A002884 -id:A002884

A346222

Number of semi-simple n X n matrices over GF(2).

+10
2

1, 2, 10, 218, 25426, 11979362, 24071588290, 195647202043778, 6352629358366433026, 829377572450912758955522, 434523953108209440907114707970, 911402584183760891982341170891585538, 7638756947617134519287879000741815013863426, 256253116935172010151547980961815772566257949204482

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Equivalently, number of n X n matrices over GF(2) that are diagonalizable over the algebraic closure of GF(2).

Equivalently, the number of n X n matrices over GF(2) whose minimal polynomial is a product of distinct irreducible factors, i.e., the minimal polynomial is squarefree.

LINKS

Table of n, a(n) for n=0..13.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

FORMULA

Sum_{n>=0} a(n)x^n/A002884(n) = Product_{d>=1} Sum_{j>=1} x^(j*d)/|GL_j(F_2^d)|)^A001037(d) where |GL_j(F_2^d)| is the order of the general linear group of degree j over the field with 2^d elements.

MATHEMATICA

nn = 13; q = 2; A001037 =Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; \[Gamma]q[j_, d_] :=Table[Product[(q^d)^n - (q^d)^i, {i, 0, n - 1}], {n, 1, nn}][[j]]; Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[ Series[Product[(1 + Sum[u^(j d)/\[Gamma]q[j, d], {j, 1, nn}])^

A001037[[d]], {d, 1, nn}], {u, 0, nn}], u]

CROSSREFS

Cf. A001037, A002884, A132186.

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Jul 11 2021

STATUS

approved

A073960

Number of commuting elements: number of ordered pairs g, h in the group GL(n,2) such that gh = hg.

+10
1

1, 18, 1008, 282240, 269982720, 1209522585600, 19170449172725760, 1315623687226078003200, 342810031916266844848128000, 367173017574548024679647831654400, 1534674653372294809728193910618770636800, 26108462572660693961035262279317764270194688000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..30

FORMULA

a(n) = A002884(n) * A006951(n).

CROSSREFS

Cf. A002884, A006951.

KEYWORD

nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

EXTENSIONS

More terms from Eric M. Schmidt, May 05 2013

STATUS

approved

A086157

Sum of the orders of the elements in the group GL(n,2).

+10
1

1, 13, 715, 137047, 151063807, 473437527487, 8759301830031103, 476009298383159238655, 124804541542651896036171775, 116591664734096391445571274997759, 528358596232540118601239210575638560767, 7144378940562638409529946721196600858719027199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..20

PROG

(GAP) A086157 := n -> Sum(ConjugacyClasses(GL(n, 2)), cc->Size(cc) * Order(Representative(cc))); # Eric M. Schmidt, May 18 2013

CROSSREFS

Cf. A002884.

KEYWORD

nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 25 2003

EXTENSIONS

More terms from Vladeta Jovovic, Aug 31 2003

a(6)-a(10) corrected by and more terms from Eric M. Schmidt, May 18 2013

STATUS

approved

A087540

Let A(n) be the matrix in the group GL(n,2) such that for 1 <= i, j <= n: A[i,j] = 1 if i+j = n+1 A[i,j] = 0 if i+j != n+1. a(n) is the size of the centralizer of A(n) in GL(n,2).

+10
1

1, 2, 8, 96, 1536, 86016, 5505024, 1321205760, 338228674560, 335522845163520, 343575393447444480, 1385295986380096143360, 5674172360212873803202560, 92239345887620476544860815360, 1511249443022773887710999598858240, 98654363640526679389774053813465907200

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The formula was given by Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) in this thread from sci.math: http://mathforum.org/discuss/sci.math/t/538859.

LINKS

Table of n, a(n) for n=1..16.

FORMULA

For even n = 2m, a(n) = 2^(m^2) * |GL(m, 2)| = 2^(m^2) * A002884(m).

For odd n = 2m+1, a(n) = 2^(m^2+2m) * |GL(m, 2)| = 2^(m^2+2m) * A002884(m).

MATHEMATICA

a[n_] := With[{m = Quotient[n, 2]}, 2^(2 m^2 + 2 m Boole[OddQ[n]]) * QPochhammer[2^-m, 2, m]];

a /@ Range[1, 16] (* Jean-François Alcover, Sep 17 2019 *)

PROG

(GAP)

a:=function(n) local M;

M:=NullMat(n, n); for i in [1..n] do M[i][n+1-i]:=1; od;

return Size(Centralizer(GL(n, Integers mod 2), M * One(Integers mod 2)));

end; # Andrew Howroyd, Jul 13 2018

(PARI) a(n)={my(m=n\2); 2^(m*if(n%2, n+3, n)/2)*prod(i=2, m, 2^i-1)*2^binomial(m, 2)} \\ Andrew Howroyd, Jul 13 2018

CROSSREFS

Cf. A002884, A087918.

KEYWORD

nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 24 2003

EXTENSIONS

a(8)-a(16) from Andrew Howroyd, Jul 13 2018

STATUS

approved

A087918

Let A(n) be the matrix in the group GL(n,2) such that for 1<=i, j<=n: A[i,j] = 1 if i+j = n+1 A[i,j] = 0 if i+j != n+1. a(n) is the size of the conjugacy class of A(n) in GL(n,2).

+10
1

1, 3, 21, 210, 6510, 234360, 29763720, 4047865920, 2068459485120, 1092146608143360, 2235624106869457920, 4650098142288472473600, 38088953883484878031257600, 314462403262051153026062745600, 10303989567687630131204997985075200, 338960040818652280796119613717033779200

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..16.

FORMULA

a(n) = A002884(n) / A087540(n).

PROG

(GAP)

a:=function(n) local M;

M:=NullMat(n, n); for i in [1..n] do M[i][n+1-i]:=1; od;

return Size(ConjugacyClass(GL(n, Integers mod 2), M * One(Integers mod 2)));

end; # Andrew Howroyd, Jul 13 2018

(PARI) \\ here b(n) is A002884.

b(n)={prod(i=2, n, 2^i-1)*2^binomial(n, 2)}

a(n)={my(m=n\2); b(n)/(2^(m*if(n%2, n+3, n)/2)*b(m))} \\ Andrew Howroyd, Jul 13 2018

CROSSREFS

Cf. A002884, A087540.

KEYWORD

nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 26 2003

EXTENSIONS

a(8)-a(16) from Andrew Howroyd, Jul 13 2018

STATUS

approved

A220790

Product(6^n - 6^k, k=0..n-1).

+10
1

1, 5, 1050, 8127000, 2273284440000, 22906523331216000000, 8310241106635054164480000000, 108537128570336598656772717772800000000, 51032497739317419104816901041614046625792000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..35

FORMULA

a(n) = (6^n - 1)*(6^n - 6)*...*(6^n - 6^(n-1)) for n>0, a(0)=1.

a(n) = A109354(n)*A027873(n). - Bruno Berselli, Jan 30 2013

MATHEMATICA

Table[Product[6^n - 6^k, {k, 0, n-1}], {n, 0, 60}]

PROG

(Magma)

[1] cat [&*[(6^n - 6^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

(Magma) /* By the second formula: */

m:=9;

A109354 := [6^(n*(n-1) div 2): n in [0..m-1]];

A027873 := [1] cat [&*[6^i-1: i in [1..n]]: n in [1..m]];

[A109354[i]*A027873[i]: i in [1..m]]; // Bruno Berselli, Jan 30 2013

CROSSREFS

Cf. A027873, A109354.

Sequences given by product(m^n-m^k, k=0..n-1): A002884 (m=2), A053290 (m=3), A053291 (m=4), A053292 (m=5), A053293 (m=7), A052496 (m=8), A052497 (m=9), A052498 (m=11).

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Jan 28 2013

STATUS

approved

A220791

Number of nonsingular n X n matrices over GF(13).

+10
1

1, 12, 26208, 9726417792, 610296923230525440, 6471875909051511775903457280, 11598637276362103019770723830073032376320, 3512938445418644176053176560741858449740612202579886080

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..30

J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher.

MATHEMATICA

Table[Product[13^n - 13^k, {k, 0, n-1}], {n, 0, 8}]

PROG

(Magma) [1] cat [&*[(13^n - 13^k): k in [0..n-1]]: n in [1..8]]

CROSSREFS

Cf. A002884, A003787, A003789, A003790, A053290-A053293.

KEYWORD

nonn

AUTHOR

Vincenzo Librandi, Jan 29 2013

STATUS

approved

A224879

Number of equivalence classes of n X n nonsingular matrices over GF(2), up to row and column permutation.

+10
1

1, 2, 7, 51, 885, 44206, 6843555, 3373513302, 5366987461839, 27936547529976720, 482768359608369460173, 28090323163597327933723100, 5574677486781815353253212392653, 3816761688188495487649082049091445498, 9106495173413853187392282303788066742174903

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Ludovic Schwob, Table of n, a(n) for n = 1..41

Finley Freibert, The Classification of Complementary Information Set Codes of Lengths 14 and 16, Advances in Mathematics of Communications, Vol. 7, No. 3 (2013), 267-278.

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.

Ludovic Schwob, Sage program

CROSSREFS

A002884 counts all matrices nonsingular over GF(2).

A116976 counts equivalence classes of binary matrices nonsingular over the reals.

KEYWORD

nonn

AUTHOR

Finley Freibert, Jul 23 2013

EXTENSIONS

a(8) from Brendan McKay, May 25 2020

a(9) onwards from Ludovic Schwob, Sep 25 2023

STATUS

approved

A239892

List of orders of finite simple groups which are unit groups of rings.

+10
1

2, 3, 7, 31, 127, 168, 8191, 20160, 131071, 524287, 9999360, 2147483647, 20158709760, 163849992929280, 2305843009213693951, 5348063769211699200, 699612310033197642547200, 618970019642690137449562111, 366440137299948128422802227200, 162259276829213363391578010288127

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The rings are assumed to be unital, but not necessarily commutative, and ring homomorphisms send 1 to 1.

LINKS

Table of n, a(n) for n=1..20.

Christopher Davis and Tommy Occhipinti, Which finite simple groups are unit groups?, preprint, J. Pure Appl. Algebra, 218 (2014), no. 4, 743--744. MR3133704.

FORMULA

Union of {2}, A000668 and A002884.

CROSSREFS

Cf. A239891, A000668, A002884.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 03 2014

STATUS

approved

A258745

Order of general affine group AGL(n,2) (=A028365(n)) divided by (n+1).

+10
1

1, 1, 8, 336, 64512, 53329920, 184308203520, 2621599886868480, 152122702768688332800, 35820150273699719298416640, 34112245508649716682268134604800, 131089993748184007771243790830298726400, 2029650642403883210241235064170615545004032000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..100

Putnam Exam. 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.

FORMULA

a(n) = A028365(n) / (n+1) = 2^n * A002884(n) / (n+1) = 2^n * n! * A053601(n) / (n+1).

KEYWORD

nonn

AUTHOR

Max Alekseyev, Jun 08 2015

STATUS

approved