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A287021 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fifth of 1s, 2s, 3s, 4s and 5s (ordered occurrences rounded up/down if n*m != 0 mod 5).

%I A287021 #32 Mar 23 2025 23:34:04
%S A287021 1,1,1,1,1,1,1,1,90,5712,1,1,1260,416064,168168000,1,60,28440,
%T A287021 42045600,76385194200,155840192585280,1,180,415800,3216282300,
%U A287021 31168037156256,342718542439257600,3574641463338838464000,1,630,8408400,320818773240,14181456923282880,794364769671213312000,40694019408428534970822000,2416738787895064029335795945088
%N A287021 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fifth of 1s, 2s, 3s, 4s and 5s (ordered occurrences rounded up/down if n*m != 0 mod 5).
%C A287021 Computed using Polya's enumeration theorem for coloring.
%H A287021 María Merino, <a href="/A287021/b287021.txt">Rows n=0..38 of triangle, flattened</a>
%H A287021 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 A287021 G.f.: g(x1,x2,x3,x4,x5)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
%F A287021 (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
%F A287021 (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;  (y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=Sum_{i=1..5} x_i, y2=Sum_{i=1..5} x_i^2, and occurrences of numbers are ceiling(m*n/5) for the first k numbers and floor(m*n/5) for the last (5-k) numbers, if m*n = k mod 5.
%e A287021 For n = 5 and m = 2 the T(5,2) = 28440 solutions are colorings of 5 X 2 matrices in 5 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2).
%e A287021 Triangle begins:
%e A287021 ============================================================
%e A287021 n\m | 0  1    2      3          4             5
%e A287021 ----|-------------------------------------------------------
%e A287021 0   | 1
%e A287021 1   | 1  1
%e A287021 2   | 1  1    1
%e A287021 3   | 1  1    90     5712
%e A287021 4   | 1  1    1260   416064     168168000
%e A287021 5   | 1  60   28440  42045600   76385194200   155840192585280
%Y A287021 Cf. A283435, A286892, A287020, A287022, A287377, A287378, A287383, A287384.
%K A287021 nonn,tabl
%O A287021 0,9
%A A287021 _María Merino_, Imanol Unanue, May 18 2017