A362209 Irregular triangle read by rows: T(n, k) is the number of k X k matrices using all the integers from 1 to k^2 and having trace equal to n, with 1 <= k <= A003056(n).

1, 0, 0, 4, 0, 4, 0, 8, 0, 4, 4320, 0, 4, 4320, 0, 0, 8640, 0, 0, 12960, 0, 0, 17280, 11496038400, 0, 0, 21600, 11496038400, 0, 0, 30240, 22992076800, 0, 0, 30240, 34488115200, 0, 0, 34560, 57480192000, 0, 0, 34560, 68976230400, 291948240981196800000

Offset: 1

Author

Stefano Spezia, Apr 11 2023

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins:
    1;
    0;
    0, 4;
    0, 4;
    0, 8;
    0, 4,  4320;
    0, 4,  4320;
    0, 0,  8640;
    0, 0, 12960;
    0, 0, 17280, 11496038400;
    0, 0, 21600, 11496038400;
    0, 0, 30240, 22992076800;
    0, 0, 30240, 34488115200;
    0, 0, 34560, 57480192000;
    0, 0, 34560, 68976230400, 291948240981196800000;
    ...
T(5,2) = 8 since we have:
    [1, 2]  [1, 3]  [4, 2]  [4, 3]
    [3, 4], [2, 4], [3, 1], [2, 1],
.
    [2, 1]  [2, 4]  [3, 1]  [3, 4]
    [4, 3], [1, 3], [4, 2], [1, 2].
		

Crossrefs

Cf. A000290, A003056 (row lengths), A345132, A362187, A362208.

Programs

Mathematica

A362208[n_,k_] := Length[Select[Join@@Permutations/@Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@#&], Length[#]==k&]]; Table[(k^2-k)!A362208[n,k],{n,15},{k,Floor[(Sqrt[8n+1]-1)/2]}]//Flatten

Formula

T(n, k) = A362187(k)*A362208(n, k).