ANHessDerang2023,
  title={Splitting the cohomology of {H}essenberg varieties and e-positivity of chromatic symmetric functions},
  author={Abreu, Alex and Nigro, Antonio},
  journal={arXiv preprint arXiv:2304.10644},
  year={2023}
}

@article{AHK2018,
    AUTHOR = {Adiprasito, Karim and Huh, June and Katz, Eric},
     TITLE = {Hodge theory for combinatorial geometries},
   JOURNAL = {Ann. of Math. (2)},
  FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {188},
      YEAR = {2018},
    NUMBER = {2},
     PAGES = {381--452}, 
       DOI = {10.4007/annals.2018.188.2.1}, 
}

@article{ANR2023PermRepchow,
    AUTHOR = {Angarone, Robert and Nathanson, Anastasia and Reiner, Victor},
     TITLE = {Chow rings of matroids as permutation representations},
   JOURNAL = {J. Lond. Math. Soc. (2)},
  FJOURNAL = {Journal of the London Mathematical Society. Second Series},
    VOLUME = {111},
      YEAR = {2025},
    NUMBER = {1},
     PAGES = {Paper No. e70039, 36 pp.}, 
       DOI = {10.1112/jlms.70039}, 
}

@article{ADH2023,
  title={Lagrangian geometry of matroids},
  author={Ardila, Federico and Denham, Graham and Huh, June},
  journal={Journal of the American Mathematical Society},
  volume={36},
  number={3},
  pages={727--794},
  year={2023}
}

@article {AthanasiadisGamma,
    AUTHOR = {Athanasiadis, Christos A.},
     TITLE = {Gamma-positivity in combinatorics and geometry},
   JOURNAL = {S\'{e}m. Lothar. Combin.},
  FJOURNAL = {S\'{e}minaire Lotharingien de Combinatoire},
    VOLUME = {77},
      YEAR = {[2016--2018]},
     PAGES = {Art. B77i, 64},
   MRCLASS = {05E05 (05A05 05A15 05E45 06A11 20B30 20F55)},
  MRNUMBER = {3878174},
MRREVIEWER = {Christian\ Stump},
}

@article{athanasiadis2020binomial,
  title={Binomial {E}ulerian polynomials for colored permutations},
  author={Athanasiadis, Christos A},
  journal={Journal of Combinatorial Theory, Series A},
  volume={173},
  pages={105214},
  year={2020},
  publisher={Elsevier},
  url={https://doi.org/10.1016/j.jcta.2020.105214}
}

@article{Baker2018hodge,
  title={Hodge theory in combinatorics},
  author={Baker, Matthew},
  journal={Bulletin of the American Mathematical Society},
  volume={55},
  number={1},
  pages={57--80},
  year={2018}
}

@article{behrend2022partial,
  title={Partial permutohedra},
  author={Behrend, Roger E and Castillo, Federico and Chavez, Anastasia and Diaz-Lopez, Alexander and Escobar, Laura and Harris, Pamela E and Insko, Erik},
  journal={arXiv preprint arXiv:2207.14253},
  year={2022},
  url={https://arxiv.org/abs/2207.14253}
}

@article{billey2022criteria,
  title={Criteria for smoothness of {P}ositroid varieties via pattern avoidance, {J}ohnson graphs, and spirographs},
  author={Billey, Sara C and Weaver, Jordan E},
  journal={arXiv preprint arXiv:2207.06508},
  year={2022}
}

@article {BjornerCMposet1980,
    AUTHOR = {Bj\"orner, Anders},
     TITLE = {Shellable and {C}ohen-{M}acaulay partially ordered sets},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {260},
      YEAR = {1980},
    NUMBER = {1},
     PAGES = {159--183},
   MRCLASS = {06A10 (13H10 52A25)},
  MRNUMBER = {570784},
MRREVIEWER = {P.\ McMullen},
       DOI = {10.2307/1999881},
       URL = {https://doi.org/10.2307/1999881},
}


@article{BHMPW2020,
    AUTHOR = {Braden, Tom and Huh, June and Matherne, Jacob P. and
              Proudfoot, Nicholas and Wang, Botong},
     TITLE = {A semi-small decomposition of the {C}how ring of a matroid},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {409},
      YEAR = {2022},
     PAGES = {Paper No. 108646, 49 pp.}, 
       DOI = {10.1016/j.aim.2022.108646}, 
}

@misc{BHMPW2020+,
  title={Singular {H}odge theory for combinatorial geometries},
  author={Braden, Tom and Huh, June and Matherne, Jacob P and Proudfoot, Nicholas and Wang, Botong},
  eprinttype = {arXiv},
  eprint = {2010.06088},
  year={2020}, 
}

@article{BrosnanChow2018,
title = {Unit interval orders and the dot action on the cohomology of regular semisimple {H}essenberg varieties},
journal = {Advances in Mathematics},
volume = {329},
pages = {955-1001},
year = {2018},
doi = {https://doi.org/10.1016/j.aim.2018.02.020},
url = {https://www.sciencedirect.com/science/article/pii/S0001870818300707},
author = {Patrick Brosnan and Timothy Y. Chow},
keywords = {Shareshian–Wachs conjecture, Chromatic quasisymmetric function, Stanley–Stembridge conjecture, Local invariant cycle theorem, Palindromicity, Reciprocity},
abstract = {Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the character of the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to ωXG(t), where XG(t) is the chromatic quasisymmetric function of the incomparability graph G of the corresponding natural unit interval order, and ω is the usual involution on symmetric functions. We prove the Shareshian–Wachs conjecture. Our proof uses the local invariant cycle theorem of Beilinson–Bernstein–Deligne to obtain a surjection, which we call the local invariant cycle map, from the cohomology of a regular Hessenberg variety of Jordan type λ to a space of local invariant cycles. As λ ranges over all partitions, the local invariant cycles collectively contain all the information about the dot action on a regular semisimple Hessenberg variety. We then prove a result showing that, under suitable hypotheses, the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology. (This is a general theorem, which is independent of the Hessenberg variety context.) Applying this result to the universal family of Hessenberg varieties, we show that, in our case, the surjections are actually isomorphisms, thus reducing the Shareshian–Wachs conjecture to computing the cohomology of a regular Hessenberg variety. But this cohomology has already been described combinatorially by Tymoczko, and, using a new reciprocity theorem for certain quasisymmetric functions, we show that Tymoczko's description coincides with the combinatorics of the chromatic quasisymmetric function.}
}

@article{CarrGraphAss,
  title={{C}oxeter complexes and graph-associahedra},
  author={Carr, Michael and Devadoss, Satyan L},
  journal={Topology and its Applications},
  volume={153},
  number={12},
  pages={2155--2168},
  year={2006},
  publisher={Elsevier}
}

@article{Caselli2012,
    title = {Signed mahonians on some trees and parabolic quotients},
    journal = {Journal of Combinatorial Theory, Series A},
    volume = {119},
    number = {7},
    pages = {1447-1460},
    year = {2012},
    doi = {https://doi.org/10.1016/j.jcta.2012.04.003},
    url = {https://www.sciencedirect.com/science/article/pii/S0097316512000660},
    author = {Fabrizio Caselli},
    keywords = {Permutation enumeration, Major index, Forest, Complex reflection group},
    abstract = {We study the distribution of the major index with sign on some parabolic quotients of the symmetric group, extending and generalizing simultaneously results of Gessel–Simion and Adin–Gessel–Roichman, and on the labellings of some special trees that we call rakes. We further consider and compute the distribution of the flag-major index on some parabolic quotients of wreath products and other related groups. All these distributions turn out to have very simple factorization formulas.}
}

@article{CharneyDavis1995,
author = {Ruth Charney and Michael Davis},
title = {{The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold.}},
volume = {171},
journal = {Pacific Journal of Mathematics},
number = {1},
publisher = {Pacific Journal of Mathematics, A Non-profit Corporation},
pages = {117 -- 137},
year = {1995},
}

@article{cheng2023PartialPerm,
    title={Statistics of partial permutations via {C}atalan matrices},
    author={Cheng, Yen-Jen and Eu, Sen-Peng and Hsu, Hsiang-Chun},
    journal={Advance in Applied Mathematics},
    volume={143},
    pages={102451},
    year={2023},
    publisher={Elsevier}
}

@article{ChoHongLee2020,
  title={Bases of the equivariant cohomologies of regular semisimple {H}essenberg varieties},
  author={Cho, Soojin and Hong, Jaehyun and Lee, Eunjeong},
  journal={arXiv preprint arXiv:2008.12500},
  year={2020}
}

@article{ChowErasing,
  title={The erasing mark conjecture},
  author={Chow, Timothy Y.},
  journal={arXiv preprint arXiv:2008.12500},
  year={2015}
}

@article {CGKEulerianSum2011,
    AUTHOR = {Chung, Fan and Graham, Ron and Knuth, Don},
     TITLE = {A symmetrical {E}ulerian identity},
   JOURNAL = {J. Comb.},
  FJOURNAL = {Journal of Combinatorics},
    VOLUME = {1},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {29--38},
   MRCLASS = {11B68},
  MRNUMBER = {2675920},
MRREVIEWER = {J.\ S\'{a}ndor},
       DOI = {10.4310/JOC.2010.v1.n1.a2},
       URL = {https://doi.org/10.4310/JOC.2010.v1.n1.a2},
}

@article {CGEulerianSum2012,
    AUTHOR = {Chung, Fan and Graham, Ron},
     TITLE = {Generalized {E}ulerian sums},
   JOURNAL = {J. Comb.},
  FJOURNAL = {Journal of Combinatorics},
    VOLUME = {3},
      YEAR = {2012},
    NUMBER = {3},
     PAGES = {299--316},
   MRCLASS = {11B68 (05A19 05A30 11B75)},
  MRNUMBER = {3029439},
MRREVIEWER = {J.\ S\'{a}ndor},
       DOI = {10.4310/JOC.2012.v3.n3.a3},
       URL = {https://doi.org/10.4310/JOC.2012.v3.n3.a3},
}

@book{Cox2011toric,
  title={Toric varieties},
  author={Cox, David A and Little, John B and Schenck, Henry K},
  volume={124},
  year={2011},
  publisher={American Mathematical Society}
}

@article{danilov1978,
  title={The geometry of toric varieties},
  author={Danilov, Vladimir I},
  journal={Russian Mathematical Surveys},
  volume={33},
  number={2},
  pages={97},
  year={1978},
  publisher={IOP Publishing}
}

@article{DeCociniProcesiWonderful,
  title={Wonderful models of subspace arrangements},
  author={De Concini, Corrado and Procesi, Claudio},
  journal={Selecta Mathematica},
  volume={1},
  pages={459--494},
  year={1995},
  publisher={Springer}
}

@article{DeCociniProcesiCohomology,
  title={Hyperplane arrangements and holonomy equations},
  author={De Concini, Corrado and Procesi, Claudio},
  journal={Selecta Mathematica},
  volume={1},
  pages={495--535},
  year={1995},
  publisher={Springer}
}


@inproceedings{desarmenien1983fonctions,
  title={Fonctions sym{\'e}triques associ{\'e}es {\`a} des suites classiques de nombres},
  author={D{\'e}sarm{\'e}nien, Jacques},
  booktitle={Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
  volume={16},
  number={2},
  pages={271--304},
  year={1983}
}

@article{dolgachev1994character,
  title={A Character Formula for the Representation of a Weyl Group in the Cohomology of the Associated Toric Variety},
  author={Dolgachev, I and Lunts, V},
  journal={Journal of Algebra},
  volume={168},
  number={3},
  pages={741--772},
  year={1994},
  publisher={Elsevier}
}


@article{EuFuHsuLiao2018,
    title={Signed countings of types {B} and {D} permutations and $t,q$-{E}uler numbers},
    author={Eu, Sen-Peng and Fu, Tung-Shan and Hsu, Hsiang-Chun and Liao, Hsin-Chieh},
    journal={Advance in Applied Mathematics},
    volume={97},
    pages={1--26},
    year={2018},
    publisher={Elsevier},
    url={https://doi.org/10.1016/j.aam.2018.02.004}
}

@article{EuFuHsuLiaoSun2020,
    title={Signed {M}ahonian identities on permutations with subsequence restrictions},
    author={Eu, Sen-Peng and Fu, Tung-Shan and Hsu, Hsiang-Chun and Liao, Hsin-Chieh and Sun, Wei-Liang},
    journal={Journal of Combinatorial Theory, Series A},
    volume={170},
    pages={105131},
    year={2020},
    publisher={Elsevier},
    url={https://doi.org/10.1016/j.jcta.2019.105131}
}

@article{EuFuHsuLiaoSunFPSAC2019,
    title={Signed {M}ahonian identities on permutations with subsequence restrictions},
    author={Eu, Sen-Peng and Fu, Tung-Shan and Hsu, Hsiang-Chun and Liao, Hsin-Chieh and Sun, Wei-Liang},
    journal={Séminaire Lotharingien de Combinatoire},
    volume={82B},
    pages={Article 21, 12pp},
    year={2020}
}

@article{Eur2022essence,
  title={An Essence of independence: Recent works of {J}une {H}uh on combinatorics and {H}odge theory},
  author={Eur, Christopher},
  journal={arXiv preprint arXiv:2211.05724},
  year={2022}
}

@article{EFLS2022signed,
  title={Signed permutohedra, delta-matroids, and beyond},
  author={Eur, Christopher and Fink, Alex and Larson, Matt and Spink, Hunter},
  journal={arXiv preprint arXiv:2209.06752},
  year={2022}
}

@article{EHL2022stellahedral, 
    AUTHOR = {Eur, Christopher and Huh, June and Larson, Matt},
     TITLE = {Stellahedral geometry of matroids},
   JOURNAL = {Forum Math. Pi},
  FJOURNAL = {Forum of Mathematics. Pi},
    VOLUME = {11},
      YEAR = {2023},
     PAGES = {Paper No. e24, 48 pp.}, 
       DOI = {10.1017/fmp.2023.24}, 
}

@article{Eur2023intersection,
  title={Intersection theory of polymatroids},
  author={Eur, Christopher and Larson, Matt},
  journal={arXiv preprint arXiv:2301.00831},
  year={2023}
}

@book{EwaldBook,
  title={Combinatorial convexity and algebraic geometry},
  author={Ewald, G{\"u}nter},
  volume={168},
  year={1996},
  publisher={Springer Science \& Business Media}
}

@article{FeichtnerKozlov2004,
  title={Incidence combinatorics of resolutions},
  author={Feichtner, Eva-Maria and Kozlov, Dmitry N},
  journal={Selecta Mathematica},
  volume={10},
  number={1},
  pages={37--60},
  year={2004},
  publisher={Springer}
}

@article{FeichtnerSturmfel2005,
  title={Matroid polytopes, nested sets and {B}ergman fans},
  author={Feichtner, Eva Maria and Sturmfels, Bernd},
  journal={Portugaliae Mathematica},
  volume={62},
  pages={437--468},
  year={2005}
}

@article{FeichtnerYuzvinsky2004,
    AUTHOR = {Feichtner, Eva Maria and Yuzvinsky, Sergey},
     TITLE = {Chow rings of toric varieties defined by atomic lattices},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {155},
      YEAR = {2004},
    NUMBER = {3},
     PAGES = {515--536}, 
       DOI = {10.1007/s00222-003-0327-2}, 
}

@misc{FMVChowPoly,
  title={Chow functions for partially ordered sets},
  author={Ferroni, Luis and Matherne, Jacob P and Vecchi, Lorenzo},
  eprinttype = {arXiv},
  eprint = {2411.04070}, 
  year={2024}
}

@article{Ferroni2024hilbert,
    AUTHOR = {Ferroni, Luis and Matherne, Jacob P. and Stevens, Matthew and
              Vecchi, Lorenzo},
     TITLE = {Hilbert-{P}oincar\'e{} series of matroid {C}how rings and
              intersection cohomology},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {449},
      YEAR = {2024},
     PAGES = {Paper No. 109733, 55 pp.}, 
       DOI = {10.1016/j.aim.2024.109733}, 
}

@article {FoataHan2010,
    AUTHOR = {Foata, Dominique and Han, Guo-Niu},
     TITLE = {The {$q$}-tangent and {$q$}-secant numbers via basic
              {E}ulerian polynomials},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {138},
      YEAR = {2010},
    NUMBER = {2},
     PAGES = {385--393},
}

@article{garsia1979permutation,
  title={Permutation statistics and partitions},
  author={Garsia, Adriano M and Gessel, Ira},
  journal={Advances in Mathematics},
  volume={31},
  number={3},
  pages={288--305},
  year={1979},
  publisher={Academic Press}
}

@incollection {GesselSkewSchur1984,
    AUTHOR = {Gessel, Ira M.},
     TITLE = {Multipartite {$P$}-partitions and inner products of skew
              {S}chur functions},
 BOOKTITLE = {Combinatorics and algebra ({B}oulder, {C}olo., 1983)},
    SERIES = {Contemp. Math.},
    VOLUME = {34},
     PAGES = {289--317},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {1984},
   MRCLASS = {05A17 (20C30)},
  MRNUMBER = {777705},
MRREVIEWER = {J.\ D\'{e}sarm\'{e}nien},
       DOI = {10.1090/conm/034/777705},
       URL = {https://doi.org/10.1090/conm/034/777705},
}

@article{Gessel1993cycle,
  title={Counting permutations with given cycle structure and descent set},
  author={Gessel, Ira M and Reutenauer, Christophe},
  journal={Journal of Combinatorial Theory, Series A},
  volume={64},
  number={2},
  pages={189--215},
  year={1993},
  publisher={Elsevier}
}

@article{GrinbergReinerHopf2014,
  title={Hopf algebras in combinatorics},
  author={Grinberg, Darij and Reiner, Victor},
  journal={arXiv preprint arXiv:1409.8356},
  year={2014}
}

@article{Guay2013modular,
  title={A modular relation for the chromatic symmetric functions of (3+ 1)-free posets},
  author={Guay-Paquet, Mathieu},
  journal={arXiv preprint arXiv:1306.2400},
  url={https://arxiv.org/abs/1306.2400},
  year={2013}
}

@article{Guay2016secondproof,
  title={A second proof of the {S}hareshian--{W}achs conjecture, by way of a new {H}opf algebra},
  author={Guay-Paquet, Mathieu},
  journal={arXiv preprint arXiv:1601.05498},
  url={https://arxiv.org/abs/1601.05498},
  year={2016}
}

@article{HaglundLoehrRemmel2005,
title = {Statistics on wreath products, perfect matchings, and signed words},
journal = {European Journal of Combinatorics},
volume = {26},
number = {6},
pages = {835-868},
year = {2005},
note = {Combinatorics and Representation Theory},
doi = {https://doi.org/10.1016/j.ejc.2004.06.004},
url = {https://www.sciencedirect.com/science/article/pii/S0195669804000964},
author = {J. Haglund and N. Loehr and J.B. Remmel},
keywords = {Major-index statistics, Wreath products, Perfect matchings, Signed words, Rook theory},
abstract = {We introduce a natural extension of Adin, Brenti, and Roichman’s major-index statistic nmaj on signed permutations (Adv. Appl. Math. 27 (2001) 210–224) to wreath products of a cyclic group with the symmetric group. We derive “insertion lemmas” which allow us to give simple bijective proofs that our extension has the same distribution as another statistic on wreath products introduced by Adin and Roichman (European J. Combin. 22 (2001) 431–446) called the flag major index. We also use our insertion lemmas to show that nmaj, the flag major index, and an inversion statistic have the same distribution on a subset of signed permutations in bijection with perfect matchings. We show that this inversion statistic has an interpretation in terms of q-counting rook placements on a shifted Ferrers board. Many results on permutation statistics extend to results on multiset permutations (words). We derive a number of analogous results for signed words, and also words with higher-order roots of unity attached to them.}
}

@article{HameisterRaoSimpson2021,
    AUTHOR = {Hameister, Thomas and Rao, Sujit and Simpson, Connor},
     TITLE = {Chow rings of vector space matroids},
   JOURNAL = {J. Comb.},
  FJOURNAL = {Journal of Combinatorics},
    VOLUME = {12},
      YEAR = {2021},
    NUMBER = {1},
     PAGES = {55--83}, 
       DOI = {10.4310/JOC.2021.v12.n1.a3}, 
}



@article{haglund2019real,
  title={Real-rootedness of variations of {E}ulerian polynomials},
  author={Haglund, James and Zhang, Philip B},
  journal={Advance in Applied Mathematics},
  volume={109},
  pages={38--54},
  year={2019},
  publisher={Elsevier}
}

@article{Hampe2017,
title = {The intersection ring of matroids},
journal = {Journal of Combinatorial Theory, Series B},
volume = {122},
pages = {578-614},
year = {2017},
doi = {https://doi.org/10.1016/j.jctb.2016.08.004},
url = {https://www.sciencedirect.com/science/article/pii/S0095895616300648},
author = {Simon Hampe},
keywords = {Matroid theory, Matroid intersection, Nested matroids, G-invariant, Tropical intersection theory},
abstract = {We study a particular graded ring structure on the set of all loopfree matroids on a fixed labeled ground set, which occurs naturally in tropical geometry. The product is given by matroid intersection and the additive structure is defined by assigning to each matroid the indicator vector of its chains of flats. We show that this ring is generated in corank one, more precisely that any matroid can be written as a linear combination of products of corank one matroids. Moreover, we prove that a basis for the graded part of rank r matroids is given by the set of nested matroids and that the total number of these is a Eulerian number. Derksen's G-invariant then defines a Z-linear map on this ring, which implies for example that the Tutte polynomial is linear on it as well. Finally we show that the ring is the cohomology ring of the toric variety of the permutohedron and thus fulfills Poincaré duality.}
}

@article{heuer2020partial,
  title={Partial permutation and alternating sign matrix polytopes},
  author={Heuer, Dylan and Striker, Jessica},
  journal={arXiv preprint arXiv:2012.09901},
  year={2020}
}

@misc{Hoster2024ChowUniform,
  title={The Chow and augmented Chow polynomials of uniform matroids},
  author={Hoster, Elena},
  eprinttype = {arXiv},
  eprint = {2410.22329}, 
  year={2024}
}

@article{Huh2016tropical,
  title={Tropical geometry of matroids},
  author={Huh, June},
  journal={Current Developments in mathematics},
  volume={2016},
  number={1},
  pages={1--46},
  year={2016},
  publisher={International Press of Boston}
}

@article {LeungReinerToric2002,
    AUTHOR = {Leung, Naichung Conan and Reiner, Victor},
     TITLE = {The signature of a toric variety},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {111},
      YEAR = {2002},
    NUMBER = {2},
     PAGES = {253--286},
   MRCLASS = {14M25 (52B05 52B20)},
  MRNUMBER = {1882135},
MRREVIEWER = {Joseph\ Gubeladze},
       DOI = {10.1215/S0012-7094-02-11123-5},
       URL = {https://doi.org/10.1215/S0012-7094-02-11123-5},
}

@article{Liao2023FPSAC,
    title={Stembrdige codes and {C}how rings},
    author={Hsin-Chieh Liao},
    journal={Sém. Lothar. Combin.},
    volume={89B},
    pages={Art. 88, 12 pp.},
    year={2023}
}

@misc{Liao2023One,
    title={Stembridge codes, permutahedral varieties, and their extensions},
    author={Liao, Hsin-Chieh},
    eprinttype = {arXiv},
    eprint = {2403.10577}, 
    year={2024}
}


@misc{Liao2023Two, 
    title={Chow rings and augmented {C}how rings of uniform matroids and their $q$-analogues},
    author={Liao, Hsin-Chieh},
    eprinttype = {arXiv},
    eprint = {2406.19660}, 
    year={2024}
}

@article{Liao2024EqGamma,
    title={Equivariant gamma-positivity for {C}how rings and augmented {C}how rings of matroids},
    author={Liao, Hsin-Chieh},
    journal={in preparation},
    volume={ },
    pages={ },
    year={ }

}

@article{Lin2022PermVar,
  title={The geometry of the permutohedral Variety as a {H}essenberg Variety},
  author={Lin, Jan-Li},
  journal={arXiv preprint arXiv:2204.13752},
  url={https://arxiv.org/abs/2204.13752v2},
  year={2022}
}

@article{lin2019around,
  title={Around the q-binomial-{E}ulerian polynomials},
  author={Lin, Zhicong and Wang, David GL and Zeng, Jiang},
  journal={European Journal of Combinatorics},
  volume={78},
  pages={105--120},
  year={2019},
  publisher={Elsevier}
}

@article{ma2017recurrence,
  title={Recurrence relations for binomial-{E}ulerian polynomials},
  author={Ma, Jun and Ma, Shi-Mei and Yeh, Yeong-Nan},
  journal={arXiv preprint arXiv:1711.09016},
  year={2017}
}

@article{MacMahon1916,
  title={Two applications of general theorems in combinatory analysis:(1) to the theory of inversions of permutations;(2) to the ascertainment of the numbers of terms in the development of a determinant which has amongst its elements an arbitrary number of zeros},
  author={MacMahon, Percy A},
  journal={Proceedings of the London Mathematical Society},
  volume={2},
  number={1},
  pages={314--321},
  year={1917},
  publisher={Oxford University Press}
}

@article{Mastroeni2022Koszul,
  title={Chow rings of matroids are Koszul},
  author={Mastroeni, Matthew and McCullough, Jason},
  journal={Mathematische Annalen},
  pages={1--33},
  year={2022},
  publisher={Springer}
}

@book{SturmfelsCommtative2005,
  title={Combinatorial commutative algebra},
  author={Miller, Ezra and Sturmfels, Bernd},
  volume={227},
  year={2005},
  publisher={Springer Science \& Business Media}
}

@article{NV2021qdeformation,
  title={A $ q $-deformation of an algebra of {K}lyachko and {M}acdonald's reduced word formula},
  author={Nadeau, Philippe and Tewari, Vasu},
  journal={arXiv preprint arXiv:2106.03828},
  year={2021}
}

@inproceedings{NV2023remixed,
  title={Remixed {E}ulerian numbers},
  author={Nadeau, Philippe and Tewari, Vasu},
  booktitle={Forum of Mathematics, Sigma},
  volume={11},
  pages={e65},
  year={2023}
}

@article{NV2023permutahedral,
  title={The permutahedral variety, mixed {E}ulerian numbers, and principal specializations of {S}chubert polynomials},
  author={Nadeau, Philippe and Tewari, Vasu},
  journal={International Mathematics Research Notices},
  volume={2023},
  number={5},
  pages={3615--3670},
  year={2023},
  publisher={Oxford University Press}
}

@article{NV2023forest,
  title={Forest polynomials and the class of the permutahedral variety},
  author={Nadeau, Philippe and Tewari, Vasu},
  journal={arXiv preprint arXiv:2306.10939},
  year={2023}
}

@article{NelsonMatroids2018,
author = {Nelson, Peter},
title = {Almost all matroids are nonrepresentable},
journal = {Bulletin of the London Mathematical Society},
volume = {50},
number = {2},
pages = {245-248},
keywords = {05B35 (primary)},
doi = {https://doi.org/10.1112/blms.12141},
url = {https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms.12141},
eprint = {https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms.12141},
abstract = {Abstract We prove that, as n tends to infinity, the proportion of n-element matroids that are representable tends to zero.},
year = {2018}
}

@book{Oxley2006matroid,
  title={Matroid theory},
  author={Oxley, James G},
  volume={3},
  year={2006},
  publisher={Oxford University Press, USA}
}


@article{PRW2008,
  title={Faces of simple generalized permutohedra},
  author={Postnikov, Alex and Reiner, Victor and Williams, Lauren},
  journal={Journal der Deutschen Mathematiker-Vereinigung Gegr{\"u}ndet 1996},
  volume={13},
  pages={207--273},
  year={2008}
}

@ARTICLE{PostnikovGenPermu2005,
  author={Postnikov, Alexander},
  journal={International Mathematics Research Notices}, 
  title={Permutohedra, Associahedra, and Beyond}, 
  year={2009},
  volume={2009},
  number={6},
  pages={1026-1106},
  doi={10.1093/imrn/rnn153}}

@article{Postnikov2006Positroid,
  title={Total positivity, {G}rassmannians, and networks},
  author={Postnikov, Alexander},
  journal={arXiv preprint math/0609764},
  year={2006}
}

@article{Procesi1990,
	Author={Procesi, C.},	
	Title={The toric variety associated to {W}eyl chambers},
	Journal={“Mots “(M. Lothaire, Ed.)Hermes, Paris},
	Pages={153--161},
	Year={1990}}

@article{RSW2004,
title = {The cyclic sieving phenomenon},
journal = {Journal of Combinatorial Theory, Series A},
volume = {108},
number = {1},
pages = {17-50},
year = {2004},
doi = {https://doi.org/10.1016/j.jcta.2004.04.009},
url = {https://www.sciencedirect.com/science/article/pii/S0097316504000822},
author = {V. Reiner and D. Stanton and D. White},
keywords = {-Binomial coefficient, -Multinomial coefficient, Roots-of-unity, Principal specialization, Schur function, Kraskiewicz–Weyman, Singer cycle, Springer regular element, Polygon dissections, Noncrossing partitions, Hook formula, Ordered tree},
abstract = {The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge's q=−1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Pólya–Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field q-analogues.}
}

@book{Sagan2013symmetric,
  title={The symmetric group: representations, combinatorial algorithms, and symmetric functions},
  author={Sagan, Bruce E},
  volume={203},
  year={2013},
  publisher={Springer Science \& Business Media}
}

@article{SaganShareshianWachs2011,
  title={Eulerian quasisymmetric functions and cyclic sieving},
  author={Sagan, Bruce and Shareshian, John and Wachs, Michelle L},
  journal={Advances in Applied Mathematics},
  volume={46},
  number={1-4},
  pages={536--562},
  year={2011},
  publisher={Elsevier}
}
 
@article{ShareshianWachs2007,
  title={Poset homology of {R}ees products, and q-{E}ulerian polynomials},
  author={Shareshian, John and Wachs, Michelle L},
  journal={The Electronic Journal of Combinatorics},
  volume={16},
  number={2},
  pages={R20},
  year={2009}
}	

@article{ShareshianWachs2010,
  title={Eulerian quasisymmetric functions},
  author={Shareshian, John and Wachs, Michelle L},
   JOURNAL = {Adv. Math.},
  volume={225},
  number={6},
  pages={2921--2966},
  year={2010},
       DOI = {10.1016/j.aim.2010.05.009},
  publisher={Elsevier}
}

@incollection{ShareshianWachs2012chromatic,
  title={Chromatic quasisymmetric functions and {H}essenberg varieties},
  author={Shareshian, John and Wachs, Michelle L},
  booktitle={Configuration Spaces: Geometry, Combinatorics and Topology},
  pages={433--460},
  year={2012},
  publisher={Springer}
}

@article{ShareshianWachs2016Chromatic,
title = {Chromatic quasisymmetric functions},
journal = {Advances in Mathematics},
volume = {295},
pages = {497-551},
year = {2016},
doi = {https://doi.org/10.1016/j.aim.2015.12.018},
url = {https://www.sciencedirect.com/science/article/pii/S0001870816000025},
author = {Shareshian, John and Wachs, Michelle L},
keywords = {Symmetric functions, Eulerian polynomials, -Positivity, Unimodality, Hessenberg varieties},
abstract = {We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the q-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the e-positivity conjecture of Stanley and Stembridge for incomparability graphs of (3+1)-free posets.}
}
 
@article{ShareshianWachs2020,
  title={Gamma-positivity of variations of {E}ulerian polynomials},
  author={Shareshian, John and Wachs, Michelle L},
   JOURNAL = {J. Comb.},
  volume={11},
  number={1},
  pages={1--33},
  year={2020},
       DOI = {10.4310/JOC.2020.v11.n1.a1},
  publisher={International Press of Boston}
}

@article {Solomon1968,
    AUTHOR = {Solomon, Louis},
     TITLE = {A decomposition of the group algebra of a finite {C}oxeter
              group},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {9},
      YEAR = {1968},
     PAGES = {220--239},
   MRCLASS = {20.80},
  MRNUMBER = {232868},
MRREVIEWER = {T.\ Ono},
       DOI = {10.1016/0021-8693(68)90022-7},
       URL = {https://doi.org/10.1016/0021-8693(68)90022-7},
}


@article{Springer1974regular,
  title={Regular elements of finite reflection groups},
  author={Springer, Tonny Albert},
  journal={Inventiones mathematicae},
  volume={25},
  number={2},
  pages={159--198},
  year={1974},
  publisher={Springer-Verlag Berlin/Heidelberg}
}

@article {StanleyRlabel1974,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Finite lattices and {J}ordan-{H}\"older sets},
   JOURNAL = {Algebra Universalis},
  FJOURNAL = {Algebra Universalis},
    VOLUME = {4},
      YEAR = {1974},
     PAGES = {361--371},
   MRCLASS = {06A30 (05A99)},
  MRNUMBER = {354473},
MRREVIEWER = {O.\ Frink},
       DOI = {10.1007/BF02485748},
       URL = {https://doi.org/10.1007/BF02485748},
}

@article{Stanley1979invariants,
  title={Invariants of finite groups and their applications to combinatorics},
  author={Stanley, Richard P},
  journal={Bulletin of the American Mathematical Society},
  volume={1},
  number={3},
  pages={475--511},
  year={1979}
}

@article{Stanley1982GroupPoset,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {Some aspects of groups acting on finite posets},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {32},
      YEAR = {1982},
    NUMBER = {2},
     PAGES = {132--161}, 
       DOI = {10.1016/0097-3165(82)90017-6}, 
}

@article{Stanley1989,
	title={Log-concave and unimodal sequences in algebra, combinatorics, and geometry},
	author={Stanley, Richard P},
	journal={Annals of the New York Academy of Sciences},
	volume={576},
	number={1},
	pages={500--535},
	year={1989}}

@article{StanleyStembridge1993,
    title = {On immanants of {J}acobi-{T}rudi matrices and permutations with restricted position},
    journal = {Journal of Combinatorial Theory, Series A},
    volume = {62},
    number = {2},
    pages = {261-279},
    year = {1993},
    doi = {https://doi.org/10.1016/0097-3165(93)90048-D},
    url = {https://www.sciencedirect.com/science/article/pii/009731659390048D},
    author = {Richard P Stanley and John R Stembridge},
    abstract = {Let χ be a character of the symmetric group Ln. The immanant of an n × n matrix A = [aij] with respect to χ is Σw ϵ Sn χ(w) a1,w(1) … an,w(n). Goulden and Jackson conjectured, and Greene recently proved, that immanants of Jacobi-Trudi matrices are polynomials with nonnegative integer coefficients. This led one of us (Stembridge) to formulate a series of conjectures involving immanants, some of which amount to stronger versions of the original Goulden-Jackson conjecture. In this paper, we prove some special cases of one of the stronger conjectures. One of the special cases we prove develops from a generalization of the theory of permutations with restricted position which takes into account the cycle structure of the permutations. We also present two refinements of the immanant conjectures, as well as a related conjecture on the number of ways to partition a partially ordered set into chains.}
}

@article{StanleyChromaticSymmetric1995,
  title={A symmetric function generalization of the chromatic polynomial of a graph},
  author={Stanley, Richard P},
  journal={Advances in Mathematics},
  volume={111},
  number={1},
  pages={166--194},
  year={1995},
  publisher={Elsevier}
}

@book{Stanley2007comutative,
  title={Combinatorics and commutative algebra},
  author={Stanley, Richard P},
  volume={41},
  year={2007},
  publisher={Springer Science \& Business Media}
}

@incollection {StanleyAlternatingSurvey,
    AUTHOR = {Stanley, Richard P.},
     TITLE = {A survey of alternating permutations},
 BOOKTITLE = {Combinatorics and graphs},
    SERIES = {Contemp. Math.},
    VOLUME = {531},
     PAGES = {165--196},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2010},
   MRCLASS = {05-02 (05A05 05E05 05E10)},
  MRNUMBER = {2757798},
MRREVIEWER = {Domenico\ Senato},
       DOI = {10.1090/conm/531/10466},
       URL = {https://doi.org/10.1090/conm/531/10466},
}

@book{StanleyEC1,
    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 pp.}, 
}

@book{StanleyEC2,
    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 pp.}, 
       DOI = {10.1017/CBO9780511609589}, 
}

@article{Steinberg1951geometric,
  title={A geometric approach to the representations of the full linear group over a Galois field},
  author={Steinberg, Robert},
  journal={Transactions of the American Mathematical Society},
  volume={71},
  number={2},
  pages={274--282},
  year={1951}
}

 
@article{Stembridge1992,
  title={Eulerian numbers, tableaux, and the {B}etti numbers of a toric variety},
  author={Stembridge, John R},
  journal={Discrete Mathematics},
  volume={99},
  number={1-3},
  pages={307--320},
  year={1992},
  publisher={Elsevier}
}

@article{Stembridge1994,
  title={Some permutation representations of {W}eyl groups associated with the cohomology of toric varieties},
  author={Stembridge, John R},
  journal={Advances in Mathematics},
  volume={106},
  pages={244--301},
  year={1994},
  publisher={Elsevier}
}

@misc{StumpChow2024,
  title={Chow and augmented {C}how polynomials as evaluations of {P}oincar\'{e}-extended ab-indices},
  author={Stump, Christian},
  eprinttype = {arXiv},
  eprint = {2406.18932}, 
  year={2024}
}

@article{DorpalenMaglioneStump2023ab,
    AUTHOR = {Dorpalen-Barry, Galen and Maglione, Joshua and Stump,
              Christian},
     TITLE = {The {P}oincar\'e-extended {$\mathbf{ab}$}-index},
      NOTE = {With an appendix by Ricky Ini Liu},
   JOURNAL = {J. Lond. Math. Soc. (2)},
  FJOURNAL = {Journal of the London Mathematical Society. Second Series},
    VOLUME = {111},
      YEAR = {2025},
    NUMBER = {1},
     PAGES = {Paper No. e70054, 33 pp.}, 
       DOI = {10.1112/jlms.70054}, 
}

@article{TymoczkoGKM,
  title={An introduction to equivariant cohomology and homology, following {G}oresky, {K}ottwitz, and {M}ac{P}herson, {S}nowbird},
  author={Tymoczko, J},
  journal={Contemporary Mathematics: Snowbird Lectures in Algebraic Geometry},
  volume={388},
  pages={169-188},
  year={2005}
}

@article{TymoczkoEqCoho2007,
  title={Permutation actions on equivariant cohomology},
  author={Tymoczko, Julianna S},
  journal={arXiv:0706.0460},
  year={2007}
}

@incollection{WachsTool2007,
    AUTHOR = {Wachs, Michelle L.},
     TITLE = {Poset topology: tools and applications},
 BOOKTITLE = {Geometric combinatorics},
    SERIES = {IAS/Park City Math. Ser.},
    VOLUME = {13},
     PAGES = {497--615},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2007}, 
       DOI = {10.1090/pcms/013/09}, 
}

@inproceedings{Zheng2019face,
  title={Face enumeration on flag complexes and flag spheres},
  author={Zheng, Hailun},
  booktitle={Algebraic and Geometric Combinatorics on Lattice Polytopes: Proceedings of the Summer Workshop on Lattice Polytopes},
  pages={435--449},
  year={2019},
  organization={World Scientific}
}


@book{ZieglerPolytopeBook,
  title={Lectures on polytopes},
  author={Ziegler, G{\"u}nter M},
  volume={152},
  year={2012},
  publisher={Springer Science \& Business Media}
}