@misc{cho2021bases,
      title={Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties}, 
      author={Soojin Cho and Jaehyun Hong and Eunjeong Lee},
      year={2021},
      eprint={2008.12500},
      archivePrefix={arXiv},
      primaryClass={math.AG}
}


@article{shareshian_wachs_2016, title={Chromatic quasisymmetric functions}, volume={295}, DOI={10.1016/j.aim.2015.12.018}, journal={Advances in Mathematics}, author={Shareshian, John and Wachs, Michelle L.}, year={2016}, pages={497–551}}

@article{stanley_1995, title={A Symmetric Function Generalization of the Chromatic Polynomial of a Graph}, volume={111}, DOI={10.1006/aima.1995.1020}, number={1}, journal={Advances in Mathematics}, author={Stanley, R.P.}, year={1995}, pages={166–194}}


@book{bjorner_anders_brenti_2010, place={New York, NY}, title={Combinatorics of coxeter groups}, publisher={Springer}, author={Björner Anders and Brenti, Francesco}, year={2010}}

 @misc{sweeting, title={Decomposing Parabolic Subgroups}, url={https://scholar.harvard.edu/files/naomisweeting/files/flag_varieties_-_naomi_sweeting.pdf}, author={Sweeting, Naomi}} 
 
  @article{mari_procesi_shayman_1992, title={Hessenberg varieties}, volume={332}, DOI={10.2307/2154181}, number={2}, journal={Transactions of the American Mathematical Society}, author={Mari, F. De and Procesi, C. and Shayman, M. A.}, year={1992}, pages={529}} 
  
   @article{reiner_stanton_white_2004, title={The cyclic sieving phenomenon}, volume={108}, DOI={10.1016/j.jcta.2004.04.009}, number={1}, journal={Journal of Combinatorial Theory, Series A}, author={Reiner, V. and Stanton, D. and White, D.}, year={2004}, pages={17–50}} 
   
    @misc{ajose_2007, title={Applications of the q-Binomial Coefficients to Counting Problems}, url={https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1194&context=hmc_theses}, journal={HMC Senior Theses}, author={Ajose, Jonathan}, year={2007}} 
    
     @inbook{bóna_branden_2015, place={Boca Raton}, title={Unimodality, Log-concavity, Real-rootedness and Beyond}, booktitle={Handbook of Enumerative Combinatorics}, publisher={CRC Press/Taylor \&; Francis Group}, author={Bóna, Miklós and Branden, Petter}, year={2015}} 
     
@article{awan_bernardi_2020,
title = {Tutte polynomials for directed graphs},
journal = {Journal of Combinatorial Theory, Series B},
volume = {140},
pages = {192-247},
year = {2020},
issn = {0095-8956},
doi = {https://doi.org/10.1016/j.jctb.2019.05.006},
url = {https://www.sciencedirect.com/science/article/pii/S0095895619300553},
author = {Jordan Awan and Olivier Bernardi},
keywords = {Digraph invariant, Potts model for digraphs, Chromatic polynomial of mixed graphs, Chromatic Symmetric Function, Quasisymmetric functions, Combinatorial reciprocity, Digraph generating functions},
abstract = {The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name the B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B-polynomial, and try to answer the following questions:•what properties of the digraph can be detected from its B-polynomial (acyclicity, length of directed paths, number of strongly connected components, etc.)?•which of the marvelous properties of the Tutte polynomial carry over to the directed graph setting? The B-polynomial generalizes the strict chromatic polynomial of mixed graphs introduced by Beck, Bogart and Pham. We also consider a quasisymmetric function version of the B-polynomial which simultaneously generalizes the Tutte symmetric function of Stanley and the quasisymmetric chromatic function of Shareshian and Wachs.}
}

@misc{Ellzey_2017,
  doi = {10.48550/ARXIV.1709.00454},
  url = {https://arxiv.org/abs/1709.00454},
  author = {Ellzey, Brittney},
  keywords = {Combinatorics (math.CO), FOS: Mathematics, FOS: Mathematics, 05E05, 05A05},
  title = {A directed graph generalization of chromatic quasisymmetric functions},
  publisher = {arXiv},
  year = {2017},
  copyright = {arXiv.org perpetual, non-exclusive license}
}

@Misc{joke1,
  author       = {Kyle Celano},
  howpublished = {{Private Communication}},
  year         = {2022},
}

@Misc{joke2,
  author       = {Nicholas Sieger},
  howpublished = {{Private Communication}},
  year         = {2022},
}

@Misc{joke3,
  author       = {Sam Spiro},
  howpublished = {{Private Communication}},
  year         = {2022},
}

@article{Stanley-LogConcave,
author = {STANLEY, RICHARD P.},
title = {Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometrya},
journal = {Annals of the New York Academy of Sciences},
volume = {576},
number = {1},
pages = {500-535},
doi = {https://doi.org/10.1111/j.1749-6632.1989.tb16434.x},
url = {https://nyaspubs.onlinelibrary.wiley.com/doi/abs/10.1111/j.1749-6632.1989.tb16434.x},
eprint = {https://nyaspubs.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1749-6632.1989.tb16434.x},
year = {1989}
}
@book{stanley_2011, place={Cambridge}, edition={2}, series={Cambridge Studies in Advanced Mathematics}, title={Enumerative Combinatorics}, volume={1}, DOI={10.1017/CBO9781139058520}, publisher={Cambridge University Press}, author={Stanley, Richard P.}, year={2011}, collection={Cambridge Studies in Advanced Mathematics}}

@misc{poznanovic2023descent,
      title={Descent polynomials for labeled tree}, 
      author={Svetlana Poznanović and Maria Rodriguez Hertz and Solomon Valore-Caplan and David Wichmann},
      year={2023},
      eprint={2305.00148},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}

@misc{diazlopez2017descent,
      title={Descent polynomials}, 
      author={Alexander Diaz-Lopez and Pamela E. Harris and Erik Insko and Mohamed Omar and Bruce E. Sagan},
      year={2017},
      eprint={1710.11033},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}

@article{ARCHER2020112041,
title = {Counting acyclic and strong digraphs by descents},
journal = {Discrete Mathematics},
volume = {343},
number = {11},
pages = {112041},
year = {2020},
issn = {0012-365X},
doi = {https://doi.org/10.1016/j.disc.2020.112041},
url = {https://www.sciencedirect.com/science/article/pii/S0012365X20302272},
author = {Kassie Archer and Ira M. Gessel and Christina Graves and Xuming Liang},
keywords = {Acyclic digraph, Strong digraph, Strong tournament, Descent, Eulerian generating function, Graphic generating function},
abstract = {A descent of a labeled digraph is a directed edge (s,t) with s>t. We count strong tournaments, strong digraphs, acyclic digraphs, and forests by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.}
}

@article{awan2022tutte,
  title={Tutte polynomials for oriented matroids},
  author={Awan, Jordan and Bernardi, Olivier},
  journal={arXiv preprint arXiv:2204.00162},
  year={2022}
}

@article{DEGENHARDT200049,
title = {Weighted-Inversion Statistics and Their Symmetry Groups},
journal = {Journal of Combinatorial Theory, Series A},
volume = {90},
number = {1},
pages = {49-103},
year = {2000},
issn = {0097-3165},
doi = {https://doi.org/10.1006/jcta.1999.3019},
url = {https://www.sciencedirect.com/science/article/pii/S0097316599930194},
author = {Sheldon L. Degenhardt and Stephen C. Milne},
keywords = {weighted-inversion statistics, permutations, multipermutations, core group, dihedral group , bijections, involutions, inversion patterns, column space, null space, rank},
abstract = {A statistic w on Sn is a weighted-inversion (w-i) statistic if there exist weights wi, j such that w(σ)=∑i<jχ[σi>σj]wi, j for each σ∈Sn. Two well-known examples are the major index and inversion count statistics. These two statistics share the same distribution over Sn, and many bijections Sn→Sn have been described to prove this. These bijections thus have the property that they map a certain w-i statistic to another. This paper presents the results of our search for bijections φ: Sn→Sn with an even stronger property: given any w-i statistic w, the statistic w∘φ is also a w-i statistic. Such a set of bijections forms a group, which we call the core group of Sn. We exhibit a subgroup of the core group of Sn which is isomorphic to the dihedral group Dn+1. We extend these ideas to other sets of objects, including subsets of Sn and sets of permutations of a multiset. As examples, we develop a family of subsets of Sn which has a core group isomorphic to a Weyl group of order 2n·n!, and we show that the set of permutations of the multiset {0k, 1n−k} has a core group containing Sk×Sn−k as a subgroup. We demonstrate that the core group of a set A is the group of permutations of the rows of a certain matrix H (depending only on the inversion patterns of the objects in A) which preserve the column space of H. This allows us to compute the core group with no knowledge of the actual w-i statistics involved.}
}

@article{FOATA199679,
title = {Graphical major indices},
journal = {Journal of Computational and Applied Mathematics},
volume = {68},
number = {1},
pages = {79-101},
year = {1996},
issn = {0377-0427},
doi = {https://doi.org/10.1016/0377-0427(95)00254-5},
url = {https://www.sciencedirect.com/science/article/pii/0377042795002545},
author = {Dominique Foata and Doron Zeilberger},
keywords = {Major index, Bipartitional relation, MacMahon Verfahren},
abstract = {A generalization of the classical statistics “maj” and “inv” (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equidistributed “inv” and “maj” is posed and answered.
Résumé
On introduit une généralisation des statistiques classiques que sont “maj” et “inv” (l'indice majeur et le nombre d'inversions) sur les mots, qui est paramétrisée par des graphes arbitraires sur l'alphabet sous-jacent. La question de caractériser ces graphes conduisant à des statistiques “inv” et “maj” qui soient équidistribuées est posée et résolue.}
}

@article{JOSUATVERGES20121613,
title = {The algebraic combinatorics of snakes},
journal = {Journal of Combinatorial Theory, Series A},
volume = {119},
number = {8},
pages = {1613-1638},
year = {2012},
issn = {0097-3165},
doi = {https://doi.org/10.1016/j.jcta.2012.05.002},
url = {https://www.sciencedirect.com/science/article/pii/S0097316512000738},
author = {Matthieu Josuat-Vergès and Jean-Christophe Novelli and Jean-Yves Thibon},
keywords = {Noncommutative symmetric functions, Euler numbers, Snakes},
abstract = {Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.}
}

@book{foata2006theorie,
  title={Theorie Geometrique des Polynomes Euleriens},
  author={Foata, D. and Sch{\"u}tzenberger, M.P.},
  isbn={9783540362944},
  series={Lecture Notes in Mathematics},
  url={https://books.google.com/books?id=oGJ6CwAAQBAJ},
  year={2006},
  publisher={Springer Berlin Heidelberg}
}


@misc{foata2005theorie,
      title={Th\'eorie G\'eom\'etrique des Polyn\^omes Eul\'eriens}, 
      author={Dominique Foata and Marcel-Paul Schützenberger},
      year={2005},
      eprint={math/0508232},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}

@incollection{FOATA1973173,
title = {Nombres d'Euler et Permutations Alternantes},
editor = {JAGDISH N. SRIVASTAVA},
booktitle = {A Survey of Combinatorial Theory},
publisher = {North-Holland},
pages = {173-187},
year = {1973},
isbn = {978-0-7204-2262-7},
doi = {https://doi.org/10.1016/B978-0-7204-2262-7.50021-1},
url = {https://www.sciencedirect.com/science/article/pii/B9780720422627500211},
author = {D. FOATA and M.-P. SCHÜTZENBERGER}
}

@misc{stanley2009survey,
      title={A Survey of Alternating Permutations}, 
      author={Richard P. Stanley},
      year={2009},
      eprint={0912.4240},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}

@article{Even-Zohar2017,
author = {Even-Zohar, Chaim},
title = {The writhe of permutations and random framed knots},
journal = {Random Structures \& Algorithms},
volume = {51},
number = {1},
pages = {121-142},
keywords = {writhe, permutation statistics, framed knot, random knot, circular rank correlation, method of moments},
doi = {https://doi.org/10.1002/rsa.20704},
abstract = {ABSTRACT We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non-Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied with Hass, Linial, and Nowik (Discrete Comput Geom, 2016). The distribution of the framing in this model is equivalent to the writhe of random permutations. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 121–142, 2017},
year = {2017}
}

@article{azose2007applications,
  title={Applications of the q-Binomial Coefficients to Counting Problems},
  author={Azose, Jonathan},
  year={2007}
}

@article{KALAI2002412,
title = {A Fourier-theoretic perspective on the Condorcet paradox and Arrow's theorem},
journal = {Advances in Applied Mathematics},
volume = {29},
number = {3},
pages = {412-426},
year = {2002},
issn = {0196-8858},
doi = {https://doi.org/10.1016/S0196-8858(02)00023-4},
url = {https://www.sciencedirect.com/science/article/pii/S0196885802000234},
author = {Gil Kalai},
abstract = {We describe a Fourier-theoretic formula for the probability of rational outcomes for a social choice function on three alternatives. Several applications are given.}
}

@book{karp2010reducibility,
  title={Reducibility among combinatorial problems},
  author={Karp, Richard M},
  year={2010},
  publisher={Springer}
}

@article{stanley1973acyclic,
  title={Acyclic orientations of graphs},
  author={Stanley, Richard P},
  journal={Discrete Mathematics},
  volume={5},
  number={2},
  pages={171--178},
  year={1973},
  publisher={Elsevier}
}
@book{beck2018combinatorial,
  title={Combinatorial reciprocity theorems},
  author={Beck, Matthias and Sanyal, Raman},
  volume={195},
  year={2018},
  publisher={American Mathematical Soc.}
}
@book{flajolet2009analytic,
  title={Analytic combinatorics},
  author={Flajolet, Philippe and Sedgewick, Robert},
  year={2009},
  publisher={cambridge University press}
}

@article{CHUNG201762,
title = {The drop polynomial of a weighted digraph},
journal = {Journal of Combinatorial Theory, Series B},
volume = {126},
pages = {62-82},
year = {2017},
issn = {0095-8956},
doi = {https://doi.org/10.1016/j.jctb.2017.04.003},
url = {https://www.sciencedirect.com/science/article/pii/S0095895617300254},
author = {Fan Chung and Ron Graham},
}

@article{KADELL198522,
title = {Weighted inversion numbers, restricted growth functions, and standard young tableaux},
journal = {Journal of Combinatorial Theory, Series A},
volume = {40},
number = {1},
pages = {22-44},
year = {1985},
issn = {0097-3165},
doi = {https://doi.org/10.1016/0097-3165(85)90044-5},
url = {https://www.sciencedirect.com/science/article/pii/0097316585900445},
author = {Kevin W.J Kadell},
abstract = {We give a family of weighted inversion numbers with the same generating function which interpolate between the inversion number and MacMahon's major index. Foata's bijection is obtained in a natural way from a simple involution. An alternative proof uses q-difference equations which yield some new results. We obtain a new generating function for restricted growth functions and two q-analogs of a formula for the number of standard Young tableaux of a given shape. While the first really goes back to MacMahon, the second uses one of our weighted inversion numbers and appears to be new.}
}

@book{lehmann1975nonparametrics,
  title={Nonparametrics: statistical methods based on ranks.},
  author={Lehmann, Erich Leo and D'Abrera, Howard J},
  year={1975},
  publisher={Holden-day}
}
@article{BELARDO2018288,
title = {Wreath product of a complete graph with a cyclic graph: Topological indices and spectrum},
journal = {Applied Mathematics and Computation},
volume = {336},
pages = {288-300},
year = {2018},
issn = {0096-3003},
doi = {https://doi.org/10.1016/j.amc.2018.05.015},
url = {https://www.sciencedirect.com/science/article/pii/S0096300318304181},
author = {Francesco Belardo and Matteo Cavaleri and Alfredo Donno},
keywords = {Wreath product, Complete graph, Cyclic graph, Wiener index, Adjacency matrix, Spectrum},
abstract = {In this manuscript we continue the investigations related to the wreath product of graphs by considering the compound graph of a clique with a circuit. This product shows nice combinatorial and algebraic properties which permit with reasonable effort to compute some topological indices and the (adjacency) spectrum.}
}

@article{godsil1978new,
  title={A new graph product and its spectrum},
  author={Godsil, CD and McKay, BD},
  journal={Bulletin of the Australian Mathematical Society},
  volume={18},
  number={1},
  pages={21--28},
  year={1978},
  publisher={Cambridge University Press}
}

@book{macmahon2001combinatory,
  title={Combinatory analysis, volumes I and II},
  author={MacMahon, Percy Alexander},
  volume={1},
  year={2001},
  publisher={American Mathematical Soc.}
}

@article{macmahon1913indices,
  title={The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects},
  author={MacMahon, Percy Alexander},
  journal={American Journal of Mathematics},
  volume={35},
  number={3},
  pages={281--322},
  year={1913},
  publisher={JSTOR}
}

@article{Ellzey2019OnEO,
  title={On enumerators of Smirnov words by descents and cyclic descents},
  author={Brittney Ellzey and Michelle L. Wachs},
  journal={Journal of Combinatorics},
  year={2019},
  url={https://api.semanticscholar.org/CorpusID:119715932}
}

 @article{stembridge_1992, title={Eulerian numbers, tableaux, and the {B}etti numbers of a Toric variety}, volume={99}, DOI={10.1016/0012-365x(92)90378-s}, number={1-3}, journal={Discrete Mathematics}, author={Stembridge, J. R.}, year={1992}, pages={307–320}} 

@misc{celano2023eulerian,
      title={Eulerian Polynomials for Digraphs}, 
      author={Kyle Celano and Nicholas Sieger and Sam Spiro},
      year={2023},
      eprint={2309.07240},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}