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A002884 Number of nonsingular n X n matrices over GF(2) (order of the group GL(n,2)); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).
(Formerly M4302 N1798)
88
1, 1, 6, 168, 20160, 9999360, 20158709760, 163849992929280, 5348063769211699200, 699612310033197642547200, 366440137299948128422802227200, 768105432118265670534631586896281600, 6441762292785762141878919881400879415296000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Also number of bases for GF(2^n) over GF(2).
Also (apparently) number of n X n matrices over GF(2) having permanent = 1. - Hugo Pfoertner, Nov 14 2003
The previous comment is true because over GF(2) permanents and determinants are the same. - Joerg Arndt, Mar 07 2008
The number of automorphisms of (Z_2)^n (the direct product of n copies of Z_2). - Peter Eastwood, Apr 06 2015
Note that n! divides a(n) since the subgroup of GL(n,2) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). - Jianing Song, Oct 29 2022
REFERENCES
Carter, Roger W. Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28. John Wiley & Sons, London-New York-Sydney, 1972. viii+331pp. MR0407163 (53 #10946). See page 2.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.
Horadam, K. J., Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. xiv+263 pp. See p. 132.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..57 (first 30 terms from T. D. Noe)
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Zong Duo Dai, Solomon W. Golomb, and Guang Gong, Generating all linear orthomorphisms without repetition, Discrete Math. 205 (1999), 47-55.
P. F. Duvall, Jr. and P. W. Harley, III, A note on counting matrices, SIAM J. Appl. Math., 20 (1971), 374-377.
N. Ilievska and D. Gligoroski, Error-Detecting Code Using Linear Quasigroups, ICT Innovations 2014, Advances in Intelligent Systems and Computing Volume 311, 2015, pp 309-318.
A. Meyerowitz & N. J. A. Sloane, Correspondence 1979
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Volume 29, 2005 - Issue 1.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
FORMULA
a(n) = Product_{i=0..n-1} (2^n-2^i).
a(n) = 2^(n*(n-1)/2) * Product_{i=1..n} (2^i - 1).
a(n) = A203303(n+1)/A203303(n). - R. J. Mathar, Jan 06 2012
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)) for n > 2. - Seiichi Manyama, Oct 20 2016
a(n) ~ A048651 * 2^(n^2). - Vaclav Kotesovec, May 19 2020
a(n) = A006125(n) * A005329(n). - John Keith, Jun 30 2021
EXAMPLE
PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.
MAPLE
# First program
A002884:= n-> product(2^n - 2^i, i=0..n-1);
seq(A002884(n), n = 0..12);
# Second program
A002884:= n-> 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n);
seq(A002884(n), n=0..12);
MATHEMATICA
Table[Product[2^n-2^i, {i, 0, n-1}], {n, 0, 13}] (* Harvey P. Dale, Aug 07 2011 *)
Table[2^(n*(n-1)/2) QPochhammer[2, 2, n] // Abs, {n, 0, 11}] (* Jean-François Alcover, Jul 15 2017 *)
PROG
(PARI) a(n)=prod(i=2, n, 2^i-1)<<binomial(n, 2) \\ Charles R Greathouse IV, Jan 13 2012
(Magma) [1] cat [(&*[2^n -2^k: k in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Aug 31 2023
(SageMath) [product(2^n -2^j for j in range(n)) for n in range(16)] # G. C. Greubel, Aug 31 2023
CROSSREFS
Column k=2 of A316622 and A316623.
Sequence in context: A181013 A003720 A306837 * A198176 A264358 A264796
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified July 20 16:13 EDT 2024. Contains 374459 sequences. (Running on oeis4.)