A286397 -id:A286397 - 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.

A287261 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4, with one-tenth 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^2 != 0 mod 10).

Original entry on oeis.org

1, 1, 1, 1, 40864828320, 7792009289281728000, 187746872107299580970294400000, 614005731326101652800803825889630961295360, 176445174659483893854948844253232539237396497554309120000, 7090469783239448892319287907564531885316857076509137838529329991091840000

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(4) = 40864828320 solutions are colorings of 4 X 4 matrices in 10 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2 x7^1 x8^1 x9^1 x10^1).

María Merino, Table of n, a(n) for n = 0..29
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286397, A082963, A286447, A286525, A286526, A287239, A287245, A287249, A287250.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10) = (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..10} x_i, y2=Sum_{i=1..10} x_i^2, y4=Sum_{i=1..10} x_i^4 and occurrences of numbers are ceiling(n^2/10) for the first k numbers and floor(n^2/10) for the last (10-k) numbers, if n^2 = k mod 10.

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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