The references for Sage, sorted alphabetically by citation key.




[ABBR2012]A. Abad, R. Barrio, F. Blesa, M. Rodriguez. Algorithm 924. ACM Transactions on Mathematical Software, 39 no. 1 (2012), 1-28.
[ADKF1970]V. Arlazarov, E. Dinic, M. Kronrod, and I. Faradzev. ‘On Economical Construction of the Transitive Closure of a Directed Graph.’ Dokl. Akad. Nauk. SSSR No. 194 (in Russian), English Translation in Soviet Math Dokl. No. 11, 1970.
[AH2002]R. J. Aumann and S. Hart, Elsevier, eds. Computing equilibria for two-person games. (2002)
[AHK2015]Karim Adiprasito, June Huh, and Eric Katz. Hodge theory for combinatorial geometries. Arxiv 1511.02888.
[AHU1974]A. Aho, J. Hopcroft, and J. Ullman. ‘Chapter 6: Matrix Multiplication and Related Operations.’ The Design and Analysis of Computer Algorithms. Addison-Wesley, 1974.
[Aj1996]M. Ajtai. Generating hard instances of lattice problems (extended abstract). STOC, pp. 99–108, ACM, 1996.
[Al1947]A. A. Albert, A Structure Theory for Jordan Algebras. Annals of Mathematics, Second Series, Vol. 48, No. 3 (Jul., 1947), pp. 546–567.
[AL1978]A. O. L. Atkin and Wen-Ch’ing Winnie Li, Twists of newforms and pseudo-eigenvalues of \(W\)-operators. Inventiones math. 48 (1978), 221-243.
[AL2015]M. Aguiar and A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras. Algebra Number Theory, v.9, 2015, n.3, 2015.
[AM1974]J. F. Adams and H. R. Margolis, “Sub-Hopf-algebras of the Steenrod algebra,” Proc. Cambridge Philos. Soc. 76 (1974), 45-52.
[Ap1997]T. Apostol, Modular functions and Dirichlet series in number theory, Springer, 1997 (2nd ed), section 3.7–3.9.
[APR2001]George E. Andrews, Peter Paule, Axel Riese, MacMahon’s partition analysis: the Omega package, European J. Combin. 22 (2001), no. 7, 887–904.
[Ar2006]D. Armstrong. Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc., 2006.
[AR2012]D. Armstrong and B. Rhoades. “The Shi arrangement and the Ish arrangement”. Transactions of the American Mathematical Society 364 (2012), 1509-1528. Arxiv 1009.1655
[AS-Bessel]F. W. J. Olver: 9. Bessel Functions of Integer Order, in Abramowitz and Stegun: Handbook of Mathematical Functions.
[AS-Spherical]H. A. Antosiewicz: 10. Bessel Functions of Fractional Order, in Abramowitz and Stegun: Handbook of Mathematical Functions.
[AS-Struve]M. Abramowitz: 12. Struve Functions and Related Functions, in Abramowitz and Stegun: Handbook of Mathematical Functions.
[AS1964]M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series, 55. 1964. See also
[As2008]Sami Assaf. A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs. FPSAC 2008, 141-152, Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., (2008). Arxiv 0804.1587v1
[AS2011]R.B.J.T Allenby and A. Slomson, “How to count”, CRC Press (2011)
[ASD1971]A. O. L. Atkin and H. P. F. Swinnerton-Dyer, “Modular forms on noncongruence subgroups”, Proc. Symp. Pure Math., Combinatorics (T. S. Motzkin, ed.), vol. 19, AMS, Providence 1971
[Av2000]D. Avis, A revised implementation of the reverse search vertex enumeration algorithm. Polytopes-combinatorics and computation. Birkhauser Basel, 2000.


[Ba1994]Kaushik Basu. The Traveler’s Dilemma: Paradoxes of Rationality in Game Theory. The American Economic Review (1994): 391-395.
[Bar1970]Barnette, “Diagrams and Schlegel diagrams”, in Combinatorial Structures and Their Applications, Proc. Calgary Internat. Conference 1969, New York, 1970, Gordon and Breach.
[Bar2006]G. Bard. ‘Accelerating Cryptanalysis with the Method of Four Russians’. Cryptography E-Print Archive (, 2006.
[BB1997]Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups. Invent. Math. 129 (1997). No. 3, 445-470.
[BB2009]Tomas J. Boothby and Robert W. Bradshaw. Bitslicing and the Method of Four Russians Over Larger Finite Fields. arXiv:0901.1413v1, 2009. Arxiv 0901.1413
[BBLSW1999]Babson, Bjorner, Linusson, Shareshian, and Welker, “Complexes of not i-connected graphs,” Topology 38 (1999), 271-299
[BBS1982]L. Blum, M. Blum, and M. Shub. Comparison of Two Pseudo-Random Number Generators. Advances in Cryptology: Proceedings of Crypto ‘82, pp.61–78, 1982.
[BBS1986]L. Blum, M. Blum, and M. Shub. A Simple Unpredictable Pseudo-Random Number Generator. SIAM Journal on Computing, 15(2):364–383, 1986.
[BC1977]R. E. Bixby, W. H. Cunningham, Matroids, Graphs, and 3-Connectivity. In Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, ON, 1977), 91-103
[BC2003]A. Biryukov and C. D. Canniere Block Ciphers and Systems of Quadratic Equations; in Proceedings of Fast Software Encryption 2003; LNCS 2887; pp. 274-289, Springer-Verlag 2003.
[BC2012]Mohamed Barakat and Michael Cuntz. “Coxeter and crystallographic arrangements are inductively free.” Adv. in Math. 229 Issue 1 (2012). pp. 691-709. doi:10.1016/j.aim.2011.09.011, Arxiv 1011.4228.
[BCCCNSY2010]Charles Bouillaguet, Hsieh-Chung Chen, Chen-Mou Cheng, Tung Chou, Ruben Niederhagen, Adi Shamir, and Bo-Yin Yang. Fast exhaustive search for polynomial systems in GF(2). In Stefan Mangard and François-Xavier Standaert, editors, CHES, volume 6225 of Lecture Notes in Computer Science, pages 203–218. Springer, 2010. pre-print available at
[BDP2013]Thomas Brüstle, Grégoire Dupont, Matthieu Pérotin On Maximal Green Sequences Arxiv 1205.2050
[Bee]Robert A. Beezer, A First Course in Linear Algebra, Accessed 15 July 2010.
[Ber2008]W. Bertram : Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, Memoirs of the American Mathematical Society, vol. 192 (2008); doi:10.1090/memo/0900; Arxiv math/0502168
[Ber1991]C. Berger, “Une version effective du théorème de Hurewicz”,
[BFZ2005]A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1–52.
[BG1985]M. Blum and S. Goldwasser. An Efficient Probabilistic Public-Key Encryption Scheme Which Hides All Partial Information. In Proceedings of CRYPTO 84 on Advances in Cryptology, pp. 289–299, Springer, 1985.
[BG1988]M. Berger & B. Gostiaux : Differential Geometry: Manifolds, Curves and Surfaces, Springer (New York) (1988); doi:10.1007/978-1-4612-1033-7
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[BHS2008]Robert Bradshaw, David Harvey and William Stein. strassen_window_multiply_c. strassen.pyx, Sage 3.0, 2008.
[Big1999]Stephen J. Bigelow. The Burau representation is not faithful for \(n = 5\). Geom. Topol., 3:397–404, 1999.
[Big2003]Stephen J. Bigelow, The Lawrence-Krammer representation, Geometric Topology, 2001 Georgia International Topology Conference, AMS/IP Studies in Advanced Mathematics 35 (2003). Arxiv math/0204057v1
[Bir1975]J. Birman. Braids, Links, and Mapping Class Groups, Princeton University Press, 1975
[BK1992]U. Brehm and W. Kuhnel, “15-vertex triangulations of an 8-manifold”, Math. Annalen 294 (1992), no. 1, 167-193.
[BK2001]W. Bruns and R. Koch, Computing the integral closure of an affine semigroup. Uni. Iaggelonicae Acta Math. 39, (2001), 59-70
[BL2000]Anders Björner and Frank H. Lutz, “Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere”, Experiment. Math. 9 (2000), no. 2, 275-289.
[BL2008]Corentin Boissy and Erwan Lanneau, “Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials” (arxiv:0710.5614) to appear in Ergodic Theory and Dynamical Systems.
[BM2008]John Adrian Bondy and U.S.R. Murty, “Graph theory”, Volume 244 of Graduate Texts in Mathematics, 2nd edition, Springer, 2008.
[BM2003]Bazzi and Mitter, {it Some constructions of codes from group actions}, (preprint March 2003, available on Mitter’s MIT website).
[BN2008]Victor V. Batyrev and Benjamin Nill. Combinatorial aspects of mirror symmetry. In Integer points in polyhedra — geometry, number theory, representation theory, algebra, optimization, statistics, volume 452 of Contemp. Math., pages 35–66. Amer. Math. Soc., Providence, RI, 2008. arXiv:math/0703456v2 [math.CO].
[Bob2013]J.W. Bober. Conditionally bounding analytic ranks of elliptic curves. ANTS 10, 2013.
[BP1982]H. Beker and F. Piper. Cipher Systems: The Protection of Communications. John Wiley and Sons, 1982.
[BP2000]V. M. Bukhshtaber and T. E. Panov, “Moment-angle complexes and combinatorics of simplicial manifolds,” Uspekhi Mat. Nauk 55 (2000), 171–172.
[BP2015]P. Butera and M. Pernici “Sums of permanental minors using Grassmann algebra”, International Journal of Graph Theory and its Applications, 1 (2015), 83–96. Arxiv 1406.5337
[BPRS2009]J. Bastian, T. Prellberg, M. Rubey, C. Stump, Counting the number of elements in the mutation classes of `tilde{A}_n`-quivers, Arxiv 0906.0487
[Br1910]Bruckner, “Uber die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope)”, Verhand. Konik. Akad. Wetenschap, Erste Sectie, 10 (1910)
[Br2000]Kenneth S. Brown, Semigroups, rings, and Markov chains, Arxiv math/0006145v1.
[BS1996]Eric Bach, Jeffrey Shallit. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. MIT Press, 1996. ISBN 978-0262024051.
[BS2003]I. Bouyukliev and J. Simonis, Some new results on optimal codes over \(F_5\), Designs, Codes and Cryptography 30, no. 1 (2003): 97-111,
[BS2011]E. Byrne and A. Sneyd, On the Parameters of Codes with Two Homogeneous Weights. WCC 2011-Workshop on coding and cryptography, pp. 81-90. 2011.
[BSS2009]David Bremner, Mathieu Dutour Sikiric, Achill Schuermann: Polyhedral representation conversion up to symmetries, Proceedings of the 2006 CRM workshop on polyhedral computation, AMS/CRM Lecture Notes, 48 (2009), 45-71.
[BSV2010]M. Bolt, S. Snoeyink, E. Van Andel. “Visual representation of the Riemann map and Ahlfors map via the Kerzman-Stein equation”. Involve 3-4 (2010), 405-420.
[BW1996]Anders Bjorner and Michelle L. Wachs. Shellable nonpure complexes and posets. I. Trans. of Amer. Math. Soc. 348 No. 4. (1996)


[CB2007]Nicolas Courtois, Gregory V. Bard: Algebraic Cryptanalysis of the Data Encryption Standard, In 11-th IMA Conference, Cirencester, UK, 18-20 December 2007, Springer LNCS 4887. See also
[CDL2015]A. Canteaut, Sebastien Duval, Gaetan Leurent Construction of Lightweight S-Boxes using Feistel and MISTY Structures; in Proceedings of SAC 2015; LNCS 9566; pp. 373-393; Springer-Verlag 2015; available at
[CE2001]Raul Cordovil and Gwihen Etienne. A note on the Orlik-Solomon algebra. Europ. J. Combinatorics. 22 (2001). pp. 165-170.
[Cer1994]D. P. Cervone, “Vertex-minimal simplicial immersions of the Klein bottle in three-space”, Geom. Ded. 50 (1994) 117-141,
[CEW2011]Georgios Chalkiadakis, Edith Elkind, and Michael Wooldridge. Computational Aspects of Cooperative Game Theory. Morgan & Claypool Publishers, (2011). ISBN 9781608456529, doi:10.2200/S00355ED1V01Y201107AIM016.
[CGW2013]Daniel Cabarcas, Florian Göpfert, and Patrick Weiden. Provably Secure LWE-Encryption with Uniform Secret. Cryptology ePrint Archive, Report 2013/164. 2013. 2013/164.
[CGMRV16]A. Conte, R. Grossi, A. Marino, R. Rizzi, L. Versari, “Directing Road Networks by Listing Strong Orientations.”, Combinatorial Algorithms, Proceedings of 27th International Workshop, IWOCA 2016, August 17-19, 2016, pages 83–95.
[Ch2012]Cho-Ho Chu. Jordan Structures in Geometry and Analysis. Cambridge University Press, New York. 2012. IBSN 978-1-107-01617-0.
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[ChenDB]Eric Chen, Online database of two-weight codes,
[CK1999]David A. Cox and Sheldon Katz. Mirror symmetry and algebraic geometry, volume 68 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999.
[CK2001]M. Casella and W. Kühnel, “A triangulated K3 surface with the minimum number of vertices”, Topology 40 (2001), 753–772.
[CK2015]J. Campbell and V. Knight. On testing degeneracy of bi-matrix games. (2015)
[CL2013]Maria Chlouveraki and Sofia Lambropoulou. The Yokonuma-Hecke algebras and the HOMFLYPT polynomial. (2015) Arxiv 1204.1871v4.
[CLRS2001]Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein, Section 22.4: Topological sort, Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, 2001, 549-552, ISBN 0-262-03293-7.
[CLS2011]David A. Cox, John Little, and Hal Schenck. Toric Varieties. Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011.
[CMO2011]C. Chun, D. Mayhew, J. Oxley, A chain theorem for internally 4-connected binary matroids. J. Combin. Theory Ser. B 101 (2011), 141-189.
[CMO2012]C. Chun, D. Mayhew, J. Oxley, Towards a splitter theorem for internally 4-connected binary matroids. J. Combin. Theory Ser. B 102 (2012), 688-700.
[CMR2005]C. Cid, S. Murphy, M. Robshaw Small Scale Variants of the AES; in Proceedings of Fast Software Encryption 2005; LNCS 3557; Springer Verlag 2005; available at
[CMR2006]C. Cid, S. Murphy, and M. Robshaw Algebraic Aspects of the Advanced Encryption Standard; Springer Verlag 2006
[Co1984]J. Conway, Hexacode and tetracode - MINIMOG and MOG. Computational group theory, ed. M. Atkinson, Academic Press, 1984.
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[Coh2007]Henri Cohen, Number Theory, Volume II. Graduate Texts in Mathematics 240. Springer, 2007.
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[CPdA2014]Maria Chlouveraki and Loic Poulain d’Andecy. Representation theory of the Yokonuma-Hecke algebra. (2014) Arxiv 1302.6225v2.
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[DDLL2013]Léo Ducas, Alain Durmus, Tancrède Lepoint and Vadim Lyubashevsky. Lattice Signatures and Bimodal Gaussians; in Advances in Cryptology – CRYPTO 2013; Lecture Notes in Computer Science Volume 8042, 2013, pp 40-56
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[DLHK2007]J. A. De Loera, D. C. Haws, M. Köppe, Ehrhart polynomials of matroid polytopes and polymatroids. Discrete & Computational Geometry, Volume 42, Issue 4. Arxiv 0710.4346, doi:10.1007/s00454-008-9120-8
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