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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). 3
1, 1, 1, 22680, 10216251360, 288592936632000000, 675888739586283307003920000, 150403128386758194407881602780164966400, 2270715491453850844620503532869818724155487772912000, 2190916399747036514334089808617857198357442887303702763561256837120 (list; graph; refs; listen; history; text; internal format)
Computed using Polya's enumeration theorem for coloring.
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
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.
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).
Sequence in context: A279443 A287377 A179726 * A263238 A263198 A263236
María Merino, Imanol Unanue, May 22 2017

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Last modified December 8 11:19 EST 2023. Contains 367678 sequences. (Running on oeis4.)