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).

1, 1, 6, 168, 20160, 9999360, 20158709760, 163849992929280, 5348063769211699200, 699612310033197642547200, 366440137299948128422802227200, 768105432118265670534631586896281600, 6441762292785762141878919881400879415296000, 216123289355092695876117433338079655078664339456000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

The number of boolean operations on n bits, or quantum operations on n qubits, that can be constructed using only CNOT (controlled NOT) gates. - David Radcliffe, Jul 06 2025

Examples

			PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.
		

References

• Roger W. Carter, 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.
• K. J. Horadam, 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.
• Nataša Ilievska and Danilo Gligoroski , Error-Detecting Code Using Linear Quasigroups, ICT Innovations 2014, Advances in Intelligent Systems and Computing Volume 311, 2015, pp 309-318.
• Aaron 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.
• Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.
• I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
• Index entries for sequences related to binary matrices.
• Index entries for sequences related to groups.
• Index to divisibility sequences.

Crossrefs

Column k=2 of A316622 and A316623.

Cf. A000409, A000410, A002820, A005329, A006125, A028365, A046747.

Cf. A006516, A048651, A203303. Row sums of A381854.

Programs

Magma

[1] cat [(&*[2^n -2^k: k in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Aug 31 2023

Maple

# First program
A002884:= n-> mul(2^n - 2^i, i=0..n-1);
seq(A002884(n), n = 0..12);
# Second program
A002884:= n-> 2^(n*(n-1)/2) * mul( 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 *)

PARI

a(n)=prod(i=2,n,2^i-1)<Charles R Greathouse IV, Jan 13 2012

SageMath

[product(2^n -2^j for j in range(n)) for n in range(16)] # G. C. Greubel, Aug 31 2023

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

a(n) = Product_{k=1..n} A006516(k). - Amiram Eldar, Jul 06 2025