A006380 -id:A006380 - cp's OEIS frontend

A006148 Number of 4 X n binary matrices up to row and column permutations.

1, 5, 22, 87, 317, 1053, 3250, 9343, 25207, 64167, 155004, 357009, 787586, 1670643, 3419552, 6774765, 13027340, 24372942, 44462456, 79240762, 138204782, 236258358, 396409924, 653639898, 1060379169, 1694174350, 2668300758, 4146300078, 6361709115, 9644583474

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Index entries for linear recurrences with constant coefficients, signature (6, -12, 6, 6, -6, 22, -54, 33, -4, 12, 60, -125, 54, -54, 70, 87, -132, 64, -132, 87, 70, -54, 54, -125, 60, 12, -4, 33, -54, 22, -6, 6, 6, -12, 6, -1).
Index entries for two-way infinite sequences

Cf. A002623, A002727, A006380.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

CoefficientList[Series[(x^20 - x^19 + 4 x^18 + 9 x^17 + 23 x^16 + 39 x^15 + 90 x^14 + 131 x^13 + 204 x^12 + 238 x^11 + 252 x^10 + 238 x^9 + 204 x^8 + 131 x^7 + 90 x^6 + 39 x^5 + 23 x^4 + 9 x^3 + 4 x^2 - x + 1)/((1 - x^4)^3 (1 - x^3)^4 (1 - x^2)^3 (1 - x)^6), {x, 0, 45}], x] (* Vincenzo Librandi, Oct 13 2015 *)
LinearRecurrence[{6,-12,6,6,-6,22,-54,33,-4,12,60,-125,54,-54,70,87,-132,64,-132,87,70,-54,54,-125,60,12,-4,33,-54,22,-6,6,6,-12,6,-1},{1,5,22,87,317,1053,3250,9343,25207,64167,155004,357009,787586,1670643,3419552,6774765,13027340,24372942,44462456,79240762,138204782,236258358,396409924,653639898,1060379169,1694174350,2668300758,4146300078,6361709115,9644583474,14456861538,21439125178,31471971903,45755970759,65915132560,94129925265},30] (* Harvey P. Dale, Jun 22 2021 *)

Vec(G(4, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^20 - x^19 + 4*x^18 + 9*x^17 + 23*x^16 + 39*x^15 + 90*x^14 + 131*x^13 + 204*x^12 + 238*x^11 + 252*x^10 + 238*x^9 + 204*x^8 + 131*x^7 + 90*x^6 + 39*x^5 + 23*x^4 + 9*x^3 + 4*x^2 - x + 1)/((1 - x^4)^3*(1 - x^3)^4*(1 - x^2)^3*(1 - x)^6). - Vladeta Jovovic, Feb 04 2000

More terms from Vladeta Jovovic, Feb 04 2000
Definition corrected by Max Alekseyev, Feb 05 2010
More terms from Vincenzo Librandi, Oct 13 2015

A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 4, 7, 19, 32, 68, 114, 210, 336, 562, 862, 1349, 1987, 2950, 4201, 5991, 8278, 11422, 15386, 20660, 27218, 35718, 46158, 59401, 75475, 95494, 119545, 149035, 184118, 226562, 276620, 336470, 406490, 489344, 585572, 698397, 828549, 979896

Offset: 0

N. J. A. Sloane

Also the number of ways in which to label the vertices of the cube (or faces of the octahedron) with nonnegative integers summing to n, where labelings that differ only by rotation or reflection are considered the same. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004

			Representatives of the seven classes of 3 X 3 binary matrices are:
[ 1 1 1 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 1 ] [ 0 1 1 ] [ 0 1 1 ] [ 0 1 1 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 0 ] [ 1 0 0 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 1 1 ] [ 1 0 0 ].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1052.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,2,-5,7,-4,2,2,-8,10,-8,2,2,-4,7,-5,2,-1,-2,3,-1).
Index entries for sequences related to binary matrices
Index entries for two-way infinite sequences

Column k=3 of A363349.

Cf. A000601, A005232, A006382, A006380, A002727, A006148.

Vec((1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48 + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48.

G.f.: (x^14 - 2*x^13 + 3*x^12 - 2*x^11 + 5*x^10 - 4*x^9 + 7*x^8 - 4*x^7 + 7*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 2*x + 1)/(x^6 - 1)/(x^2 + 1)^2/(x^2 + x + 1)/(x + 1)^3/(x - 1)^7.

Entry revised by Vladeta Jovovic, Aug 05 2000

Definition corrected by Max Alekseyev, Feb 05 2010

A006382 Number of n X 4 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 5, 11, 41, 101, 301, 757, 1981, 4714, 11133, 24763, 53818, 111941, 226857, 444260, 848620, 1576226, 2862426, 5077454, 8827758, 15043096, 25183794, 41434222, 67108437, 107051463, 168402958, 261384026, 400684767, 606936536

Offset: 0

N. J. A. Sloane

nonn

			Representatives of the five classes of 2 X 4 binary matrices are:
[ 1 1 1 1 ] [ 1 1 1 0 ] [ 1 1 0 1 ] [ 1 0 1 1 ] [ 0 1 1 1 ]
[ 1 1 1 1 ] [ 1 1 1 1 ] [ 1 1 1 0 ] [ 1 1 0 0 ] [ 1 0 0 0 ]

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -13, 10, 4, -14, 25, -46, 53, -36, 8, 44, -111, 138, -123, 106, -54, -66, 181, -238, 259, -220, 98, 36, -150, 280, -352, 280, -150, 36, 98, -220, 259, -238, 181, -66, -54, 106, -123, 138, -111, 44, 8, -36, 53, -46, 25, -14, 4, 10, -13, 6, -1).
Index entries for sequences related to binary matrices
Index entries for two-way infinite sequences

Column k=4 of A363349.

Cf. A005232, A006380, A006381, A002727, A006148.

G.f. : (1/(1 - x^1)^16 + 51/(1 - x^2)^8 + 12/(1 - x^1)^8/(1 - x^2)^4 + 84/(1 - x^4)^4 + 12/(1 - x^1 )^4/(1 - x^2)^6 + 32/(1 - x^1)^4/(1 - x^3)^4 + 96/(1 - x^2)^2/(1 - x^6)^2 + 48/(1 - x^1)^2/(1 - x^2)^1/(1 - x^4)^3 + 48/(1 - x^8)^2)/384.

Entry revised by Vladeta Jovovic, Aug 05 2000

A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 3, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 6, 11, 19, 10, 4, 1, 1, 1, 7, 16, 41, 32, 16, 4, 1, 1, 1, 8, 23, 81, 101, 68, 20, 5, 1, 1, 1, 9, 31, 153, 299, 301, 114, 29, 5, 1, 1, 1, 10, 41, 273, 849, 1358, 757, 210, 35, 6, 1

Offset: 0

Andrew Howroyd, May 28 2023

T(n,k) is also the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows and columns.

			Array begins:
======================================================
n/k| 0 1  2   3    4     5      6       7        8 ...
---+--------------------------------------------------
0  | 1 1  1   1    1     1      1       1        1 ...
1  | 1 1  1   1    1     1      1       1        1 ...
2  | 1 2  3   4    5     6      7       8        9 ...
3  | 1 2  4   7   11    16     23      31       41 ...
4  | 1 3  8  19   41    81    153     273      468 ...
5  | 1 3 10  32  101   299    849    2290     5901 ...
6  | 1 4 16  68  301  1358   6128   27008   114763 ...
7  | 1 4 20 114  757  5567  43534  343656  2645494 ...
8  | 1 5 29 210 1981 23350 319119 4633380 67013431 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

A259344 is the same array without the first row and column read by upward antidiagonals.
Columns k=0..6 are A000012, A004526(n+2), A005232, A006381, A006382, A056204, A056205.
Rows n=2..4 are A000027(n+1), A000601, A006380.
Main diagonal is A006383.
Cf. A028657, A241956, A362905.

\\ Compare A028657.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
T(n, k)={if(n==0, 1, my(s=0); forpart(q=n, my(e=1<

A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 6, 16, 81, 299, 1358, 5567, 23350, 91998, 351058, 1269907, 4394634, 14495236, 45779246, 138567568, 403282017, 1130773069, 3062535192, 8028046724, 20411824364, 50429813556, 121280243676, 284360432241, 650972702410

Offset: 0

Vladeta Jovovic, Aug 05 2000

nonn

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 148.

Column k=5 of A363349.

Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

G.f.: 1/3840*(1/(1 - x^1)^32 + 231/(1 - x^2)^16 + 20/(1 - x^1)^16/(1 - x^2)^8 + 520/(1 - x^4)^8 + 60/(1 - x^1)^8/(1 - x^2)^12 + 80/(1 - x^1)^8/(1 - x^3)^8 + 720/(1 - x^2)^4/(1 - x^6)^4 + 160/(1 - x^1)^4/(1 - x^2)^2/(1 - x^3)^4/(1 - x^6)^2 + 320/(1 - x^4)^2/(1 - x^12)^2 + 240/(1 - x^1)^4/(1 - x^2)^2/(1 - x^4)^6 + 480/(1 - x^8)^4 + 240/(1 - x^2)^4/(1 - x^4)^6 + 384/(1 - x^1)^2/(1 - x^5)^6 + 384/(1 - x^2)^1/(1 - x^10)^3).

A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 7, 23, 153, 849, 6128, 43534, 319119, 2255466, 15307395, 98349144, 597543497, 3430839916, 18653684881, 96273409815, 473010823993, 2218614773950, 9961651259869, 42927432229913, 177963663264430

Offset: 0

Vladeta Jovovic, Aug 05 2000

nonn

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans.Computers, 22 (1973), 1048-1051.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 312.

Column k=6 of A363349.

Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

G.f.: 1/46080*(1/(1 - x^1)^64 + 1053/(1 - x^2)^32 + 30/(1 - x^1)^32/(1 - x^2)^16 + 4920/(1 - x^4)^16 + 180/(1 - x^1)^16/(1 - x^2)^24 + 120/(1 - x^1)^8/(1 - x^2)^28 + 160/(1 - x^1)^16/(1 - x^3)^16 + 5280/(1 - x^2)^8/(1 - x^6)^8 + 960/(1 - x^1)^8/(1 - x^2)^4/(1 - x^3)^8/(1 - x^6)^4 + 3840/(1 - x^4)^4/(1 - x^12)^4 + 640/(1 - x^1)^4/(1 - x^3)^20 + 1920/(1 - x^2)^2/(1 - x^6)^10 + 720/(1 - x^1)^8/(1 - x^2)^4/(1 - x^4)^12 + 5760/(1 - x^8)^8 + 2160/(1 - x^2)^8/(1 - x^4)^12 + 1440/(1 - x^1)^4/(1 - x^2)^6/(1 - x^4)^12 + 2304/(1 - x^1)^4/(1 - x^5)^12 + 6912/(1 - x^2)^2/(1 - x^10)^6 + 3840/(1 - x^1)^2/(1 - x^2)^1/(1 - x^3)^2/(1 - x^6)^9 + 3840/(1 - x^4)^1/(1 - x^12)^5).

A259344 Array read by antidiagonals: number of inequivalent m X n (0,1)-matrices under permutation of rows and permutation and/or complementation of columns.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 8, 3, 1, 6, 11, 19, 10, 4, 1, 7, 16, 41, 32, 16, 4, 1, 8, 23, 81, 101, 68, 20, 5

Offset: 1

N. J. A. Sloane, Jun 27 2015

			The first few antidiagonals are:
1,
1,2,
1,3,2,
1,4,4,3,
1,5,7,8,3,
1,6,11,19,10,4,
1,7,16,41,32,16,4,
1,8,23,81,101,68,20,5,
...

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. See Table II.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

For some rows, columns, diagonals see A006380, A006281, A006382, A006383.

The second row of the array starts 2,4,7,11,16,23, which does not identify it uniquely.

cp's OEIS Frontend

A006148 Number of 4 X n binary matrices up to row and column permutations.

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A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

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A006382 Number of n X 4 binary matrices under row and column permutations and column complementations.

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A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

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A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

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A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

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A259344 Array read by antidiagonals: number of inequivalent m X n (0,1)-matrices under permutation of rows and permutation and/or complementation of columns.

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