cp's OEIS Frontend

A287384 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-tenth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n*m != 0 mod 10).

%I A287384 #30 Jul 27 2025 17:17:48
%S A287384 1,1,1,1,1,1,1,1,1,1,1,1,1,29937600,81729648000,1,1,907200,
%T A287384 10216209600,593970221376000,15584018578345728000,1,1,29937600,
%U A287384 6252318979200,1870082229375360000,1096699334071461120000000,375493744214599112902800000000
%N A287384 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-tenth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n*m != 0 mod 10).
%C A287384 Computed using Polya's enumeration theorem for coloring.
%H A287384 María Merino, <a href="/A287384/b287384.txt">Rows n=0..32 of triangle, flattened</a>
%H A287384 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 A287384 G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m; (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m; (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..10} x_i, y2=Sum_{i=1..10} x_i^2, and occurrences of numbers are ceiling(m*n/10) for the first k numbers and floor(m*n/10) for the last (10-k) numbers, if m*n = k mod 10.
%e A287384 For n = 4 and m = 3 the T(4,3)=29937600 solutions are colorings of 4 X 3 matrices in 10 colors inequivalent under the action of the Klein group with exactly 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 occurrences of each color (coefficient of x1^2 x2^2 x3^1 x4^1 x5^1 x6^1 x7^1 x8^1 x9^1 x10^1).
%e A287384 Triangle begins:
%e A287384   ==========================================
%e A287384   n\m | 0  1  2   3           4
%e A287384   ----|-------------------------------------
%e A287384   0   | 1
%e A287384   1   | 1  1
%e A287384   2   | 1  1  1
%e A287384   3   | 1  1  1   1
%e A287384   4   | 1  1  1   29937600    81729648000
%Y A287384 Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287378, A287383.
%K A287384 nonn,tabl
%O A287384 0,14
%A A287384 _María Merino_ and Imanol Unanue, May 24 2017
%E A287384 More terms from _María Merino_, Aug 01 2017