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).

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

María Merino, Rows n=0..37 of triangle, flattened

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

Cf. A283435, A286892, A287020, A287021.

Sequence in context: A271673 A008378 A214818 * A217791 A035830 A244056

Adjacent sequences: A287019 A287020 A287021 * A287023 A287024 A287025

Keyword

nonn,tabl

Author

María Merino, Imanol Unanue, May 18 2017

Status

approved