@article {Assaf.Searles,
    AUTHOR = {Assaf, Sami and Searles, Dominic},
     TITLE = {Schubert polynomials, slide polynomials, {S}tanley symmetric
              functions and quasi-{Y}amanouchi pipe dreams},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {306},
      YEAR = {2017},
     PAGES = {89--122},
      ISSN = {0001-8708,1090-2082},
}

@article {Benedetti.Varbaro,
    AUTHOR = {Benedetti, Bruno and Varbaro, Matteo},
     TITLE = {On the dual graphs of {C}ohen-{M}acaulay algebras},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2015},
    NUMBER = {17},
     PAGES = {8085--8115},
      ISSN = {1073-7928},
}

@book {Berge,
    AUTHOR = {Berge, C.},
     TITLE = {Principles of combinatorics},
    SERIES = {Translated from the French. Mathematics in Science and
              Engineering, Vol. 72},
 PUBLISHER = {Academic Press, New York-London},
      YEAR = {1971},
     PAGES = {viii+176},
}

	
@article {Behrend,
    AUTHOR = {Behrend, Roger E.},
     TITLE = {Osculating paths and oscillating tableaux},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {15},
      YEAR = {2008},
    NUMBER = {1},
     PAGES = {Research Paper 7, 60},
      ISSN = {1077-8926},
}

@article {Bergeron.Billey,
    AUTHOR = {Bergeron, Nantel and Billey, Sara},
     TITLE = {R{C}-graphs and {S}chubert polynomials},
   JOURNAL = {Experiment. Math.},
  FJOURNAL = {Experimental Mathematics},
    VOLUME = {2},
      YEAR = {1993},
    NUMBER = {4},
     PAGES = {257--269},
}

@article {Bergeron.Sottile,
    AUTHOR = {Bergeron, Nantel and Sottile, Frank},
     TITLE = {Schubert polynomials, the {B}ruhat order, and the geometry of
              flag manifolds},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {95},
      YEAR = {1998},
    NUMBER = {2},
     PAGES = {373--423},
}

@article {Bergeron.Sottile.2002,
    AUTHOR = {Bergeron, Nantel and Sottile, Frank},
     TITLE = {Skew {S}chubert functions and the {P}ieri formula for flag
              manifolds},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {354},
      YEAR = {2002},
    NUMBER = {2},
     PAGES = {651--673},
      ISSN = {0002-9947,1088-6850},
   MRCLASS = {05E15 (06A07 14M15 14N15)},
  MRNUMBER = {1862562},
MRREVIEWER = {Mikhail\ Kogan},
       DOI = {10.1090/S0002-9947-01-02845-8},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1090/S0002-9947-01-02845-8},
}

@article{Billey.Holroyd.Young,
  title={A bijective proof of {M}acdonald's reduced word formula},
  author={Billey, Sara C. and Holroyd, Alexander E. and Young, Benjamin},
  journal={Algebraic Combin.},
  year={2019},
  volume={2},
  number={2},
  pages = {217-248},
}

@article {BJS93,
    AUTHOR = {Billey, Sara C. and Jockusch, William and Stanley, Richard P.},
     TITLE = {Some combinatorial properties of {S}chubert polynomials},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {2},
      YEAR = {1993},
    NUMBER = {4},
     PAGES = {345--374},
      ISSN = {0925-9899,1572-9192},
   MRCLASS = {05E05 (05E10 14M15 20F55)},
  MRNUMBER = {1241505},
MRREVIEWER = {Axel\ Kohnert},
       DOI = {10.1023/A:1022419800503},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1023/A:1022419800503},
}

@incollection {Billey.Vakil,
    AUTHOR = {Billey, Sara and Vakil, Ravi},
     TITLE = {Intersections of {S}chubert varieties and other permutation
              array schemes},
 BOOKTITLE = {Algorithms in algebraic geometry},
    SERIES = {IMA Vol. Math. Appl.},
    VOLUME = {146},
     PAGES = {21--54},
 PUBLISHER = {Springer, New York},
      YEAR = {2008},
}

@incollection {Brion,
    AUTHOR = {Brion, Michel},
     TITLE = {Multiplicity-free subvarieties of flag varieties},
 BOOKTITLE = {Commutative algebra ({G}renoble/{L}yon, 2001)},
    SERIES = {Contemp. Math.},
    VOLUME = {331},
     PAGES = {13--23},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2003},
}

@book {Bruns.Herzog,
    AUTHOR = {Bruns, Winfried and Herzog, J\"{u}rgen},
     TITLE = {Cohen-{M}acaulay rings},
    SERIES = {Cambridge Studies in Advanced Mathematics},
    VOLUME = {39},
 PUBLISHER = {Cambridge University Press, Cambridge},
      YEAR = {1993},
     PAGES = {xii+403},
      ISBN = {0-521-41068-1},
}

@article {Buch,
    AUTHOR = {Buch, Anders Skovsted},
     TITLE = {A {L}ittlewood-{R}ichardson rule for the {$K$}-theory of
              {G}rassmannians},
   JOURNAL = {Acta Math.},
  FJOURNAL = {Acta Mathematica},
    VOLUME = {189},
      YEAR = {2002},
    NUMBER = {1},
     PAGES = {37--78},
}

@article {Buch:twostep,
    AUTHOR = {Buch, Anders Skovsted},
     TITLE = {Mutations of puzzles and equivariant cohomology of two-step
              flag varieties},
   JOURNAL = {Ann. of Math. (2)},
  FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {182},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {173--220},
}

@article {Buch.Kresch.Purbhoo.Tamvakis,
    AUTHOR = {Buch, Anders Skovsted and Kresch, Andrew and Purbhoo, Kevin
              and Tamvakis, Harry},
     TITLE = {The puzzle conjecture for the cohomology of two-step flag
              manifolds},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {44},
      YEAR = {2016},
    NUMBER = {4},
     PAGES = {973--1007},
}

@article {Buch.Samuel,
    AUTHOR = {Buch, Anders Skovsted and Samuel, Matthew J.},
     TITLE = {{$K$}-theory of minuscule varieties},
   JOURNAL = {J. Reine Angew. Math.},
  FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's
              Journal]},
    VOLUME = {719},
      YEAR = {2016},
     PAGES = {133--171},
}

@article {Buciumas.Scrimshaw,
    AUTHOR = {Buciumas, Valentin and Scrimshaw, Travis},
     TITLE = {Double {G}rothendieck polynomials and colored lattice models},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2022},
    NUMBER = {10},
     PAGES = {7231--7258},
      ISSN = {1073-7928,1687-0247},
   MRCLASS = {14M15 (16T25 19L47)},
  MRNUMBER = {4418706},
MRREVIEWER = {Eric\ S.\ Egge},
       DOI = {10.1093/imrn/rnaa327},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1093/imrn/rnaa327},
}

@misc{CCRMM,
      title={K-polynomials of multiplicity-free varieties}, 
      author={Federico Castillo and Yairon Cid-Ruiz and Fatemeh Mohammadi and Jonathan Montaño},
      year={2025},
      eprint={2212.13091},
      archivePrefix={arXiv},
      primaryClass={math.AG},
}


@article {Carlitz,
    AUTHOR = {Carlitz, L.},
     TITLE = {A combinatorial property of {$q$}-{E}ulerian numbers},
   JOURNAL = {Amer. Math. Monthly},
  FJOURNAL = {American Mathematical Monthly},
    VOLUME = {82},
      YEAR = {1975},
     PAGES = {51--54},
      ISSN = {0002-9890},
}

@article {Chan.Pflueger,
    AUTHOR = {Chan, Melody and Pflueger, Nathan},
     TITLE = {Combinatorial relations on skew {S}chur and skew stable
              {G}rothendieck polynomials},
   JOURNAL = {Algebr. Comb.},
  FJOURNAL = {Algebraic Combinatorics},
    VOLUME = {4},
      YEAR = {2021},
    NUMBER = {1},
     PAGES = {175--188},
      ISSN = {2589-5486},
   MRCLASS = {05E05},
  MRNUMBER = {4226561},
MRREVIEWER = {Sho\ Matsumoto},
       DOI = {10.5802/alco.144},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.5802/alco.144},
}
@article {Claesson,
    AUTHOR = {Claesson, Anders},
     TITLE = {Generalized pattern avoidance},
   JOURNAL = {European J. Combin.},
  FJOURNAL = {European Journal of Combinatorics},
    VOLUME = {22},
      YEAR = {2001},
    NUMBER = {7},
     PAGES = {961--971},
      ISSN = {0195-6698},
}

@article {Escobar.Meszaros,
    AUTHOR = {Escobar, Laura and M\'{e}sz\'{a}ros, Karola},
     TITLE = {Toric matrix {S}chubert varieties and their polytopes},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {144},
      YEAR = {2016},
    NUMBER = {12},
     PAGES = {5081--5096},
}

@article {Escobar.Yong,
    AUTHOR = {Escobar, Laura and Yong, Alexander},
     TITLE = {Newton polytopes and symmetric {G}rothendieck polynomials},
   JOURNAL = {C. R. Math. Acad. Sci. Paris},
  FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris},
    VOLUME = {355},
      YEAR = {2017},
    NUMBER = {8},
     PAGES = {831--834},
      ISSN = {1631-073X,1778-3569},
   MRCLASS = {14C35 (05E05 14M25 14N15)},
  MRNUMBER = {3693501},
       DOI = {10.1016/j.crma.2017.07.003},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1016/j.crma.2017.07.003},
}

@article {Fink.Rajchgot.Sullivant,
    AUTHOR = {Fink, Alex and Rajchgot, Jenna and Sullivant, Seth},
     TITLE = {Matrix {S}chubert varieties and {G}aussian conditional
              independence models},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {44},
      YEAR = {2016},
    NUMBER = {4},
     PAGES = {1009--1046},
}

@incollection {Fomin.Greene,
    AUTHOR = {Fomin, Sergey and Greene, Curtis},
     TITLE = {Noncommutative {S}chur functions and their applications},
      NOTE = {Selected papers in honor of Adriano Garsia (Taormina, 1994)},
   JOURNAL = {Discrete Math.},
  FJOURNAL = {Discrete Mathematics},
    VOLUME = {193},
      YEAR = {1998},
    NUMBER = {1-3},
     PAGES = {179--200},
}

@incollection {Fomin.Kirillov,
    AUTHOR = {Fomin, Sergey and Kirillov, Anatol N.},
     TITLE = {Grothendieck polynomials and the {Y}ang-{B}axter equation},
 BOOKTITLE = {Formal power series and algebraic combinatorics/{S}\'eries
              formelles et combinatoire alg\'ebrique},
     PAGES = {183--189},
 PUBLISHER = {DIMACS, Piscataway, NJ},
      YEAR = {1994},
}

@inproceedings {Fomin.Kirillov.96,
    AUTHOR = {Fomin, Sergey and Kirillov, Anatol N.},
     TITLE = {The {Y}ang-{B}axter equation, symmetric functions, and
              {S}chubert polynomials},
 BOOKTITLE = {Proceedings of the 5th {C}onference on {F}ormal {P}ower
              {S}eries and {A}lgebraic {C}ombinatorics ({F}lorence, 1993)},
   JOURNAL = {Discrete Math.},
  FJOURNAL = {Discrete Mathematics},
    VOLUME = {153},
      YEAR = {1996},
    NUMBER = {1-3},
     PAGES = {123--143},
}

@article {Fomin.Stanley,
    AUTHOR = {Fomin, Sergey and Stanley, Richard P.},
     TITLE = {Schubert polynomials and the nil-{C}oxeter algebra},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {103},
      YEAR = {1994},
    NUMBER = {2},
     PAGES = {196--207},
      ISSN = {0001-8708,1090-2082},
   MRCLASS = {05E05 (14M15 20F55)},
  MRNUMBER = {1265793},
MRREVIEWER = {Axel\ Kohnert},
       DOI = {10.1006/aima.1994.1009},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1006/aima.1994.1009},
}

@article {Fulton,
    AUTHOR = {Fulton, William},
     TITLE = {Flags, {S}chubert polynomials, degeneracy loci, and
              determinantal formulas},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {65},
      YEAR = {1992},
    NUMBER = {3},
     PAGES = {381--420},
}

@article {Fulton:bulletin,
    AUTHOR = {Fulton, William},
     TITLE = {Eigenvalues, invariant factors, highest weights, and
              {S}chubert calculus},
   JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},
  FJOURNAL = {American Mathematical Society. Bulletin. New Series},
    VOLUME = {37},
      YEAR = {2000},
    NUMBER = {3},
     PAGES = {209--249},
}

@book {Fulton:YT,
    AUTHOR = {Fulton, William},
     TITLE = {Young tableaux. {W}ith applications to representation theory and geometry},
    SERIES = {London Mathematical Society Student Texts},
    VOLUME = {35},
 PUBLISHER = {Cambridge University Press, Cambridge},
      YEAR = {1997},
     PAGES = {x+260},
}

	

@article {Fulton.Lascoux,
    AUTHOR = {Fulton, William and Lascoux, Alain},
     TITLE = {A {P}ieri formula in the {G}rothendieck ring of a flag bundle},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {76},
      YEAR = {1994},
    NUMBER = {3},
     PAGES = {711--729},
}

@article {Gao.Huang,
    AUTHOR = {Gao, Yibo and Huang, Daoji},
     TITLE = {The canonical bijection between pipe dreams and bumpless pipe
              dreams},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2023},
    NUMBER = {21},
     PAGES = {18629--18663},
      ISSN = {1073-7928,1687-0247},
   MRCLASS = {14M15},
  MRNUMBER = {4665632},
MRREVIEWER = {Arpita\ Nayek},
       DOI = {10.1093/imrn/rnad083},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1093/imrn/rnad083},
}


@Article{Gaetz.Gao,
  author  = {Gaetz, Christian and Gao, Yibo},
  journal = {Proc. Amer. Math. Soc.},
  title   = {A combinatorial {$\mathfrak{sl}_2$}-action and the {S}perner property for the weak order},
  year    = {2019},
  note    = {\arxiv{1811.05501}},
}

@unpublished {Gaetz.Gao:2,
author = {Gaetz, Christian and Gao, Yibo},
title = {A combinatorial duality between the weak and strong {B}ruhat orders},
year = {2018},
note = {\arxiv{1812.05126}},
pages = {14 pages},
}

@article {Hamaker.Patrias.Pechenik.Williams,
    AUTHOR = {Hamaker, Zachary and Patrias, Rebecca and Pechenik, Oliver and
              Williams, Nathan},
     TITLE = {Doppelg\"{a}ngers: bijections of plane partitions},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2020},
    NUMBER = {2},
     PAGES = {487--540},
}

@article {Hamaker.Pechenik.Speyer.Weigandt,
    AUTHOR = {Hamaker, Zachary and Pechenik, Oliver and Speyer, David E and
              Weigandt, Anna},
     TITLE = {Derivatives of {S}chubert polynomials and proof of a
              determinant conjecture of {S}tanley},
   JOURNAL = {Algebr. Comb.},
  FJOURNAL = {Algebraic Combinatorics},
    VOLUME = {3},
      YEAR = {2020},
    NUMBER = {2},
     PAGES = {301--307},
}


@article {Hamaker.Pechenik.Weigandt,
    AUTHOR = {Hamaker, Zachary and Pechenik, Oliver and Weigandt, Anna},
     TITLE = {Gr\"obner geometry of {S}chubert polynomials through ice},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {398},
      YEAR = {2022},
     PAGES = {Paper No. 108228, 29},
      ISSN = {0001-8708,1090-2082},
   MRCLASS = {05E05 (05E14 13P10 14M15)},
  MRNUMBER = {4379747},
MRREVIEWER = {Frank\ Sottile},
       DOI = {10.1016/j.aim.2022.108228},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1016/j.aim.2022.108228},
}

@article {Hammett.Pittel,
    AUTHOR = {Hammett, Adam and Pittel, Boris},
     TITLE = {How often are two permutations comparable?},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {360},
      YEAR = {2008},
    NUMBER = {9},
     PAGES = {4541--4568},
      ISSN = {0002-9947},
}

	

@article {Hardt.Wallach,
author = {Hardt, Andrew and Wallach, David},
title = {Stability of products of double {G}rothendieck polynomials},
journal = {preprint},
note =     {\arxiv{2501.14691}},
year = {2025},
pages = {14 pages}
}

@article {Hsiao,
    AUTHOR = {Hsiao, Jen-Chieh},
     TITLE = {On the {$F$}-rationality and cohomological properties of
              matrix {S}chubert varieties},
   JOURNAL = {Illinois J. Math.},
  FJOURNAL = {Illinois Journal of Mathematics},
    VOLUME = {57},
      YEAR = {2013},
    NUMBER = {1},
     PAGES = {1--15},
}


@article {Huang,
    AUTHOR = {Huang, Daoji},
     TITLE = {Schubert products for permutations with separated descents},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2023},
    NUMBER = {20},
     PAGES = {17461--17493},
      ISSN = {1073-7928,1687-0247},
   MRCLASS = {05E14 (05E05 14N15)},
  MRNUMBER = {4659856},
MRREVIEWER = {Yibo\ Gao},
       DOI = {10.1093/imrn/rnac299},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1093/imrn/rnac299},
}

@article {Huang.2023,
    AUTHOR = {Huang, Daoji},
     TITLE = {Bijective proofs of {M}onk's rule for {S}chubert and double
              {S}chubert polynomials with bumpless pipe dreams},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {30},
      YEAR = {2023},
    NUMBER = {3},
     PAGES = {Paper No. 3.4, 14},
      ISSN = {1077-8926},
   MRCLASS = {05E05 (05E14)},
  MRNUMBER = {4614538},
MRREVIEWER = {Timothy\ Y.\ Chow},
       DOI = {10.37236/11824},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.37236/11824},
}
@misc{Huang.Shimozono.Yu,
      title={Marked Bumpless Pipedreams and Compatible Pairs}, 
      author={Daoji Huang and Mark Shimozono and Tianyi Yu},
      year={2024},
      eprint={2407.18160},
      archivePrefix={arXiv},
      primaryClass={math.CO},
}

@article {Huang.Striker,
    AUTHOR = {Huang, Daoji and Striker, Jessica},
     TITLE = {A pipe dream perspective on totally symmetric
              self-complementary plane partitions},
   JOURNAL = {Forum Math. Sigma},
  FJOURNAL = {Forum of Mathematics. Sigma},
    VOLUME = {12},
      YEAR = {2024},
     PAGES = {Paper No. e17, 19},
      ISSN = {2050-5094},
   MRCLASS = {05A19},
  MRNUMBER = {4696012},
MRREVIEWER = {Sam\ Hopkins},
       DOI = {10.1017/fms.2023.131},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1017/fms.2023.131},
}

@article {Hudson,
    AUTHOR = {Hudson, Thomas},
     TITLE = {A {T}hom-{P}orteous formula for connective {$K$}-theory using
              algebraic cobordism},
   JOURNAL = {J. K-Theory},
  FJOURNAL = {Journal of K-Theory. K-Theory and its Applications in Algebra,
              Geometry, Analysis \& Topology},
    VOLUME = {14},
      YEAR = {2014},
    NUMBER = {2},
     PAGES = {343--369},
}

@article {Klein,
    AUTHOR = {Klein, Patricia},
     TITLE = {Diagonal degenerations of matrix {S}chubert varieties},
   JOURNAL = {Algebr. Comb.},
  FJOURNAL = {Algebraic Combinatorics},
    VOLUME = {6},
      YEAR = {2023},
    NUMBER = {4},
     PAGES = {1073--1094},
      ISSN = {2589-5486},
   MRCLASS = {05E14 (05E05 13P10 14M15)},
  MRNUMBER = {4635090},
       DOI = {10.5802/alco.296},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.5802/alco.296},
}

@misc{Klein.Weigandt,
      title={Bumpless pipe dreams encode Gr\"obner geometry of Schubert polynomials}, 
      author={Patricia Klein and Anna Weigandt},
      year={2025},
      eprint={2108.08370},
      archivePrefix={arXiv},
      primaryClass={math.CO},
}

@article {Kogan.Miller,
    AUTHOR = {Kogan, Mikhail and Miller, Ezra},
     TITLE = {Toric degeneration of {S}chubert varieties and
              {G}elfand-{T}setlin polytopes},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {193},
      YEAR = {2005},
    NUMBER = {1},
     PAGES = {1--17},
      ISSN = {0001-8708,1090-2082},
}

@article {Knutson:cycling,
    AUTHOR = {Knutson, Allen},
     TITLE = {Descent-cycling in {S}chubert calculus},
   JOURNAL = {Experiment. Math.},
  FJOURNAL = {Experimental Mathematics},
    VOLUME = {10},
      YEAR = {2001},
    NUMBER = {3},
     PAGES = {345--353},
}

@article{Knutson:recurrence,
author = {Knutson, Allen},
title = {A {S}chubert calculus recurrence from the noncomplex ${W}$-action on ${G/B}$},
journal = {preprint},
note = {\arxiv{math/0306304}},
year = {2003},
pages = {51 pages}
}

@inbook{Knutson:cotransition, 
place={Cambridge}, series={London Mathematical Society Lecture Note Series}, title={Schubert Polynomials, Pipe Dreams, Equivariant Classes, and a Co-transition Formula}, booktitle={Facets of Algebraic Geometry: A Collection in Honor of William Fulton’s 80th Birthday}, publisher={Cambridge University Press}, author={Knutson, Allen}, editor={Aluffi, Paolo and Anderson, David and Hering, Milena and Mustaţă, Mircea and Payne, SamEditors}, year={2022}, pages={63–83}, collection={London Mathematical Society Lecture Note Series}}

@ARTICLE{Knutson:Mobius,
       author = {Knutson, Allen},
        title = {Frobenius splitting and {M}\"obius inversion},
      journal = {preprint},
         year = {2009},
	note={\arxiv{0902.1930}},
	pages={5 pages},
}

@article{Knutson:ICM,
author = {Knutson, Allen},
title = {{S}chubert calculus in quiver varieties},
journal = {Proc. ICM},
note = {to appear, \url{https://pi.math.cornell.edu/~allenk/Knutson2022.pdf}},
year = {2022},
pages = {23 pages}
}

		

@article {Knutson.Miller,
    AUTHOR = {Knutson, Allen and Miller, Ezra},
     TITLE = {Gr\"obner geometry of {S}chubert polynomials},
   JOURNAL = {Ann. of Math. (2)},
  FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {161},
      YEAR = {2005},
    NUMBER = {3},
     PAGES = {1245--1318},
      ISSN = {0003-486X,1939-8980},
   MRCLASS = {05E15 (13C40 13F55 13P10 14M15 14N15)},
  MRNUMBER = {2180402},
MRREVIEWER = {Harry\ Tamvakis},
       DOI = {10.4007/annals.2005.161.1245},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.4007/annals.2005.161.1245},
}

@article {Knutson.Miller:subword,
    AUTHOR = {Knutson, Allen and Miller, Ezra},
     TITLE = {Subword complexes in {C}oxeter groups},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {184},
      YEAR = {2004},
    NUMBER = {1},
     PAGES = {161--176},
}

@article {Knutson.Miller.Yong,
    AUTHOR = {Knutson, Allen and Miller, Ezra and Yong, Alexander},
     TITLE = {Gr\"obner geometry of vertex decompositions and of flagged
              tableaux},
   JOURNAL = {J. Reine Angew. Math.},
  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's
              Journal]},
    VOLUME = {630},
      YEAR = {2009},
     PAGES = {1--31},
      ISSN = {0075-4102,1435-5345},
   MRCLASS = {13C40 (05E05 13F55 14M12)},
  MRNUMBER = {2526784},
MRREVIEWER = {Harry\ Tamvakis},
       DOI = {10.1515/CRELLE.2009.033},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1515/CRELLE.2009.033},
}

@article {Knutson.Purbhoo,
    AUTHOR = {Knutson, Allen and Purbhoo, Kevin},
     TITLE = {Product and puzzle formulae for {${\rm GL}_n$}
              {B}elkale-{K}umar coefficients},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {18},
      YEAR = {2011},
    NUMBER = {1},
     PAGES = {Paper 76, 20 pages},
}

@article {Knutson.Tao:honeycomb,
    AUTHOR = {Knutson, Allen and Tao, Terence},
     TITLE = {The honeycomb model of {${\rm GL}_n({\bf C})$} tensor
              products. {I}. {P}roof of the saturation conjecture},
   JOURNAL = {J. Amer. Math. Soc.},
  FJOURNAL = {Journal of the American Mathematical Society},
    VOLUME = {12},
      YEAR = {1999},
    NUMBER = {4},
     PAGES = {1055--1090},
}

	

@article {Knutson.Tao:HT,
    AUTHOR = {Knutson, Allen and Tao, Terence},
     TITLE = {Puzzles and (equivariant) cohomology of {G}rassmannians},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {119},
      YEAR = {2003},
    NUMBER = {2},
     PAGES = {221--260},
}

	


@article {Knutson.Tao.Woodward,
    AUTHOR = {Knutson, Allen and Tao, Terence and Woodward, Christopher},
     TITLE = {The honeycomb model of {${\rm GL}_n(\mathbb C)$} tensor products.
              {II}. {P}uzzles determine facets of the
              {L}ittlewood-{R}ichardson cone},
   JOURNAL = {J. Amer. Math. Soc.},
  FJOURNAL = {Journal of the American Mathematical Society},
    VOLUME = {17},
      YEAR = {2004},
    NUMBER = {1},
     PAGES = {19--48},
}

@article{Knutson.Zinn-Justin:1,
author = {Knutson, Allen and Zinn{-}Justin, Paul},
title = {Schubert puzzles and integrability {I}: {I}nvariant trilinear forms},
journal = {preprint},
note =     {\arxiv{1706.10019}},
year = {2017},
pages = {51 pages}
}

@article{Knutson.Zinn-Justin:2,
author = {Knutson, Allen and Zinn{-}Justin, Paul},
title = {Schubert puzzles and integrability {II}: {M}ultiplying motivic {S}egre classes},
journal = {preprint},
note =     {\arxiv{2102.00563}},
year = {2021},
pages = {77 pages}
}

@article {Kogan.Kumar,
    AUTHOR = {Kogan, Mikhail and Kumar, Abhinav},
     TITLE = {A proof of {P}ieri's formula using the generalized {S}chensted
              insertion algorithm for rc-graphs},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {130},
      YEAR = {2002},
    NUMBER = {9},
     PAGES = {2525--2534},
}

@article {Lam.Lee.Shimozono,
    AUTHOR = {Lam, Thomas and Lee, Seung Jin and Shimozono, Mark},
     TITLE = {Back stable {S}chubert calculus},
   JOURNAL = {Compos. Math.},
  FJOURNAL = {Compositio Mathematica},
    VOLUME = {157},
      YEAR = {2021},
    NUMBER = {5},
     PAGES = {883--962},
      ISSN = {0010-437X,1570-5846},
   MRCLASS = {14M15 (05E05)},
  MRNUMBER = {4252201},
MRREVIEWER = {Praise\ Adeyemo},
       DOI = {10.1112/S0010437X21007028},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1112/S0010437X21007028},
}

@article {Lam.Lee.Shimozono-K-theory,
    AUTHOR = {Lam, Thomas and Lee, Seung Jin and Shimozono, Mark},
     TITLE = {Back stable {$K$}-theory {S}chubert calculus},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2023},
    NUMBER = {24},
     PAGES = {21381--21466},
      ISSN = {1073-7928,1687-0247},
   MRCLASS = {19E20 (14N15 19D55)},
  MRNUMBER = {4681291},
       DOI = {10.1093/imrn/rnac315},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1093/imrn/rnac315},
}


@Article{Lascoux,
  author  = {Lascoux, Alain},
  journal = {preprint},
  title   = {{C}hern and {Y}ang through ice},
  year    = {2002},
}

@article {Lascoux.03,
    AUTHOR = {Lascoux, Alain},
     TITLE = {Schubert \& {G}rothendieck: un bilan bid\'{e}cennal},
   JOURNAL = {S\'{e}m. Lothar. Combin.},
  FJOURNAL = {S\'{e}minaire Lotharingien de Combinatoire},
    VOLUME = {50},
      YEAR = {2003/04},
     PAGES = {Art. B50i, 32},
      ISSN = {1286-4889},
}


@article {Lascoux.Schutzenberger,
    AUTHOR = {Lascoux, Alain and Sch{\"u}tzenberger, Marcel-Paul},
     TITLE = {Structure de {H}opf de l'anneau de cohomologie et de l'anneau
              de {G}rothendieck d'une vari\'et\'e de drapeaux},
   JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.},
  FJOURNAL = {Comptes Rendus des S\'eances de l'Acad\'emie des Sciences.
              S\'erie I. Math\'ematique},
    VOLUME = {295},
      YEAR = {1982},
    NUMBER = {11},
     PAGES = {629--633},
}

@article {Lascoux.Schutzenberger:Schubert,
    AUTHOR = {Lascoux, Alain and Sch\"{u}tzenberger, Marcel-Paul},
     TITLE = {Polyn\^{o}mes de {S}chubert},
   JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. I Math.},
  FJOURNAL = {Comptes Rendus des S\'{e}ances de l'Acad\'{e}mie des Sciences. S\'{e}rie
              I. Math\'{e}matique},
    VOLUME = {294},
      YEAR = {1982},
    NUMBER = {13},
     PAGES = {447--450},
}

@article {LOTRZ,
    AUTHOR = {Le, Tuong and Ouyang, Shuge and Tao, Leo and Restivo, Joseph
              and Zhang, Angelina},
     TITLE = {Quantum bumpless pipe dreams},
   JOURNAL = {Forum Math. Sigma},
  FJOURNAL = {Forum of Mathematics. Sigma},
    VOLUME = {13},
      YEAR = {2025},
     PAGES = {Paper No. e28, 21},
      ISSN = {2050-5094},
   MRCLASS = {05E05 (14N15)},
  MRNUMBER = {4860030},
MRREVIEWER = {Eric\ S.\ Egge},
       DOI = {10.1017/fms.2024.112},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1017/fms.2024.112},
}
@article {Lenart,
    AUTHOR = {Lenart, Cristian},
     TITLE = {Noncommutative {S}chubert calculus and {G}rothendieck
              polynomials},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {143},
      YEAR = {1999},
    NUMBER = {1},
     PAGES = {159--183},
      ISSN = {0001-8708,1090-2082},
       DOI = {10.1006/aima.1998.1795},
       URL = {https://doi.org/10.1006/aima.1998.1795},
}

@article {Lenart.2000,
	AUTHOR = {Lenart, Cristian},
	TITLE = {Combinatorial aspects of the {$K$}-theory of {G}rassmannians},
	JOURNAL = {Ann. Comb.},
	FJOURNAL = {Annals of Combinatorics},
	VOLUME = {4},
	YEAR = {2000},
	NUMBER = {1},
	PAGES = {67--82},
	ISSN = {0218-0006,0219-3094},
	DOI = {10.1007/PL00001276},
	URL = {https://doi.org/10.1007/PL00001276},
}


@article {Lenart.2003,
    AUTHOR = {Lenart, Cristian},
     TITLE = {A {$K$}-theory version of {M}onk's formula and some related
              multiplication formulas},
   JOURNAL = {J. Pure Appl. Algebra},
  FJOURNAL = {Journal of Pure and Applied Algebra},
    VOLUME = {179},
      YEAR = {2003},
    NUMBER = {1-2},
     PAGES = {137--158},
      ISSN = {0022-4049,1873-1376},
       DOI = {10.1016/S0022-4049(02)00208-6},
       URL = {https://doi.org/10.1016/S0022-4049(02)00208-6},
}

@article {Lenart.Sottile,
    AUTHOR = {Lenart, Cristian and Sottile, Frank},
     TITLE = {Skew {S}chubert polynomials},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {131},
      YEAR = {2003},
    NUMBER = {11},
     PAGES = {3319--3328},
      ISSN = {0002-9939,1088-6826},
}

@article {Lenart.Robinson.Sottile,
    AUTHOR = {Lenart, Cristian and Robinson, Shawn and Sottile, Frank},
     TITLE = {Grothendieck polynomials via permutation patterns and chains
              in the {B}ruhat order},
   JOURNAL = {Amer. J. Math.},
  FJOURNAL = {American Journal of Mathematics},
    VOLUME = {128},
      YEAR = {2006},
    NUMBER = {4},
     PAGES = {805--848},
      ISSN = {0002-9327,1080-6377},
   MRCLASS = {05E15 (14M15)},
  MRNUMBER = {2251587},
MRREVIEWER = {Julianna\ Tymoczko},
       URL = {http://muse.jhu.edu.srv-proxy1.library.tamu.edu/journals/american_journal_of_mathematics/v128/128.4lenart.pdf},
}

@article {Lesieur,
    AUTHOR = {Lesieur, L\'{e}once},
     TITLE = {Les probl\`emes d'intersection sur une vari\'{e}t\'{e} de {G}rassmann},
   JOURNAL = {C. R. Acad. Sci. Paris},
  FJOURNAL = {Comptes Rendus Hebdomadaires des S\'{e}ances de l'Acad\'{e}mie des
              Sciences},
    VOLUME = {225},
      YEAR = {1947},
     PAGES = {916--917},
}

@article {Littlewood.Richardson,
    AUTHOR = {Littlewood, D.E. and Richardson, A.R.},
     TITLE = {Group characters and algebra},
   JOURNAL = {Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.},
    VOLUME = {233},
      YEAR = {1934},
     PAGES = {99--141},
}

@book {Macdonald,
    AUTHOR = {Macdonald, I. G.},
     TITLE = {Symmetric functions and {H}all polynomials},
    SERIES = {Oxford Mathematical Monographs},
   EDITION = {Second},
      NOTE = {With contributions by A. Zelevinsky,
              Oxford Science Publications},
 PUBLISHER = {The Clarendon Press, Oxford University Press, New York},
      YEAR = {1995},
     PAGES = {x+475},
}     

@Book{Macdonald:notes,
author = {I. G. Macdonald},
title = {Notes on {S}chubert polynomials},
publisher = {Publications du LACIM, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al},
year = {1991},
volume = {6},
}


@book {Manivel,
    AUTHOR = {Manivel, Laurent},
     TITLE = {Symmetric functions, {S}chubert polynomials and degeneracy
              loci},
    SERIES = {SMF/AMS Texts and Monographs},
    VOLUME = {6},
      NOTE = {Translated from the 1998 French original by John R. Swallow,
              Cours Sp\'{e}cialis\'{e}s, 3},
 PUBLISHER = {American Mathematical Society, Providence, RI; Soci\'{e}t\'{e} Math\'{e}matique de France, Paris},
      YEAR = {2001},
     PAGES = {viii+167},
}

@incollection {Matsuki.Oshima,
    AUTHOR = {Matsuki, Toshihiko and \={O}shima, Toshio},
     TITLE = {Embeddings of discrete series into principal series},
 BOOKTITLE = {The orbit method in representation theory ({C}openhagen,
              1988)},
    SERIES = {Progr. Math.},
    VOLUME = {82},
     PAGES = {147--175},
 PUBLISHER = {Birkh\"{a}user Boston, Boston, MA},
      YEAR = {1990},
}

@article {Meszaros,
    AUTHOR = {M\'esz\'aros, Karola},
     TITLE = {Pipe dream complexes and triangulations of root polytopes
              belong together},
   JOURNAL = {SIAM J. Discrete Math.},
  FJOURNAL = {SIAM Journal on Discrete Mathematics},
    VOLUME = {30},
      YEAR = {2016},
    NUMBER = {1},
     PAGES = {100--111},
      ISSN = {0895-4801,1095-7146},
}


@article {Meszaros.StDizier,
    AUTHOR = {M\'esz\'aros, Karola and St. Dizier, Avery},
     TITLE = {From generalized permutahedra to {G}rothendieck polynomials
              via flow polytopes},
   JOURNAL = {Algebr. Comb.},
  FJOURNAL = {Algebraic Combinatorics},
    VOLUME = {3},
      YEAR = {2020},
    NUMBER = {5},
     PAGES = {1197--1229},
      ISSN = {2589-5486},
   MRCLASS = {05E10 (05C21 52B20)},
  MRNUMBER = {4166815},
MRREVIEWER = {Andrew\ Berget},
       DOI = {10.5802/alco.136},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.5802/alco.136},
}

@article{Meszaros.Setiabrata.StDizier,
author = {M\'{e}sz\'{a}ros, Karola and Setiabrata, Linus and St.{ }Dizier, Avery},
  title     = {On the Support of {G}rothendieck Polynomials},
  journal   = {Ann. Comb.},
  year      = {2024},
  doi       = {10.1007/s00026-024-00712-3},
  url       = {https://doi.org/10.1007/s00026-024-00712-3},
  issn      = {0219-3094},
}


@article {Monk,
    AUTHOR = {Monk, D.},
     TITLE = {The geometry of flag manifolds},
   JOURNAL = {Proc. London Math. Soc. (3)},
  FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
    VOLUME = {9},
      YEAR = {1959},
     PAGES = {253--286},
}

	
	
@article {Monical.Tokcan.Yong,
    AUTHOR = {Monical, Cara and Tokcan, Neriman and Yong, Alexander},
     TITLE = {Newton polytopes in algebraic combinatorics},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {25},
      YEAR = {2019},
    NUMBER = {5},
     PAGES = {Paper No. 66, 37},
      ISSN = {1022-1824,1420-9020},
   MRCLASS = {05E05 (05E10)},
  MRNUMBER = {4021852},
MRREVIEWER = {Allan\ Berele},
       DOI = {10.1007/s00029-019-0513-8},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1007/s00029-019-0513-8},
}
@article {Mulmuley.Narayanan.Sohoni,
    AUTHOR = {Mulmuley, Ketan D. and Narayanan, Hariharan and Sohoni,
              Milind},
     TITLE = {Geometric complexity theory {III}: on deciding nonvanishing of
              a {L}ittlewood-{R}ichardson coefficient},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {36},
      YEAR = {2012},
    NUMBER = {1},
     PAGES = {103--110},
}

@article {Narayanan,
    AUTHOR = {Narayanan, Hariharan},
     TITLE = {On the complexity of computing {K}ostka numbers and
              {L}ittlewood-{R}ichardson coefficients},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {24},
      YEAR = {2006},
    NUMBER = {3},
     PAGES = {347--354},
}

@article{Nenashev,
author = {Nenashev, Gleb},
title = {Differential operators on {S}chur and {S}chubert polynomials},
journal = {preprint},
note =     {\arxiv{2005.08329}},
year = {2020},
pages = {19 pages}
}

@misc { OEIS,
title={The {O}n-{L}ine {E}ncyclopedia of {I}nteger {S}equences},
author= {N. J. A. Sloane},
note={\href{http://oeis.org}{\texttt{http://oeis.org}}}
}


@article {Pechenik.Satriano,
    AUTHOR = {Pechenik, Oliver and Satriano, Matthew},
     TITLE = {Proof of a conjectured {M}\"obius inversion formula for
              {G}rothendieck polynomials},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {30},
      YEAR = {2024},
    NUMBER = {5},
     PAGES = {Paper No. 83, 8},
      ISSN = {1022-1824,1420-9020},
}

@article{Pechenik.Speyer.Weigandt,
author = {Pechenik, Oliver and Speyer, David E and Weigandt, Anna},
title = {Castelnuovo--{M}umford regularity of matrix {S}chubert varieties},
journal = {preprint},
note =     {\arxiv{2111.10681}},
year = {2021},
pages = {36 pages}
}

@article {Pechenik.Yong,
    AUTHOR = {Pechenik, Oliver and Yong, Alexander},
     TITLE = {Equivariant {$K$}-theory of {G}rassmannians},
   JOURNAL = {Forum Math. Pi},
  FJOURNAL = {Forum of Mathematics. Pi},
    VOLUME = {5},
      YEAR = {2017},
     PAGES = {e3, 128 pages},
}

@article {Petersen,
    AUTHOR = {Petersen, T. Kyle},
     TITLE = {A two-sided analogue of the {C}oxeter complex},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {25},
      YEAR = {2018},
    NUMBER = {4},
     PAGES = {Paper 4.64, 28},
}

@article {Proctor:trapezoidal,
    AUTHOR = {Proctor, Robert A.},
     TITLE = {Shifted plane partitions of trapezoidal shape},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {89},
      YEAR = {1983},
    NUMBER = {3},
     PAGES = {553--559},
}


@article{Proctor:det,
  title={Product evaluations of {L}efschetz determinants for {G}rassmannians and of determinants of multinomial coefficients},
  author={Proctor, Robert A.},
  journal={J. Combin. Theory Ser. A},
  volume={54},
  number={2},
  pages={235--247},
  year={1990},
}

@article {Purbhoo:vanishing,
    AUTHOR = {Purbhoo, Kevin},
     TITLE = {Vanishing and nonvanishing criteria in {S}chubert calculus},
   JOURNAL = {Int. Math. Res. Not.},
  FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
     PAGES = {Art. ID 24590, 38 pages},
}

@article {Purbhoo:game,
    AUTHOR = {Purbhoo, Kevin},
     TITLE = {Root games on {G}rassmannians},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {25},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {239--258},
}

	


@article {RRRSW,
    AUTHOR = {Rajchgot, Jenna and Ren, Yi and Robichaux, Colleen and St.
              Dizier, Avery and Weigandt, Anna},
     TITLE = {Degrees of symmetric {G}rothendieck polynomials and
              {C}astelnuovo-{M}umford regularity},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {149},
      YEAR = {2021},
    NUMBER = {4},
     PAGES = {1405--1416},
      ISSN = {0002-9939,1088-6826},
   MRCLASS = {13C40 (05E40 14N15)},
  MRNUMBER = {4242300},
MRREVIEWER = {Quang\ Hoa\ Tran},
       DOI = {10.1090/proc/15294},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1090/proc/15294},
}

@article {Ramanathan,
    AUTHOR = {Ramanathan, A.},
     TITLE = {Schubert varieties are arithmetically {C}ohen-{M}acaulay},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {80},
      YEAR = {1985},
    NUMBER = {2},
     PAGES = {283--294},
}

@article {Richardson.Springer,
    AUTHOR = {Richardson, R. W. and Springer, T. A.},
     TITLE = {The {B}ruhat order on symmetric varieties},
   JOURNAL = {Geom. Dedicata},
  FJOURNAL = {Geometriae Dedicata},
    VOLUME = {35},
      YEAR = {1990},
    NUMBER = {1-3},
     PAGES = {389--436},
}


@article {Smirnov.Tutubalina,
    AUTHOR = {Smirnov, E. Yu. and Tutubalina, A. A.},
     TITLE = {Slide polynomials and subword complexes},
   JOURNAL = {Sb. Math.},
    VOLUME = {212},
      YEAR = {2021},
    NUMBER = {10},
     PAGES = {1471--1490},
}

@article {Sottile,
    AUTHOR = {Sottile, Frank},
     TITLE = {Pieri's formula for flag manifolds and {S}chubert polynomials},
   JOURNAL = {Ann. Inst. Fourier (Grenoble)},
  FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier},
    VOLUME = {46},
      YEAR = {1996},
    NUMBER = {1},
     PAGES = {89--110},
}


@unpublished{Stanley:shen,
author = {Stanley, Richard P.},
title = {Some {S}chubert shenanigans},
note =     {\arxiv{1704.00851}},
year = {2017},
pages = {8 pages}
}


@article {Stanley:Weyl,
    AUTHOR = {Stanley, Richard P.},
     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},
}

@article{StDizier.Yong,
author = {St.{ }Dizier, Avery and Yong, Alexander},
title = {Generalized permutahedra and {S}chubert calculus},
journal = {preprint},
note =     {\arxiv{2010.14332}},
year = {2020},
pages = {13 pages}
}

@article {Thomas.Yong,
    AUTHOR = {Thomas, Hugh and Yong, Alexander},
     TITLE = {Multiplicity-free {S}chubert calculus},
   JOURNAL = {Canad. Math. Bull.},
  FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de
              Math\'{e}matiques},
    VOLUME = {53},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {171--186},
}

@article {Thomas.Yong:comin,
    AUTHOR = {Thomas, Hugh and Yong, Alexander},
     TITLE = {A combinatorial rule for (co)minuscule {S}chubert calculus},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {222},
      YEAR = {2009},
    NUMBER = {2},
     PAGES = {596--620},
}

@article {Weigandt,
    AUTHOR = {Weigandt, Anna},
     TITLE = {Bumpless pipe dreams and alternating sign matrices},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {182},
      YEAR = {2021},
     PAGES = {Paper No. 105470, 52},
      ISSN = {0097-3165,1096-0899},
   MRCLASS = {05E14 (05E05 14N15 19D99)},
  MRNUMBER = {4258766},
MRREVIEWER = {Edward\ E.\ Allen},
       DOI = {10.1016/j.jcta.2021.105470},
       URL = {https://doi-org.srv-proxy1.library.tamu.edu/10.1016/j.jcta.2021.105470},
}

@misc{Weigandt.Bases,
      title={Changing Bases with Pipe Dream Combinatorics}, 
      author={Anna Weigandt},
      year={2025},
      eprint={2506.07306},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}

@article {Weigandt.Yong,
    AUTHOR = {Weigandt, Anna and Yong, Alexander},
     TITLE = {The prism tableau model for {S}chubert polynomials},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {154},
      YEAR = {2018},
     PAGES = {551--582},
}

@article {Woo.Wyser,
    AUTHOR = {Woo, Alexander and Wyser, Benjamin J.},
     TITLE = {Combinatorial results on {$(1,2,1,2)$}-avoiding {${\rm
              GL}(p,\Bbb{C})\times{\rm GL}(q,\Bbb{C})$}-orbit closures on
              {${\rm GL}(p+q,\Bbb{C})/B$}},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2015},
    NUMBER = {24},
     PAGES = {13148--13193},
}

@article {Woo.Yong,
    AUTHOR = {Woo, Alexander and Yong, Alexander},
     TITLE = {A {G}r\"{o}bner basis for {K}azhdan-{L}usztig ideals},
   JOURNAL = {Amer. J. Math.},
  FJOURNAL = {American Journal of Mathematics},
    VOLUME = {134},
      YEAR = {2012},
    NUMBER = {4},
     PAGES = {1089--1137},
}

@article {Wyser,
    AUTHOR = {Wyser, Benjamin J.},
     TITLE = {Schubert calculus of {R}ichardson varieties stable under
              spherical {L}evi subgroups},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {38},
      YEAR = {2013},
    NUMBER = {4},
     PAGES = {829--850},
}

@article {Wyser.Yong,
    AUTHOR = {Wyser, Benjamin J. and Yong, Alexander},
     TITLE = {Polynomials for {${\rm GL}_p\times{\rm GL}_q$} orbit closures
              in the flag variety},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {4},
     PAGES = {1083--1110},
}

@article {Yamamoto,
    AUTHOR = {Yamamoto, Atsuko},
     TITLE = {Orbits in the flag variety and images of the moment map for
              classical groups. {I}},
   JOURNAL = {Represent. Theory},
  FJOURNAL = {Representation Theory. An Electronic Journal of the American
              Mathematical Society},
    VOLUME = {1},
      YEAR = {1997},
     PAGES = {329--404},
}

@book {Bjorner.Brenti,
    AUTHOR = {Bj\"orner, Anders and Brenti, Francesco},
     TITLE = {Combinatorics of {C}oxeter groups},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {231},
 PUBLISHER = {Springer, New York},
      YEAR = {2005},
     PAGES = {xiv+363},
      ISBN = {978-3540-442387; 3-540-44238-3},
}

@Comment{jabref-meta: databaseType:bibtex;}
