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Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4, with a fifth of 1's, 2's, 3's, 4's and 5's (ordered occurrences rounded up/down if n^2 != 0 mod 5).
6

%I #24 Apr 29 2019 05:42:23

%S 1,1,1,2874,84086160,77920099694640,1787320731699689472000,

%T 1208369393947533515948886636000,

%U 22022604563875220592723146462014970246400,10631042739086498005729294276105510004209560426195000,136864426940639977623403211038729959780835360788855628470904385280

%N Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4, with a fifth of 1's, 2's, 3's, 4's and 5's (ordered occurrences rounded up/down if n^2 != 0 mod 5).

%H María Merino, <a href="/A286526/b286526.txt">Table of n, a(n) for n = 0..37</a>

%H M. Merino and I. Unanue, <a href="https://doi.org/10.1387/ekaia.17851">Counting squared grid patterns with Pólya Theory</a>, EKAIA, 34 (2018), 289-316 (in Basque).

%F G.f.: g(x1,x2,x3,x4,x5) = 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..5} x_i, y2 = Sum_{i=1..5} x_i^2, y4 = Sum_{i=1..5} x_i^4 and occurrences of numbers are ceiling(n^2/5) for the first k numbers and floor(n^2/5) for the last (5-k) numbers, if n^2 = k mod 5.

%e For n=3 the a(3)=2874 solutions are colorings of 3 X 3 matrices in 5 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 2, 2, 1 of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^1).

%Y Cf. A054752, A082963, A286447, A286525.

%K nonn

%O 0,4

%A _María Merino_, Imanol Unanue, May 11 2017