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.
[ACFLSS04]F. N. Abu-Khzam, R. L. Collins, M. R. Fellows, M. A. Langston, W. H. Suters, and C. T. Symons: Kernelization Algorithm for the Vertex Cover Problem: Theory and Experiments. SIAM ALENEX/ANALCO 2004: 62-69.
[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.
[ADKLPY2014]M. R. Albrecht, B. Driessen, E. B. Kavun, G. Leander, C. Paar, and T. Yalcin, Block ciphers - focus on the linear layer (feat. PRIDE); in CRYPTO, (2014), pp. 57-76.
[AGHJLPR2017]Benjamin Assarf, Ewgenij Gawrilow, Katrin Herr, Michael Joswig, Benjamin Lorenz, Andreas Paffenholz, and Thomas Rehn, Computing convex hulls and counting integer points with polymake, Math. Program. Comput. 9 (2017), no. 1, 1–38,
[AguSot05]Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Advances in Mathematics, Volume 191, Issue 2, 1 March 2005, pp. 225–275, Arxiv math/0203282v2.
[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.
[AHMP2008]J.-P. Aumasson, L. Henzen, W. Meier, and R. C-W Phan, Sha-3 proposal blake; in Submission to NIST, (2008).
[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.
[AIKMMNT2001]K. Aoki, T. Ichikawa, M. Kanda, M. Matsui, S. Moriai, J. Nakajima, and T. Tokita, Camellia: A 128-bit block cipher suitable for multiple platforms - Design and analysis; in SAC, (2000), pp. 39-56.
[Aj1996]M. Ajtai. Generating hard instances of lattice problems (extended abstract). STOC, pp. 99–108, ACM, 1996.
[AK1994]S. Ariki and K. Koike. A Hecke algebra of \((\mathbb{Z}/r\mathbb{Z})\wr\mathfrak{S}_n\) and construction of its irreducible representations. Adv. Math. 106 (1994), 216–243. mathscinet:\(MR1279219\)
[AJL2011]S. Ariki, N. Jacon, and C. Lecouvey. The modular branching rule for affine Hecke algebras of type A. Adv. Math. 228:481-526, 2011.
[Aki1980]J. Akiyama. and G. Exoo and F. Harary. Covering and packing in graphs. III: Cyclic and acyclic invariants. Mathematical Institute of the Slovak Academy of Sciences. Mathematica Slovaca vol 30, n 4, pages 405–417, 1980
[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.
[AM2000]S. Ariki and A. Mathas. The number of simple modules of the Hecke algebras of type G(r,1,n). Math. Z. 233 (2000), no. 3, 601–623. MathSciNet MR1750939
[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
[Ariki1996]S. Ariki. On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\). J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MathSciNet MR1443748
[Ariki2001]S. Ariki. On the classification of simple modules for cyclotomic Hecke algebras of type \(G(m,1,n)\) and Kleshchev multipartitions. Osaka J. Math. 38 (2001), 827–837. MathSciNet MR1864465
[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.
[Ava2017]R. Avanzi, The QARMA block cipher family; in ToSC, (2017.1), pp. 4-44.


[Ba1994]Kaushik Basu. The Traveler’s Dilemma: Paradoxes of Rationality in Game Theory. The American Economic Review (1994): 391-395.
[BAK1998]E. Biham, R. J. Anderson, and L. R. Knudsen, Serpent: A new block cipher proposal; in FSE, (1998), pp. 222-238.
[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 0901.1413, 2009.
[BBISHAR2015]S. Banik, A. Bogdanov, T. Isobe, K. Shibutani, H. Hiwatari, T. Akishita, and F. Regazzoni, Midori: A block cipher for low energy; in ASIACRYPT, (2015), pp. 411-436.
[BBKMW2013]B. Bilgin, A. Bogdanov, M, Knezevic, F. Mendel, and Q. Wang, Fides: Lightweight authenticated cipher with side-channel resistance for constrained hardware; in CHES, (2013), pp. 142-158.
[BBLSW1999]Babson, Bjorner, Linusson, Shareshian, and Welker, “Complexes of not i-connected graphs,” Topology 38 (1999), 271-299
[BPPSST2017]Banik, Pandey, Peyrin, Sasaki, Sim, and Todo, GIFT : A Small Present Towards Reaching the Limit of Lightweight Encryption. Cryptographic Hardware and Embedded Systems - CHES 2017, 2017.
[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.
[BIANCO]L. Bianco, P. Dell‘Olmo, S. Giordani An Optimal Algorithm to Find the Jump Number of Partially Ordered Sets Computational Optimization and Applications, 1997, Volume 8, Issue 2, pp 197–210, doi:10.1023/A:1008625405476
[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
[BCGKKKLNPRRTY2012]J. Borghoff, A. Canteaut, T. Güneysu, E. B. Kavun, M. Knezevic, L. R. Knudsen, G. Leander, V. Nikov, C. Paar, C. Rechberger, P. Rombouts, S. S. Thomsen, and T. Yalcin, PRINCE - A low-latency block cipher for pervasive computing applications; in ASIACRYPT, (2012), pp. 208-225.
[BCHOPSY2017]G. Benkart, L. Colmenarejo, P. E. Harris, R. Orellana, G. Panova, A. Schilling, M. Yip. A minimaj-preserving crystal on ordered multiset partitions. Advances in Applied Math. 95 (2018) 96-115, doi:10.1016/j.aam.2017.11.006. Arxiv 1707.08709v2.
[BCM15]Michele Borassi, Pierluigi Crescenzi, and Andrea Marino, Fast and Simple Computation of Top-k Closeness Centralities. Arxiv 1507.01490.
[BCN1989]Andries E. Brouwer, Arjeh M. Cohen, and Arnold Neumaier. Distance-Regular Graphs, Springer, 1989.
[BdJ2008]Besser, Amnon, and Rob de Jeu. “Li^(p)-Service? An Algorithm for Computing p-Adic Polylogarithms.” Mathematics of Computation (2008): 1105-1134.
[BD2004]M. Becker and A. Desoky. A study of the DVD content scrambling system (CSS) algorithm; in Proceedings of ISSPIT, (2004), pp. 353-356.
[BDP2013]Thomas Brüstle, Grégoire Dupont, Matthieu Pérotin On Maximal Green Sequences Arxiv 1205.2050
[BDMW2010]K. A. Browning, J. F. Dillon, M. T. McQuistan, and A. J. Wolfe, An APN permutation in dimension six; in Finite Fields: Theory and Applications - FQ9, volume 518 of Contemporary Mathematics, pages 33–42. AMS, 2010.
[Bec1992]Bernhard Beckermann. “A reliable method for computing M-Padé approximants on arbitrary staircases”. J. Comput. Appl. Math., 40(1):19-42, 1992.
[BeCoMe]Frits Beukers, Henri Cohen, Anton Mellit, Finite hypergeometric functions, Arxiv 1505.02900
[Bee]Robert A. Beezer, A First Course in Linear Algebra, Accessed 15 July 2010.
[Bel2011]Belarusian State University, Information technologies. Data protection. Cryptograpic algorithms for encryption and integrity control; in STB 34.101.31-2011, (2011).
[Benasque2009]Fernando Rodriguez Villegas, The L-function of the quintic,
[Ber1987]M. Berger, Geometry I, Springer (Berlin) (1987); doi:10.1007/978-3-540-93815-6
[Ber1991]C. Berger, “Une version effective du théorème de Hurewicz”,
[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
[BerZab05]Nantel Bergeron, Mike Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, J. of Algebra and its Applications (8)(2009), No 4, pp. 581–600, doi:10.1142/S0219498809003485, Arxiv math/0509265v3.
[BeukersHeckman]F. Beukers and G. Heckman, Monodromy for the hypergeometric function `{}_n F_{n-1}`, Invent. Math. 95 (1989)
[BF1999]Thomas Britz, Sergey Fomin, Finite posets and Ferrers shapes, Advances in Mathematics 158, pp. 86-127 (2001), Arxiv math/9912126 (the arXiv version has fewer errors).
[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.
[BG1980]R. L. Bishop and S. L. Goldberg, Tensor analysis on Manifolds, Dover (New York) (1980)
[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
[Bil2011]N. Billerey. Critères d’irréductibilité pour les représentations des courbes elliptiques. Int. J. Number Theory, 7 (2011); doi:10.1142/S1793042111004538
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[BH2017]Georgia Benkart and Tom Halverson. Partition algebras \(\mathsf{P}_k(n)\) with \(2k > n\) and the fundamental theorems of invariant theory for the symmetric group \(\mathsf{S}_n\). Preprint (2017). Arxiv 1707.1410
[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
[Bj1980]Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159-183, doi:10.1090/S0002-9947-1980-0570784-2
[BJKLMPSSS2016]C. Beierle, J. Jean, S. Kölbl, G. Leander, A. Moradi, T. Peyrin, Y. Sasaki, P. Sasdrich, and S. M. Sim, The SKINNY family of block ciphers and its low-latency variant MANTIS; in CRYPTO, (2016), pp. 123-153.
[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
[BK2008]J. Brundan and A. Kleshchev. Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. Invent. Math. 178 (2009), no. 3, 451–484. MathSciNet MR2551762
[BK2009]J. Brundan and A. Kleshchev. Graded decomposition numbers for cyclotomic Hecke algebras. Adv. Math. 222 (2009), 1883–1942. MathSciNet MR2562768
[BK2017]Pascal Baseilhac and Stefan Kolb. Braid group action and root vectors for the \(q\)-Onsager algebra. Preprint, (2017) Arxiv 1706.08747.
[BKK2000]Georgia Benkart, Seok-Jin Kang, Masaki Kashiwara. Crystal bases for the quantum superalgebra \(U_q(\mathfrak{gl}(m,n))\), J. Amer. Math. Soc. 13 (2000), no. 2, 295-331.
[BKLPPRSV2007]A. Bogdanov, L. Knudsen, G. Leander, C. Paar, A. Poschmann, M. Robshaw, Y. Seurin, C. Vikkelsoe. PRESENT: An Ultra-Lightweight Block Cipher; in Proceedings of CHES 2007; LNCS 7427; pp. 450-466; Springer Verlag 2007; available at
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[BM2003]Bazzi and Mitter, {it Some constructions of codes from group actions}, (preprint March 2003, available on Mitter’s MIT website).
[BM2012]N. Bruin and A. Molnar, Minimal models for rational functions in a dynamical setting, LMS Journal of Computation and Mathematics, Volume 15 (2012), pp 400-417.
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