A287250 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4, with one-ninth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's and 9's (ordered occurrences rounded up/down if n^2 != 0 mod 9).
1, 1, 1, 45360, 20432427120, 1731557619792000000, 17601269260059379482191694720, 11370476506038919496334983007474778275840, 944848320304251231447932170156537415535539635814400000, 6641336088298446224006555306105706090482482272285249518936232000000000
Offset: 0
Keywords
Examples
For n = 3 the a(3) = 45360 solutions are colorings of 3 X 3 matrices in 9 colors inequivalent under the action of D_4 with exactly 1 occurrence of each color (coefficient of x1^1 x2^1 x3^1 x4^1 x5^1 x6^1 x7^1 x8^1 x9^1).
Links
- María Merino, Table of n, a(n) for n = 0..31
- 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) = (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..9} x_i, y2=Sum_{i=1..9} x_i^2, y4=Sum_{i=1..9} x_i^4 and occurrences of numbers are ceiling(n^2/9) for the first k numbers and floor(n^2/9) for the last (9-k) numbers, if n^2 = k mod 9.
Comments