A286394 -id:A286394 - cp's OEIS frontend

A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 21, 102, 1, 0, 1, 5, 55, 2862, 8548, 1, 0, 1, 6, 120, 34960, 5398083, 4211744, 1, 0, 1, 7, 231, 252375, 537157696, 105918450471, 8590557312, 1, 0, 1, 8, 406, 1284066, 19076074375, 140738033618944, 18761832172500795, 70368882591744, 1, 0

Offset: 0

Andrew Howroyd, Apr 14 2021

			Array begins:
====================================================================
n\k | 0 1       2            3               4                 5
----+---------------------------------------------------------------
  0 | 1 1       1            1               1                 1 ...
  1 | 0 1       2            3               4                 5 ...
  2 | 0 1       6           21              55               120 ...
  3 | 0 1     102         2862           34960            252375 ...
  4 | 0 1    8548      5398083       537157696       19076074375 ...
  5 | 0 1 4211744 105918450471 140738033618944 37252918396015625 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860

Rows 0..5 are A000012, A001477, A002817, A217331, A217338, A283033.
Columns 0..10 are A000007, A000012, A054247, A054739, A054751, A054752, A286392, A286393, A286394, A286396, A286397.
Main diagonal is A001326.
Cf. A246106, A343095, A343875.

T(n,k) = {(k^(n^2) + 2*k^((n^2 + 3*(n%2))/4) + k^((n^2 + (n%2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n+n%2)/2) )/8}

T(n,k) = (k^(n^2) + 2*k^((n^2 + 3*(n mod 2))/4) + k^((n^2 + (n mod 2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n + n mod 2)/2) )/8.

A286396 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 9, 1035, 48700845, 231628411446741, 89737248564744874067889, 2816049943117424212512789695666175, 7158021121277935153545945911617993395398302485, 1473773072217322896440109113309952350877179744639518847951721

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 08 2017

nonn

Burnside's orbit-counting lemma.

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

Column k=9 of A343097.

Cf. A054247, A054739, A054751, A054752, A286392, A286393, A286394.

Table[1/8*(9^(n^2) + 2*9^((n^2 + 3 #)/4) + (3 - 2 #)*9^((n^2 + #)/2) + (2 + 2 #)*9^((n^2 + n)/2)) &@ Boole@ OddQ@ n, {n, 0, 7}] (* Michael De Vlieger, May 12 2017 *)

a(n) = (1/8)*(9^(n^2) + 2*9^(n^2/4) + 3*9^(n^2/2) + 2*9^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(9^(n^2) + 2*9^((n^2 + 3)/4) + 9^((n^2 + 1)/2) + 4*9^((n^2 + n)/2)) if n is odd.

A286397 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 10, 1540, 125512750, 1250002537502500, 1250000000501250002500000, 125000000000000250375000000250000000, 1250000000000000000005001250000000002500000000000

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 08 2017

nonn

Burnside's orbit-counting lemma.

María Merino, Table of n, a(n) for n = 0..20
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Column k=10 of A343097.

Cf. A054247, A054739, A054751, A054752, A286392, A286393, A286394, A286395.

Table[1/8*(10^(n^2) + 2*10^((n^2 + 3 #)/4) + (3 - 2 #)*10^((n^2 + #)/2) + (2 + 2 #)*10^((n^2 + n)/2)) &@ Boole@ OddQ@ n, {n, 7}] (* Michael De Vlieger, May 12 2017 *)

a(n) = (1/8)*(10^(n^2) + 2*10^(n^2/4) + 3*10^(n^2/2) + 2*10^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(10^(n^2) + 2*10^((n^2 + 3)/4) + 10^((n^2 + 1)/2) + 4*10^((n^2 + n)/2)) if n is odd.

A287249 Number of inequivalent n X n matrices over GF(8) under action of dihedral group of the square D_4, 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^2 != 0 mod 8).

Original entry on oeis.org

1, 1, 1, 22680, 10216251360, 288592936632000000, 675888739586283307003920000, 150403128386758194407881602780164966400, 2270715491453850844620503532869818724155487772912000, 2190916399747036514334089808617857198357442887303702763561256837120

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(4) = 10216251360 solutions are colorings of 4 X 4 matrices in 8 colors inequivalent under the action of D_4 with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2 x7^2 x8^2).

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

Cf. A286394, A082963, A286447, A286525, A286526, A287239, A287245.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8) = (1/8)*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and (1/8)*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..8} x_i, y2=Sum_{i=1..8} x_i^2, y4=Sum_{i=1..8} x_i^4 and occurrences of numbers are ceiling(n^2/8) for the first k numbers and floor(n^2/8) for the last (8-k) numbers, if n^2 = k mod 8.

cp's OEIS Frontend

A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

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A286396 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4.

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A286397 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4.

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Mathematica

Formula

A287249 Number of inequivalent n X n matrices over GF(8) under action of dihedral group of the square D_4, 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^2 != 0 mod 8).

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