login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A287378 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-eighth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's and 8's (ordered occurrences rounded up/down if n*m != 0 mod 8). 8
1, 1, 1, 1, 1, 1, 1, 1, 1, 45360, 1, 1, 10080, 7484544, 20432442240, 1, 1, 226800, 2554075440, 29331862801920, 577185873264000000 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Computed using Polya's enumeration theorem for coloring.

LINKS

María Merino, Rows n=0..35 of triangle, flattened

M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

FORMULA

g(x1,x2,x3,x4,x5,x6,x7,x8) = (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 the coefficients y1 and y2 correspond to y1 = Sum_{i=1..8} x_i and y2 = Sum_{i=1..8} x_i^2. Occurrences of numbers are ceiling(m*n/8) for the first k numbers and floor(m*n/8) for the last (8-k) numbers, if m*n = k mod 8.

EXAMPLE

For n = 4 and m = 2, the T(4,2) = 10080 solutions are colorings of 4 X 2 matrices in 8 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, x7^1, x8^1).

Triangle begins:

=================================================================

n\m | 0  1  2       3           4               5

----|------------------------------------------------------------

0   | 1

1   | 1  1

2   | 1  1  1

3   | 1  1  1       45360

4   | 1  1  10080   7484544     20432442240

5   | 1  1  226800  2554075440  29331862801920  577185873264000000

CROSSREFS

Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287383, A287384.

Sequence in context: A162894 A031851 A185927 * A163816 A055355 A210618

Adjacent sequences:  A287375 A287376 A287377 * A287379 A287380 A287381

KEYWORD

nonn,tabl

AUTHOR

María Merino, Imanol Unanue, May 24 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 19 04:56 EST 2022. Contains 350464 sequences. (Running on oeis4.)