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A287022 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with a sixth of 1s, 2s, 3s, 4s, 5s and 6s (ordered occurrences rounded up/down if n*m != 0 mod 6).

%I A287022 #32 Mar 23 2025 18:44:44
%S A287022 1,1,1,1,1,1,1,1,180,11358,1,1,2520,1872000,1009008000,1,1,56712,
%T A287022 189197280,814774020480,4058338214422800,1,360,1871640,34306401600,
%U A287022 811667639890800,22208161516294279680,667544434159390230643200
%N A287022 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with a sixth of 1s, 2s, 3s, 4s, 5s and 6s (ordered occurrences rounded up/down if n*m != 0 mod 6).
%C A287022 Computed using Polya's enumeration theorem for coloring.
%H A287022 María Merino, <a href="/A287022/b287022.txt">Rows n=0..37 of triangle, flattened</a>
%H A287022 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 A287022 G.f.: g(x1,x2,x3,x4,x5,x6)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
%F A287022 (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
%F A287022 (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m; where coefficient correspond to y1=Sum_{i=1..6} x_i, y2=Sum_{i=1..6} x_i^2, and occurrences of numbers are ceiling(m*n/6) for the first k numbers and floor(m*n/6) for the last (6-k) numbers, if m*n = k mod 6.
%e A287022 For n=3 and m=2 the T(3,2)=180 solutions are colorings of 3 X 2 matrices in 6 colors inequivalent under the action of the Klein group with exactly 1 occurrence of each color (coefficient of x1^1 x2^1 x3^1 x4^1 x5^1 x6^1).
%e A287022 Triangle begins:
%e A287022 ============================================================
%e A287022 n\m | 0  1  2      3           4             5
%e A287022 ----|-------------------------------------------------------
%e A287022 0   | 1
%e A287022 1   | 1  1
%e A287022 2   | 1  1  1
%e A287022 3   | 1  1  180    11358
%e A287022 4   | 1  1  2520   1872000     1009008000
%e A287022 5   | 1  1  56712  189197280   814774020480  4058338214422800
%Y A287022 Cf. A283435, A286892, A287020, A287021.
%K A287022 nonn,tabl
%O A287022 0,9
%A A287022 _María Merino_, Imanol Unanue, May 18 2017