A287377 -id:A287377 - cp's OEIS frontend

A283435 Triangle read by rows: T(n,m) is the number of binary pattern classes in the (n,m)-rectangular grid with half 1's and half 0's: two patterns are in same class if one can be obtained by a reflection or 180-degree rotation of the other (ordered occurrences rounded up/down if m*n is odd).

1, 1, 1, 1, 1, 3, 1, 2, 6, 39, 1, 4, 22, 252, 3270, 1, 6, 66, 1675, 46448, 1302196, 1, 10, 246, 12300, 676732, 38786376, 2268820290, 1, 19, 868, 88900, 10032648, 1134474924, 134564842984, 15801337532526

Offset: 0

María Merino and Imanol Unanue, May 15 2017

Computed using Polya's enumeration theorem for colorings.

			For n = 3 and m = 2 the T(3,2) = 6 solutions are colorings of 3 X 2 matrices in 2 colors inequivalent under the action of the Klein group with exactly 3 occurrences of each color (coefficient of x1^3 x2^3).
Triangle begins:
  ======================================
  n\m | 0   1   2   3      4       5
  ----|---------------------------------
  0   | 1
  1   | 1   1
  2   | 1   1   3
  3   | 1   2   6   39
  4   | 1   4   22  252    3270
  5   | 1   6   66  1675   46448   1302196

María Merino, Rows n=0..59 of triangle, flattened
María Merino and Imanol Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286892, A287020, A287021, A287022, A287377, A287378, A287383, A287384.

G.f.: g(x1,x2)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
(y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=x1+x2, y2=x1^2+x2^2 and occurrences of numbers are ceiling(m*n/2) for 0's and floor(m*n/2) for 1's.

A286892 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-third each of 1s, 2s and 3s (ordered occurrences rounded up/down if m*n != 0 mod 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 27, 438, 1, 6, 140, 8766, 504504, 1, 16, 1056, 189774, 33258880, 6573403050, 1, 48, 8730, 4292514, 2366403930, 1387750992012, 846182953495152, 1, 108, 63108, 99797220, 159511561440, 282061024690536, 530143167401850960, 976645996512669379710

Offset: 0

María Merino, Imanol Unanue, May 15 2017

Computed using Polya's enumeration theorem for coloring.

			For n = 3 and m = 2 the T(3,2) = 27 solutions are colorings of 3 X 2 matrices in 3 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2).
Triangle begins:
=================================================
n\m | 0    1   2      3       4         5
----|--------------------------------------------
0   | 1
1   | 1    1
2   | 1    1   3
3   | 1    3   27     438
4   | 1    6   140    8766    504504
5   | 1    16  1056   189774  33258880   6573403050

María Merino, Rows n=0..47 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A283435, A287020, A287021, A287022, A287377, A287378, A287383, A287384.

G.f.: g(x1,x2,x3)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
(y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=x1+x2+x3, y2=x1^2+x2^2+x3^2, and occurrences of numbers are ceiling(m*n/3) for  the first k numbers and floor(m*n/3) for the last (3-k) numbers, if m*n = k mod 3.

A287020 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fourth each of 1s, 2s, 3s and 4s (ordered occurrences rounded up/down if n*m != 0 mod 4).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 46, 1926, 1, 12, 648, 92544, 15767640, 1, 30, 6312, 3943710, 2933201376, 2061379857600, 1, 90, 92400, 192994200, 577186150464, 1605824110657800, 5363188066566330000, 1, 318, 1051140, 10266445476, 118129589107200, 1340797019145183600

Offset: 0

María Merino, Imanol Unanue, May 18 2017

Computed using Polya's enumeration theorem for coloring.

			For n = 4 and m = 2 the T(4,2) = 648 solutions are colorings of 4 X 2 matrices in 4 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2).
Triangle begins:
========================================================
n\m | 0   1    2     3         4            5
----|---------------------------------------------------
0   | 1
1   | 1   1
2   | 1   1    6
3   | 1   1    46    1926
4   | 1   12   648   92544     15767640
5   | 1   30   6312  3943710   2933201376   2061379857600

María Merino, Rows n=0..42 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A283435, A286892, A287021, A287022, A287377, A287378, A287383, A287384.

G.f.: g(x1,x2,x3,x4)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
(y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=x1+x2+x3+x4, y2=x1^2+x2^2+x3^2+x4^2, and occurrences of numbers are ceiling(m*n/4) for the first k numbers and floor(m*n/4) for the last (4-k) numbers, if m*n = k mod 4.

A287021 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fifth of 1s, 2s, 3s, 4s and 5s (ordered occurrences rounded up/down if n*m != 0 mod 5).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 90, 5712, 1, 1, 1260, 416064, 168168000, 1, 60, 28440, 42045600, 76385194200, 155840192585280, 1, 180, 415800, 3216282300, 31168037156256, 342718542439257600, 3574641463338838464000, 1, 630, 8408400, 320818773240, 14181456923282880, 794364769671213312000, 40694019408428534970822000, 2416738787895064029335795945088

Offset: 0

María Merino, Imanol Unanue, May 18 2017

Computed using Polya's enumeration theorem for coloring.

			For n = 5 and m = 2 the T(5,2) = 28440 solutions are colorings of 5 X 2 matrices in 5 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2).
Triangle begins:
============================================================
n\m | 0  1    2      3          4             5
----|-------------------------------------------------------
0   | 1
1   | 1  1
2   | 1  1    1
3   | 1  1    90     5712
4   | 1  1    1260   416064     168168000
5   | 1  60   28440  42045600   76385194200   155840192585280

María Merino, Rows n=0..38 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A283435, A286892, A287020, A287022, A287377, A287378, A287383, A287384.

G.f.: g(x1,x2,x3,x4,x5)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;  (y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=Sum_{i=1..5} x_i, y2=Sum_{i=1..5} x_i^2, and occurrences of numbers are ceiling(m*n/5) for the first k numbers and floor(m*n/5) for the last (5-k) numbers, if m*n = k mod 5.

A287378 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-eighth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's and 8's (ordered occurrences rounded up/down if n*m != 0 mod 8).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 45360, 1, 1, 10080, 7484544, 20432442240, 1, 1, 226800, 2554075440, 29331862801920, 577185873264000000

Offset: 0

María Merino, Imanol Unanue, May 24 2017

Computed using Polya's enumeration theorem for coloring.

			For n = 4 and m = 2, the T(4,2) = 10080 solutions are colorings of 4 X 2 matrices in 8 colors inequivalent under the action of the Klein group with exactly 1 occurrence of each color (coefficient of x1^1, x2^1, x3^1, x4^1, x5^1, x6^1, x7^1, x8^1).
Triangle begins:
=================================================================
n\m | 0  1  2       3           4               5
----|------------------------------------------------------------
0   | 1
1   | 1  1
2   | 1  1  1
3   | 1  1  1       45360
4   | 1  1  10080   7484544     20432442240
5   | 1  1  226800  2554075440  29331862801920  577185873264000000

María Merino, Rows n=0..35 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287383, A287384.

g(x1,x2,x3,x4,x5,x6,x7,x8) = (y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
(y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m, where the coefficients y1 and y2 correspond to y1 = Sum_{i=1..8} x_i and y2 = Sum_{i=1..8} x_i^2. Occurrences of numbers are ceiling(m*n/8) for the first k numbers and floor(m*n/8) for the last (8-k) numbers, if m*n = k mod 8.

A287383 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-ninth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's and 9's (ordered occurrences rounded up/down if n*m != 0 mod 9).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 90720, 1, 1, 1, 14968800, 40864824000, 1, 1, 453600, 5108114880, 131993382447360, 3463115239584000000

Offset: 0

María Merino, Imanol Unanue, May 24 2017

Computed using Polya's enumeration theorem for coloring.

			For n = 3 and m = 3 the T(3,3) = 90720 solutions are colorings of 3 X 3 matrices in 9 colors inequivalent under the action of the Klein group with exactly 1 occurrence of each color (coefficient of x1^1, x2^1, x3^1, x4^1, x5^1, x6^1, x7^1, x8^1, x9^1).
Triangle begins:
===================================================================
n\m | 0  1  2       3           4                5
----|--------------------------------------------------------------
0   | 1
1   | 1  1
2   | 1  1  1
3   | 1  1  1       90720
4   | 1  1  1       14968800    40864824000
5   | 1  1  453600  5108114880  131993382447360  3463115239584000000

María Merino, Rows n=0..34 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287378, A287384.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9) = (y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m; (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m; (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m; (y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where the coefficients y1 and y2 correspond to y1 = Sum_{i=1..9} x_i and y2 = Sum_{i=1..9} x_i^2. Occurrences of numbers are ceiling(m*n/9) for the first k numbers and floor(m*n/9) for the last (9-k) numbers, if m*n = k mod 9.

A287384 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-tenth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n*m != 0 mod 10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29937600, 81729648000, 1, 1, 907200, 10216209600, 593970221376000, 15584018578345728000, 1, 1, 29937600, 6252318979200, 1870082229375360000, 1096699334071461120000000, 375493744214599112902800000000

Offset: 0

María Merino and Imanol Unanue, May 24 2017

Computed using Polya's enumeration theorem for coloring.

			For n = 4 and m = 3 the T(4,3)=29937600 solutions are colorings of 4 X 3 matrices in 10 colors inequivalent under the action of the Klein group with exactly 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 occurrences of each color (coefficient of x1^2 x2^2 x3^1 x4^1 x5^1 x6^1 x7^1 x8^1 x9^1 x10^1).
Triangle begins:
  ==========================================
  n\m | 0  1  2   3           4
  ----|-------------------------------------
  0   | 1
  1   | 1  1
  2   | 1  1  1
  3   | 1  1  1   1
  4   | 1  1  1   29937600    81729648000

María Merino, Rows n=0..32 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287378, A287383.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m; (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m; (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m; (y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=Sum_{i=1..10} x_i, y2=Sum_{i=1..10} x_i^2, and occurrences of numbers are ceiling(m*n/10) for the first k numbers and floor(m*n/10) for the last (10-k) numbers, if m*n = k mod 10.

More terms from María Merino, Aug 01 2017

cp's OEIS Frontend

A283435 Triangle read by rows: T(n,m) is the number of binary pattern classes in the (n,m)-rectangular grid with half 1's and half 0's: two patterns are in same class if one can be obtained by a reflection or 180-degree rotation of the other (ordered occurrences rounded up/down if m*n is odd).

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A286892 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-third each of 1s, 2s and 3s (ordered occurrences rounded up/down if m*n != 0 mod 3).

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A287020 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fourth each of 1s, 2s, 3s and 4s (ordered occurrences rounded up/down if n*m != 0 mod 4).

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A287021 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fifth of 1s, 2s, 3s, 4s and 5s (ordered occurrences rounded up/down if n*m != 0 mod 5).

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A287378 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-eighth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's and 8's (ordered occurrences rounded up/down if n*m != 0 mod 8).

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A287383 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-ninth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's and 9's (ordered occurrences rounded up/down if n*m != 0 mod 9).

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A287384 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-tenth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n*m != 0 mod 10).

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