A287245 Number of inequivalent n X n matrices over GF(7) under action of dihedral group of the square D_4, with one-seventh each of 1's, 2's, 3's, 4's, 5's, 6's and 7's (ordered occurrences rounded up/down if n^2 != 0 mod 7).
1, 1, 1, 11340, 2270280240, 27055587870486000, 21628439666761521875561280, 920451958269648700957746787694592000, 1914192808178753950843058828570207003149548000000, 216425158352284448578663515683744576588775769063470820304640000
Offset: 0
Keywords
Examples
For n = 3 the a(3) = 11340 solutions are colorings of 3 X 3 matrices in 7 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 1, 1, 1, 1, 1 of each color (coefficient of x1^2 x2^2 x3^1 x4^1 x5^1 x6^1 x7^1).
Links
- María Merino, Table of n, a(n) for n = 0..33
- 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)=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..7} x_i, y2=Sum_{i=1..7} x_i^2, y4=Sum_{i=1..7} x_i^4 and occurrences of numbers are ceiling(n^2/7) for the first k numbers and floor(n^2/7) for the last (7-k) numbers, if n^2 = k mod 7.
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