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A283435 Triangle read by rows: T(n,m) is the number of binary pattern classes in the (n,m)-rectangular grid with half 1's and half 0's: two patterns are in same class if one can be obtained by a reflection or 180-degree rotation of the other (ordered occurrences rounded up/down if m*n is odd).

%I A283435 #50 Jun 11 2024 09:39:45
%S A283435 1,1,1,1,1,3,1,2,6,39,1,4,22,252,3270,1,6,66,1675,46448,1302196,1,10,
%T A283435 246,12300,676732,38786376,2268820290,1,19,868,88900,10032648,
%U A283435 1134474924,134564842984,15801337532526
%N A283435 Triangle read by rows: T(n,m) is the number of binary pattern classes in the (n,m)-rectangular grid with half 1's and half 0's: two patterns are in same class if one can be obtained by a reflection or 180-degree rotation of the other (ordered occurrences rounded up/down if m*n is odd).
%C A283435 Computed using Polya's enumeration theorem for colorings.
%H A283435 María Merino, <a href="/A283435/b283435.txt">Rows n=0..59 of triangle, flattened</a>
%H A283435 María Merino and Imanol 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 A283435 G.f.: g(x1,x2)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
%F A283435 (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
%F A283435 (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
%F A283435 (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=x1+x2, y2=x1^2+x2^2 and occurrences of numbers are ceiling(m*n/2) for 0's and floor(m*n/2) for 1's.
%e A283435 For n = 3 and m = 2 the T(3,2) = 6 solutions are colorings of 3 X 2 matrices in 2 colors inequivalent under the action of the Klein group with exactly 3 occurrences of each color (coefficient of x1^3 x2^3).
%e A283435 Triangle begins:
%e A283435   ======================================
%e A283435   n\m | 0   1   2   3      4       5
%e A283435   ----|---------------------------------
%e A283435   0   | 1
%e A283435   1   | 1   1
%e A283435   2   | 1   1   3
%e A283435   3   | 1   2   6   39
%e A283435   4   | 1   4   22  252    3270
%e A283435   5   | 1   6   66  1675   46448   1302196
%Y A283435 Cf. A286892, A287020, A287021, A287022, A287377, A287378, A287383, A287384.
%K A283435 nonn,tabl
%O A283435 0,6
%A A283435 _María Merino_ and Imanol Unanue, May 15 2017