A363349 -id:A363349 - cp's OEIS frontend

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

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

A006380 Number of equivalence classes of 4 X n binary matrices when one can permute rows, permute columns and complement columns.

Original entry on oeis.org

1, 3, 8, 19, 41, 81, 153, 273, 468, 774, 1240, 1930, 2933, 4356, 6341, 9064, 12743, 17643, 24093, 32479, 43270, 57019, 74377, 96103, 123089, 156354, 197081, 246622, 306519, 378520, 464614, 567028, 688276, 831169, 998845, 1194793, 1422899, 1687447, 1993182

Offset: 0

N. J. A. Sloane

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
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Index entries for sequences related to binary matrices
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-2,2,5,-8,6,-8,5,2,-2,2,-5,4,-1).
Index entries for two-way infinite sequences

Row n=4 of A363349.

Cf. A000601, A006148, A006383.

LinearRecurrence[{4,-5,2,-2,2,5,-8,6,-8,5,2,-2,2,-5,4,-1},{1,3,8,19,41,81,153,273,468,774,1240,1930,2933,4356,6341,9064},40] (* Harvey P. Dale, Nov 23 2024 *)

Vec((1 - x + x^2 + x^4 + x^6 - x^7 + x^8)/((1 - x)^8*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2) + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1 - x + x^2 + x^4 + x^6 - x^7 + x^8)/((1 - x)^8*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2). - Andrew Howroyd, May 30 2023

Terms a(7) onwards from Max Alekseyev, Feb 05 2010

A006383 Number of equivalence classes of n X n binary matrices when one can permute rows, permute columns and complement columns.

Original entry on oeis.org

1, 1, 3, 7, 41, 299, 6128, 343656, 67013431, 45770163273, 108577103160005, 886929528971819040, 24943191706060101926577, 2425246700258693990625775794, 820270898724825121532156178527106

Offset: 0

N. J. A. Sloane

			a(2) = 3:
00 10 11
00 00 00

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..50 (terms 0..35 from Sean A. Irvine)
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)
Index entries for sequences related to binary matrices

Main diagonal of A363349.

Cf. A002724, A005232, A006381, A006382, A056204, A056205.

Definition corrected by Brendan McKay, Jan 07 2007

Terms a(7) onward from Max Alekseyev, Feb 05 2010

A362905 Array read by antidiagonals: T(n,k) is the number of n element multisets of length k vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 8, 5, 3, 1, 1, 1, 16, 15, 11, 3, 1, 1, 1, 32, 51, 50, 14, 4, 1, 1, 1, 64, 187, 276, 99, 24, 4, 1, 1, 1, 128, 715, 1768, 969, 232, 30, 5, 1, 1, 1, 256, 2795, 12496, 11781, 3504, 429, 45, 5, 1, 1, 1, 512, 11051, 93600, 162877, 73440, 10659, 835, 55, 6, 1

Offset: 0

Andrew Howroyd, May 27 2023

Equivalently, T(n,k) is the number multisets with n elements drawn from {0..2^k-1} such that the bitwise-xor of all the elements gives zero.
T(n,k) is 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.
T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and complementation of columns.

			Array begins:
=========================================
n/k| 0 1  2   3     4      5        6 ...
---+-------------------------------------
0  | 1 1  1   1     1      1        1 ...
1  | 1 1  1   1     1      1        1 ...
2  | 1 2  4   8    16     32       64 ...
3  | 1 2  5  15    51    187      715 ...
4  | 1 3 11  50   276   1768    12496 ...
5  | 1 3 14  99   969  11781   162877 ...
6  | 1 4 24 232  3504  73440  1878976 ...
7  | 1 4 30 429 10659 394383 18730855 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..4 are A000012, A004526(n+2), A053307, A362906, A363350.
Rows n=2..3 are A000079, A007581.
Main diagonal is A363351.
Cf. A054724, A340030, A340312, A363349.

A362905[n_,k_]:=(Binomial[2^k+n-1,n]+If[EvenQ[n],(2^k-1)Binomial[2^(k-1)+n/2-1,n/2],0])/2^k;Table[A362905[n-k,k],{n,0,15},{k,n,0,-1}] (* Paolo Xausa, Nov 19 2023 *)

T(n,k)={(binomial(2^k+n-1, n) + if(n%2==0, (2^k-1)*binomial(2^(k-1)+n/2-1,n/2)))/2^k}

T(n,k) = binomial(2^k+n-1, n)/2^k for odd n;
T(n,k) = (binomial(2^k+n-1, n) + (2^k-1)*binomial(2^(k-1)+n/2-1, n/2))/2^k for even n.
G.f. of column k: (1/(1-x)^(2^k) + (2^k-1)/(1-x^2)^(2^(k-1)))/2^k.

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).

cp's OEIS Frontend

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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A006380 Number of equivalence classes of 4 X n binary matrices when one can permute rows, permute columns and complement columns.

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A006383 Number of equivalence classes of n X n binary matrices when one can permute rows, permute columns and complement columns.

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A362905 Array read by antidiagonals: T(n,k) is the number of n element multisets of length k vectors over GF(2) that sum to zero.

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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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