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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).
9
1, 1, 1, 1, 1, 1, 1, 1, 180, 11358, 1, 1, 2520, 1872000, 1009008000, 1, 1, 56712, 189197280, 814774020480, 4058338214422800, 1, 360, 1871640, 34306401600, 811667639890800, 22208161516294279680, 667544434159390230643200
OFFSET
0,9
COMMENTS
Computed using Polya's enumeration theorem for coloring.
LINKS
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
FORMULA
G.f.: g(x1,x2,x3,x4,x5,x6)=(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; 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.
EXAMPLE
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).
Triangle begins:
============================================================
n\m | 0 1 2 3 4 5
----|-------------------------------------------------------
0 | 1
1 | 1 1
2 | 1 1 1
3 | 1 1 180 11358
4 | 1 1 2520 1872000 1009008000
5 | 1 1 56712 189197280 814774020480 4058338214422800
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
María Merino, Imanol Unanue, May 18 2017
STATUS
approved