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@Misc{M2,
          author = {Grayson, D.~R. and Stillman, M.~E.},
          title = {Macaulay2, a software system for research in algebraic geometry},
          howpublished = {Available at \url{http://www.math.uiuc.edu/Macaulay2/}}
        }

@book {Abhyankar,
    AUTHOR = {Abhyankar, S.~S.},
     TITLE = {Enumerative combinatorics of {Y}oung tableaux},
    SERIES = {Monographs and Textbooks in Pure and Applied Mathematics},
    VOLUME = {115},
 PUBLISHER = {Marcel Dekker, Inc., New York},
      YEAR = {1988},
     PAGES = {xx+509},
}

@article {AbKul,
    AUTHOR = {Abhyankar, S. and Kulkarni, D.~M.},
     TITLE = {On {H}ilbertian ideals},
   JOURNAL = {Linear Algebra Appl.},
  FJOURNAL = {Linear Algebra and its Applications},
    VOLUME = {116},
      YEAR = {1989},
     PAGES = {53--79},
      ISSN = {0024-3795},
}



@article {Benedetti.Varbaro,
    AUTHOR = {Benedetti, B. and Varbaro, M.},
     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},
       VOLUME = {2015},
    NUMBER = {17},
     PAGES = {8085--8115},
      ISSN = {1073-7928},
   MRCLASS = {14N20 (05E15)},
  MRNUMBER = {3404010},
MRREVIEWER = {Enrico Carlini},
}

@article {BS87,
    AUTHOR = {Bayer, D. and Stillman, M.},
     TITLE = {A criterion for detecting {$m$}-regularity},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {87},
      YEAR = {1987},
    NUMBER = {1},
     PAGES = {1--11},
      ISSN = {0020-9910},
}

@article {BergeronBilley,
    AUTHOR = {Bergeron, N. and Billey, S.},
     TITLE = {R{C}-graphs and {S}chubert polynomials},
   JOURNAL = {Experiment. Math.},
  FJOURNAL = {Experimental Mathematics},
    VOLUME = {2},
      YEAR = {1993},
    NUMBER = {4},
     PAGES = {257--269},
      ISSN = {1058-6458}
}
	
@article{BJS,
	Author = {Billey, S. and Jockusch, W. and Stanley, R.~P.},
    Journal = {J.~Algebraic Combin.},
	Number = {4},
	Pages = {345--374},
	Title = {Some combinatorial properties of {S}chubert polynomials},
	Volume = {2},
	Year = {1993}}



@Book{BH,
  author    = {Bruns, W. and Herzog, J.},
  publisher = {Cambridge University Press, Cambridge},
  title     = {Cohen-{M}acaulay rings},
  year      = {1993},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {39},
  pages     = {xii+403},
}

@article{Buch,
author = {Buch, A.~S.},
title = {{A Littlewood-Richardson rule for the K-theory of Grassmannians}},
volume = {189},
journal = {Acta Mathematica},
number = {1},
publisher = {Institut Mittag-Leffler},
pages = {37 -- 78},
year = {2002},
}



@Article{Buc02,
  author   = {Buch, A.~S.},
  journal  = {Duke Math. J.},
  title    = {Grothendieck classes of quiver varieties},
  year     = {2002},
  issn     = {0012-7094},
  number   = {1},
  pages    = {75--103},
  volume   = {115},
  fjournal = {Duke Mathematical Journal},
  mrclass  = {14C35 (05E10 19E08)},
}

@article {ConcaHerzog,
    AUTHOR = {Conca, A. and Herzog, J.},
     TITLE = {Ladder determinantal rings have rational singularities},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {132},
      YEAR = {1997},
    NUMBER = {1},
     PAGES = {120--147},
      ISSN = {0001-8708},
}
		
@article {Conca2,
    AUTHOR = {Conca, A.},
     TITLE = {Gorenstein ladder determinantal rings},
   JOURNAL = {J. London Math. Soc. (2)},
  FJOURNAL = {Journal of the London Mathematical Society. Second Series},
    VOLUME = {54},
      YEAR = {1996},
    NUMBER = {3},
     PAGES = {453--474},
      ISSN = {0024-6107},
}
		
@article {Conca,
    AUTHOR = {Conca, A.},
     TITLE = {Ladder determinantal rings},
   JOURNAL = {J. Pure Appl. Algebra},
  FJOURNAL = {Journal of Pure and Applied Algebra},
    VOLUME = {98},
      YEAR = {1995},
    NUMBER = {2},
     PAGES = {119--134},
      ISSN = {0022-4049},
}


      
@book {Eisenbud,
    AUTHOR = {Eisenbud, D.},
     TITLE = {The geometry of syzygies},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {229},
      NOTE = {A second course in commutative algebra and algebraic geometry},
 PUBLISHER = {Springer-Verlag, New York},
      YEAR = {2005},
     PAGES = {xvi+243},
}     
     
@Book{EisenbudCA,
  author     = {Eisenbud, D.},
  publisher  = {Springer-Verlag, New York},
  title      = {Commutative algebra},
  year       = {1995},
  note       = {With a view toward algebraic geometry},
  series     = {Graduate Texts in Mathematics},
  volume     = {150},
  pages      = {xvi+785},
}




@article{Fan.Guo,
title = {{Set-valued Rothe tableaux and Grothendieck polynomials}},
journal = {Advances in Applied Mathematics},
volume = {128},
pages = {102203},
year = {2021},
issn = {0196-8858},
author = {Fan, N.~J.~Y. and Guo, P.~L.},
}


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

@Article{Fulton.Flags,
  author  = {Fulton, W.},
  journal = {Duke Math. J.},
  title   = {Flags, {S}chubert polynomials, degeneracy loci, and determinantal formulas},
  year    = {1992},
  number  = {3},
  pages   = {381--420},
  volume  = {65},
}

@article{Gao,
title = {Principal specializations of {S}chubert polynomials and pattern containment},
journal = {European Journal of Combinatorics},
volume = {94},
pages = {103291},
year = {2021},
issn = {0195-6698},
author = {Gao, Y.}}


@article {GMN,
    AUTHOR = {Gorla, E. and Migliore, J.~C. and Nagel, U.},
     TITLE = {Gr\"{o}bner bases via linkage},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {384},
      YEAR = {2013},
     PAGES = {110--134},
      ISSN = {0021-8693},
}
		
@article {Gorla,
    AUTHOR = {Gorla, E.},
     TITLE = {Mixed ladder determinantal varieties from two-sided ladders},
   JOURNAL = {J. Pure Appl. Algebra},
  FJOURNAL = {Journal of Pure and Applied Algebra},
    VOLUME = {211},
      YEAR = {2007},
    NUMBER = {2},
     PAGES = {433--444},
      ISSN = {0022-4049},
}

@article {GonLak,
    AUTHOR = {Gonciulea, N. and Lakshmibai, V.},
     TITLE = {Singular loci of ladder determinantal varieties and {S}chubert
              varieties},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {229},
      YEAR = {2000},
    NUMBER = {2},
     PAGES = {463--497},
      ISSN = {0021-8693},
}

@article {GK15,
    AUTHOR = {Ghorpade, S.~R. and Krattenthaler, C.},
     TITLE = {Computation of the {$a$}-invariant of ladder determinantal
              rings},
   JOURNAL = {J. Algebra Appl.},
  FJOURNAL = {Journal of Algebra and its Applications},
    VOLUME = {14},
      YEAR = {2015},
    NUMBER = {9},
     PAGES = {1540014, 24},
      ISSN = {0219-4988},
}

@article {GM,
    AUTHOR = {Gonciulea, N. and Miller, C.},
     TITLE = {Mixed ladder determinantal varieties},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {231},
      YEAR = {2000},
    NUMBER = {1},
     PAGES = {104--137},
      ISSN = {0021-8693},
}

@article {Graham.Kreiman,
    AUTHOR = {Graham, W. and Kreiman, V.},
     TITLE = {Excited {Y}oung diagrams, equivariant {$K$}-theory, and
              {S}chubert varieties},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {367},
      YEAR = {2015},
    NUMBER = {9},
     PAGES = {6597--6645},
      ISSN = {0002-9947},
}


@misc{Knutson-Frob,
        AUTHOR = {Knutson, A.},
        EPRINT = {0911.4941},
    EPRINTTYPE = {arxiv},
         TITLE = {{Frobenius splitting, point-counting, and degeneration}},
          YEAR = {2009}
          }

@article {Kazhdan-Lusztig,
    AUTHOR = {Kazhdan, D. and Lusztig, G.},
     TITLE = {Representations of {C}oxeter groups and {H}ecke algebras},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {53},
      YEAR = {1979},
    NUMBER = {2},
     PAGES = {165--184},
      ISSN = {0020-9910},
}

@Article{KM.Subword,
  author   = {Knutson, A. and Miller, E.},
  journal  = {Adv. Math.},
  title    = {Subword complexes in {C}oxeter groups},
  year     = {2004},
  number   = {1},
  volume   = {184},  fjournal = {Advances in Mathematics},
}

@Article{KM,
  author  = {Knutson, A. and Miller, E.},
  journal = {Ann. of Math.},
  title   = {Gr\"obner geometry of {S}chubert polynomials},
  year    = {2005},
  number  = {3},
  pages   = {1245--1318},
  volume  = {161}
}

@Article{KMY,
  author  = {Knutson, A. and Miller, E. and Yong, A.},
  journal = {J. Reine Angew. Math.},
  title   = {Gr\"obner geometry of vertex decompositions and of flagged tableaux},
  year    = {2009},
  pages   = {1--31},
  volume  = {630}
}

@Article{KY,
  author  = {Knutson, A. and Yong, A.},
  journal = {International Mathematics Research Notices},
  title   = {{A formula for K-theory truncation Schubert calculus}},
    volume = {2004},
    number = {70},
    pages = {3741--3756},
    year = {2004},
    month = {01},
    issn = {1073--7928}
}

@Article{KLSS,
  author   = {Kummini, M. and Lakshmibai, V. and Sastry, P. and Seshadri, C.~S.},
  journal  = {Pacific J. Math.},
  title    = {Free resolutions of some {S}chubert singularities},
  year     = {2015},
  issn     = {0030-8730},
  number   = {1-2},
  pages    = {299--328},
  volume   = {279},
  fjournal = {Pacific Journal of Mathematics},
}

@Article{Lascoux78,
  author   = {Lascoux, A.},
  journal  = {Adv. in Math.},
  title    = {Syzygies des vari\'{e}t\'{e}s d\'{e}terminantales},
  year     = {1978},
  issn     = {0001-8708},
  number   = {3},
  pages    = {202--237},
  volume   = {30},
  fjournal = {Advances in Mathematics},
}

@Article{LS82,
  author  = {Lascoux, A. and Sch\"{u}tzenberger, M.~P.},
  journal = {C.~R.~Acad.~Sci. Paris S\'{e}r.~I Math.},
  title   = {Structure de {H}opf de l'anneau de cohomologie et de l'anneau de {G}rothendieck d’une vari\'et\'e de drapeaux},
  year    = {1982},
  number  = {11},
  pages   = {629--633},
  volume  = {295},
}

@incollection {Lascoux.Transition,
    AUTHOR = {Lascoux, A.},
     TITLE = {Transition on {G}rothendieck polynomials},
 BOOKTITLE = {Physics and combinatorics, 2000 ({N}agoya)},
     PAGES = {164--179},
 PUBLISHER = {World Sci. Publ., River Edge, NJ},
      YEAR = {2001},
}

@Article{Le00,
  author  = {Lenart, C.},
  journal = {Ann. Comb.},
  title   = {Combinatorial aspects of the {K}-theory of {G}rassmanians},
  year    = {2000},
  number  = {1},
  pages   = {67--82},
  volume  = {4},
}

@Book{Manivel,
  author    = {Manivel, L.},
  publisher = {American Mathematical Society, Providence},
  title     = {Symmetric functions, {S}chubert polynomials and degeneracy loci},
  year      = {2001},
  note      = {Translated from the 1998 French original by John R. Swallow.},
  series    = {SMF/AMS Texts and Monographs},
}

@article{Mats20,
author = {Matsumura, T.},
year = {2020},
month = {03},
pages = {},
title = {A {T}ableau {F}ormula of {D}ouble {G}rothendieck {P}olynomials for 321-{A}voiding {P}ermutations},
volume = {24},
journal = {Annals of Combinatorics},
}


@book {Miller.Sturmfels,
    AUTHOR = {Miller, E. and Sturmfels, B.},
     TITLE = {Combinatorial commutative algebra},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {227},
 PUBLISHER = {Springer-Verlag, New York},
      YEAR = {2005},
     PAGES = {xiv+417},
}

@article {MPP,
    AUTHOR = {Morales, A.~H. and Pak,~I. and Panova,~G.},
     TITLE = {Hook formulas for skew shapes {I}. {$q$}-analogues and
              bijections},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {154},
      YEAR = {2018},
     PAGES = {350--405},
      ISSN = {0097-3165},
}

@article {HT,
    AUTHOR = {Herzog, J. and Trung, N.~V.},
     TITLE = {Gr\"{o}bner bases and multiplicity of determinantal and {P}faffian
              ideals},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {96},
      YEAR = {1992},
    NUMBER = {1},
     PAGES = {1--37},
      ISSN = {0001-8708},
}

@article {Narasimhan,
    AUTHOR = {Narasimhan, H.},
     TITLE = {The irreducibility of ladder determinantal varieties},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {102},
      YEAR = {1986},
    NUMBER = {1},
     PAGES = {162--185},
      ISSN = {0021-8693},
}

@misc{Neye,
      title={A {G}r\"obner basis for {S}chubert patch ideals}, 
      author={Neye, E.},
      year={2021},
      eprint={2111.13778},
      EPRINTTYPE={arXiv},
}




@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},
      ISSN = {0020-9910},
}

o

@misc{Rob321,
        AUTHOR = {Robichaux, C.},
        EPRINT = {2308.14208},
    EPRINTTYPE = {arxiv},
         TITLE = {{Castelnuovo--Mumford regularity for 321-avoiding Kazhdan--Lusztig
  varieties}},
          YEAR = {2023}
          }

@article {RRRSW,
    AUTHOR = {Rajchgot, J. and Ren, Y. and Robichaux, C. and St.
              Dizier, A. and Weigandt, A.},
     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},
}

@article {Weigandt.BPD,
    AUTHOR = {Weigandt, A.},
     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},
}
	
	
@misc{Hafner,
      title={{Vexillary Grothendieck Polynomials via Bumpless Pipe Dreams}}, 
      author={Hafner, E.},
      year={2022},
      eprint={2201.12432},
      EPRINTTYPE={arXiv},
}		





@misc{PSW,
      title={{Castelnuovo-Mumford regularity of matrix Schubert varieties}}, 
      author={ Pechenik, O. and Speyer, D. and Weigandt, A.},
      year={2021},
      eprint={2111.10681},
      EPRINTTYPE={arXiv},
}


@misc{WooYongsurvey,
      title={{Schubert geometry and combinatorics}}, 
      author={ Woo, A. and Yong, A.},
      year={2023},
      eprint={2303.01436},
      EPRINTTYPE={arXiv},
}


@article{RRW,
title = {{Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties}},
journal = {Journal of Algebra},
volume = {617},
pages = {160--191},
year = {2023},
issn = {0021--8693},
author = {Rajchgot, J. and Robichaux, C. and Weigandt, A.},
}

@Book{Weyman,
  author    = {Weyman, J.},
  publisher = {Cambridge University Press, Cambridge},
  title     = {Cohomology of vector bundles and syzygies},
  year      = {2003},
  series    = {Cambridge Tracts in Mathematics},
  volume    = {149},
  pages     = {xiv+371},
}

@Article{WooYongGrobner,
  author  = {Woo, A. and Yong, A.},
  journal = {Amer. J. Math.},
  title   = {A {G}r\"obner basis for {K}azhdan-{L}usztig ideals},
  year    = {2012},
  number  = {4},
  pages   = {1089--1137},
  volume  = {134},
}


@article {WooYongSings,
    AUTHOR = {Woo, A. and Yong, A.},
     TITLE = {Governing singularities of {S}chubert varieties},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {320},
      YEAR = {2008},
    NUMBER = {2},
     PAGES = {495--520},
      ISSN = {0021-8693},
}
@Comment{jabref-meta: databaseType:bibtex;}

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