A002884 -id:A002884 - cp's OEIS frontend

A346214 Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices over GF(2) with index k, 1 <= k <= n, n >= 1.

1, 1, 3, 1, 21, 42, 1, 315, 1260, 2520, 1, 6975, 104160, 312480, 624960, 1, 373023, 23436000, 104993280, 314979840, 629959680, 1, 32252031, 9175162752, 121912197120, 426692689920, 1280078069760, 2560156139520, 1, 6619979775, 9978120069120, 421755245936640, 1989607056998400, 6963624699494400, 20890874098483200, 41781748196966400

Offset: 1

Geoffrey Critzer, Jul 10 2021

The index of a nilpotent matrix A is the smallest positive integer k such that A^k = 0.

Define the co-index of an n X n matrix A to be n - index(A). Let X_n be the random variable that assigns to each nilpotent n X n matrix over GF(2) the value j in {0,1,...,n-1} of its co-index. Conjecture: lim_{n->inf} P(X_n = j) = Product_{i>=1}1-1/2^i * 2^((j-1)^2)/A002884(j). Moreover, for j < 2n, T(n,n-j) = A002884(n)/(A002884(j)*2^(n - (j-1)^2)). - Geoffrey Critzer, Jun 10 2025

			  1,
  1,    3,
  1,   21,     42,
  1,  315,   1260,   2520,
  1, 6975, 104160, 312480, 624960

Cf. A083402 (main diagonal), A053763 (row sums), A002884, A048651.

nn = 8;  q = 2;
b[p_, i_] := Count[p, i];
d[p_, i_] :=  Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];
aut[deg_, p_] := Product[Product[ q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}];
l = Level[Table[IntegerPartitions[n],  {n,  0,  nn}],  {2}];
\[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}];
g[u_,  v_,  deg_,  partitions_] := Total[Map[v^If[# == {},  0,  Max[#]] u^(deg Total[#])/aut[deg,  #] &, partitions]];
Map[Select[#,  # > 0 &] &, Drop[Table[\[Gamma][n,  q],  {n,  0,  nn}] CoefficientList[     Series[g[u,  v,  1,  l],  {u,  0,  nn}],  {u,  v}],  1]] // Grid

More terms from Geoffrey Critzer, Jun 10 2025

A348622 Triangular array read by rows: T(n,k) is the number of periodic n X n matrices over GF(2) having rank k, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 6, 6, 1, 28, 168, 168, 1, 120, 3360, 20160, 20160, 1, 496, 59520, 1666560, 9999360, 9999360, 1, 2016, 999936, 119992320, 3359784960, 20158709760, 20158709760, 1, 8128, 16386048, 8127479808, 975297576960, 27308332154880, 163849992929280, 163849992929280

Offset: 0

Geoffrey Critzer, Oct 25 2021

A matrix T is periodic if and only image(T) = image(T^2). Cf. A348015.

			Triangle begins:
  1;
  1,   1;
  1,   6,     6;
  1,  28,   168,     168;
  1, 120,  3360,   20160,   20160;
  1, 496, 59520, 1666560, 9999360, 9999360;
  ...

Alois P. Heinz, Rows n = 0..57, flattened
Eric Weisstein's World of Mathematics, Periodic Matrix

Cf. A348015 (row sums).

Main diagonal gives A002884.

b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
T:= (n, k)-> b(n)/b(n-k):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Oct 30 2021

nn = 7; q = 2; b[p_, i_] := Count[p, i];s[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] := Product[Product[q^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1,nn}]; l[greatestpart_]:=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}];
g1[u_, v_, deg_] :=Total[Map[v^(Length[#]) u^(deg Total[#])/aut[deg, #] &, l[1]]];
g2[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &,l[nn]]];
Map[Reverse, Map[Select[#, # > 0 &] &,Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g1[u, v, 1] g2[u, 1, 1]^(q - 1) Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u,v}]]] // Flatten

T(n,k) = A002884(n)/A002884(n-k).
T(n,k) = T(n,k-1)*T(n-k+1,1).
Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*g(y*x) where e(x) = Sum_{n>=0} x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)*x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - Geoffrey Critzer, Jan 03 2025

Title improved by Geoffrey Critzer, Sep 16 2022

A060704 Singular n X n matrices over GF(2).

Original entry on oeis.org

1, 10, 344, 45376, 23555072, 48560766976, 399099960492032, 13098680304497852416, 1718239329196060706865152, 901210462928281273073900978176, 1890350559451566075272982533664407552

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001

nonn

Also (apparently) number of n X n matrices over GF(2) having permanent = 0. - Hugo Pfoertner, Nov 14 2003

Harry J. Smith, Table of n, a(n) for n = 1..57
Index entries for sequences related to binary matrices

Cf. A002884, A002416.

for n from 1 to 20 do printf(`%d,`,2^(n^2) - product(2^n - 2^j, j=0..n-1)) od:

a(n)={2^(n^2) - prod(i=0, n-1, 2^n - 2^i)} \\ Harry J. Smith, Jul 09 2009

For n >= 1 a(n) = 2^(n^2) - A002884(n) = A002416(n) - A002884(n) = 2^(n^2) - Product_{i=0..n-1} (2^n - 2^i).

More terms from James Sellers, Apr 23 2001

A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

Original entry on oeis.org

1, 2, 288, 1886976, 489104179200, 4755360379856486400, 1695944421638473850132889600, 21967113634648374162210646578639667200, 10286692771039109536373764545035369981946101760000, 173770439600109774111384717714984362383506603790098046648320000

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
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.

Column k=6 of A316622.

Cf. A002884, A048651, A053290, A065128, A100220.

a[n_] := 6^(n^2)*Product[(1 - 1/2^k)*(1 - 1/3^k), { k, 1, n} ]; Table[ a[n], {n, 0, 9} ]

a(n) = 6^(n^2) * Product_{k=1..n} ((1 - 1/2^k)(1 - 1/3^k)).
a(n) = A002884(n)*A053290(n). - Geoffrey Critzer, Jan 26 2018
a(n) ~ c * 6^(n^2), where c = A048651 * A100220 = 0.161757743053... . - Amiram Eldar, Jul 06 2025

More terms from Robert G. Wilson v, Nov 28 2001

A070731 Size of largest conjugacy class in the group GL(n,2).

Original entry on oeis.org

1, 3, 56, 3360, 833280, 959938560, 3901190307840, 63667425823948800, 4759267415191820697600, 1246395024829755538853068800

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 15 2002

nonn

V. Jovovic, Cycle indices of general linear group GL(n,2)

Cf. A006951, A002884.

More terms from Vladeta Jovovic, Jun 03 2002

A085430 a(n) is the minimal m such that the group GL(m,2) has an element of order n.

Original entry on oeis.org

2, 2, 3, 4, 4, 3, 5, 6, 6, 10, 5, 12, 5, 4, 9, 8, 8, 18, 7, 5, 12, 11, 7, 20, 14

Offset: 2

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 18 2003

For n > 1, a(prime(n)) = A014664(n). Also, a(n) <= n. - Eric M. Schmidt, May 17 2013

Cf. A002884.

A085430 := function(n) local m; if IsPrime(n) and n>2 then return Order(2*Z(n)^0); fi; m := 1; while true do if ForAny(ConjugacyClasses(GL(m,2)), cc->Order(Representative(cc))=n) then return m; fi; m := m + 1; od; end; # Eric M. Schmidt, May 17 2013

Sequence extended and corrected by Eric M. Schmidt, May 17 2013

A086812 Number of symmetric invertible n X n matrices over GF(2).

Original entry on oeis.org

1, 4, 28, 448, 13888, 888832, 112881664, 28897705984, 14766727757824, 15121129224011776, 30952951521552105472, 126783289432277424013312, 1038481923739784380093038592, 17014487838552627283444344291328

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 15 2003

nonn

R. P. Brent and B. D. McKay, Determinants of random symmetric matrices over Zm, Ars Combinatoria, 26-A (1988) 57-64.

R. P. Brent and B. D. McKay, Determinants of random symmetric matrices over Zm, arXiv:1004.5440 [math.CO], 2010.

Cf. A002884.

for n from 1 to 31 do k := ceil(n/2); a[n] := 2^(n*(n+1)/2)*product(1-(1/2)^j,j=1..2*k)/product(1-(1/4)^j,j=1..k); od:seq(a[j],j=1..31); # Sascha Kurz, Sep 19 2003

m = 14; For[n = 1, n <= m, n++, k = Ceiling[n/2]; a[n] = 2^(n*(n+1)/2)* Product[1-(1/2)^j, {j, 1, 2k}]/Product[1-(1/4)^j, {j, 1, k}]];
Array[a, m] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Let k = ceiling(n/2). Then a(n) = 2^(n*(n+1)/2) * (Product_{j=1..2k} (1 - (1/2)^j)) / Product_{j=1..k} (1 - (1/4)^j).

More terms from Ray Chandler and Sascha Kurz, Sep 19 2003

A224879 Number of equivalence classes of n X n nonsingular matrices over GF(2), up to row and column permutation.

Original entry on oeis.org

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

Offset: 1

Finley Freibert, Jul 23 2013

nonn

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.
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.
Ludovic Schwob, Sage program
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 16.

A002884 counts all matrices nonsingular over GF(2).

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

a(8) from Brendan McKay, May 25 2020

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

A288853 Triangle read by rows: T(n,k) is the number of surjective linear mappings from an n-dimensional vector space over F_2 onto a k-dimensional vector space, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 1, 7, 42, 168, 1, 15, 210, 2520, 20160, 1, 31, 930, 26040, 624960, 9999360, 1, 63, 3906, 234360, 13124160, 629959680, 20158709760, 1, 127, 16002, 1984248, 238109760, 26668293120, 2560156139520, 163849992929280, 1, 255, 64770, 16322040, 4047865920, 971487820800, 217613271859200, 41781748196966400, 5348063769211699200

Offset: 0

Geoffrey Critzer, Jun 18 2017

The (q = 2) analog of A008279.
A022166(m,k)*T(n,k) is the number of m X n matrices over F_2 that have rank k.
a(n) is the number of n X n matrices over F_2 in Green's R class containing A where rank(A) = k. - Geoffrey Critzer, Oct 05 2022

			  1;
  1,  1;
  1,  3,   6;
  1,  7,  42,   168;
  1, 15, 210,  2520,  20160;
  1, 31, 930, 26040, 624960, 9999360;
  ...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, 2010-2023, Example 2.3.6.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Wikipedia, Green's relations.

Columns k=0-10 give: A000012, A000225, 6*A006095, 168*A006096, 20160*A006097, 9999360*A006110, 20158709760*A022189, 163849992929280*A022190, 5348063769211699200*A022191, 699612310033197642547200*A022192, 366440137299948128422802227200*A022193.
Main diagonal gives A002884.
Cf. A022166.

Table[Table[Product[q^n - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0,8}] // Grid

T(n,k) = Product_{j=0..k-1} (2^n - 2^j).
T(n,k) = A002884(k)*A022166(n,k).
Let g_m(x) = Sum_{n>=0} (2^m*x)^n/A005329(n) and e(x) = Sum_{n>=0} x^n/A005329(n).  Then Sum_{k>=0} T(n,k)*x^k/A005329(k) = g_n(x)/e(x). - Geoffrey Critzer, Jun 01 2024

A345463 Number of n X n matrices over GF(2) whose characteristic polynomial is irreducible.

Original entry on oeis.org

2, 2, 48, 4032, 1935360, 2879815680, 23222833643520, 629183972848435200, 76669842195418919731200, 35461948770962722105432473600, 69793654310697320331920603401420800, 526981774867699711240400039137800880128000, 16622838761287803491875715175557341313583022080000

Offset: 1

Geoffrey Critzer, Jun 20 2021

nonn

Cf. A000225, A001037, A002884.

nn = 12; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1,
    nn}] Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 1, nn}]/ Table[2^n - 1, {n, 1, nn}]

a(n) = A001037(n)*A002884(n)/A000225(n).

cp's OEIS Frontend

A346214 Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices over GF(2) with index k, 1 <= k <= n, n >= 1.

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A348622 Triangular array read by rows: T(n,k) is the number of periodic n X n matrices over GF(2) having rank k, n>=0, 0<=k<=n.

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A060704 Singular n X n matrices over GF(2).

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A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

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Mathematica

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A070731 Size of largest conjugacy class in the group GL(n,2).

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A085430 a(n) is the minimal m such that the group GL(m,2) has an element of order n.

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A086812 Number of symmetric invertible n X n matrices over GF(2).

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A224879 Number of equivalence classes of n X n nonsingular matrices over GF(2), up to row and column permutation.

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