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We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.}, author = {Katie Gedeon and Nicholas Proudfoot and Benjamin Young}, doi = {https://doi.org/10.1016/j.jcta.2017.03.007}, issn = {0097-3165}, journal = {Journal of Combinatorial Theory, Series A}, keywords = {Kazhdan--Lusztig polynomials, Matroids, Equivariant matroids, Representation theory, Symmetric functions, Log concavity, Linear species, Schubert calculus, Representation stability}, pages = {267-294}, title = {The equivariant Kazhdan--Lusztig polynomial of a matroid}, url = {https://www.sciencedirect.com/science/article/pii/S0097316517300341}, volume = {150}, year = {2017}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0097316517300341}, bdsk-url-2 = {https://doi.org/10.1016/j.jcta.2017.03.007} } @misc{braden2020singular, title={Singular {H}odge theory for combinatorial geometries}, author={Braden, Tom and Huh, June and Matherne, Jacob P and Proudfoot, Nicholas and Wang, Botong}, eprint={2010.06088}, eprinttype={arxiv}, year={2020} } @incollection{harima2013lefschetz, title={Lefschetz properties}, author={Harima, Tadahito and Maeno, Toshiaki and Morita, Hideaki and Numata, Yasuhide and Wachi, Akihito and Watanabe, Junzo}, booktitle={The Lefschetz Properties}, pages={97--140}, year={2013}, publisher={Springer} } @article{hara2008determinants, title={The determinants of certain matrices arising from the {B}oolean lattice}, author={Hara, Masao and Watanabe, Junzo}, journal={Discrete mathematics}, volume={308}, number={23}, pages={5815--5822}, year={2008}, publisher={Elsevier} } @article{Krattenthaler, author = {Krattenthaler, C.}, title = {Combinatorial Proof of the Log-Concavity of the Sequence of Matching Numbers}, year = {1996}, issue_date = {May 1996}, publisher = {Academic Press, Inc.}, address = {USA}, volume = {74}, number = {2}, issn = {0097-3165}, url = {https://doi.org/10.1006/jcta.1996.0058}, doi = {10.1006/jcta.1996.0058}, abstract = {Fork lwe construct an injection from the set of pairs of matchings in a given graphGof sizesl 1 andk+1 into the set of pairs of matchings inGof sizeslandk. This provides a combinatorial proof of the log-concavity of the sequence of matching numbers of a graph. Besides, this injection implies that a certain weighted version of the matching numbers is stronglyx-log-concave in the sense of Sagan (Discrete Math.99(1992), 289 306).}, journal = {Journal of Combinatorial Theory, Series A}, month = may, pages = {351–354}, numpages = {4} } @book{Stanley2001Enumerative2, abstract = {This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and representation theory. An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.}, author = {Stanley, Richard P.}, file = {:home/anhduc/Desktop/Dropbox/Mendeley Desktop/Enumerative Combinatorics, Vol. 2 - Stanley - 2001.djvu:djvu}, isbn = {9780521789875}, publisher = {Cambridge University Press}, title = {{Enumerative Combinatorics, Vol. 2}}, url = {https://www.dropbox.com/sh/v4blv7sito3opzn/AACPseYqmcib7rg5-BMmt5A4a/Enumerative Combinatorics, Vol. 2 - Stanley - 2001.djvu?dl=0}, year = {2001} } @misc{matherne2021equivariant, title={Equivariant log concavity and representation stability}, author={Matherne, Jacob P and Miyata, Dane and Proudfoot, Nicholas and Ramos, Eric}, eprint={2104.00715}, eprinttype={arxiv}, year={2021} } @article{stanley1980weyl, author = {Stanley, Richard P.}, doi = {10.1137/0601021}, journal = {SIAM Journal on Algebraic Discrete Methods}, number = {2}, pages = {168-184}, title = {Weyl Groups, the Hard {L}efschetz Theorem, and the {S}perner Property}, volume = {1}, year = {1980}, bdsk-url-1 = {https://doi.org/10.1137/0601021}} @article{Proudfoot2018Z, author = {Nicholas Proudfoot and Yuan Xu and Benjamin Young}, title = {The {Z}-polynomial of a Matroid}, journal = {Electronic Journal of Combinatorics}, volume = {25}, number = {1}, pages = {1}, year = {2018}, url = {https://doi.org/10.37236/7105}, doi = {10.37236/7105}, timestamp = {Fri, 10 Jun 2022 10:35:09 +0200}, biburl = {https://dblp.org/rec/journals/combinatorics/ProudfootXY18.bib}, bibsource = {dblp computer science bibliography, https://dblp.org} } @article{stanley2013algebraic, title={Algebraic {C}ombinatorics}, author={Stanley, Richard P}, journal={Springer}, volume={20}, pages={22}, year={2013}, publisher={Springer} } @article{hamidoune1990, abstract = {Let G be a graph without induced K1,3. The number of independent k-subsets of V(G) will be denoted by sk. We prove that the sequence (sk) is log concave and hence unimodal. These results are known for line graphs as a consequence of a result of Heilman and Lieb.}, author = {Yahya Ould Hamidoune}, doi = {https://doi.org/10.1016/0095-8956(90)90079-F}, issn = {0095-8956}, journal = {Journal of Combinatorial Theory, Series B}, number = {2}, pages = {241-244}, title = {On the numbers of independent k-sets in a claw free graph}, url = {https://www.sciencedirect.com/science/article/pii/009589569090079F}, volume = {50}, year = {1990}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/009589569090079F}, bdsk-url-2 = {https://doi.org/10.1016/0095-8956(90)90079-F} } @inproceedings{stanley1983combinatorial, title={Combinatorial applications of the hard {L}efschetz theorem}, author={Stanley, Richard P}, booktitle={Proceedings of the International Congress of Mathematicians}, volume={1}, pages={2}, year={1983} }