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).

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

Offset

0,5

Comments

Computed using Polya's enumeration theorem for coloring.

Links

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).

Formula

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.

Example

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).

Crossrefs

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

Sequence in context: A256842 A225886 A199428 * A172631 A172721 A259350

Adjacent sequences: A287258 A287259 A287260 * A287262 A287263 A287264

Keyword

nonn

Author

María Merino, Imanol Unanue, May 22 2017

Status

approved