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proposed

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

T(n, k) = A362176A362187(k)*A362208(n, k).

Cf. A000290, A003056 (row lengths), A345132, A362176, A362187, A362208.

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].

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;

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;

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;

...

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, (Sqrt[8n+1]-1)/2}]//Flatten

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, (Sqrt[8n+1]-1)/2}]//Flatten

allocated for Stefano SpeziaIrregular 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

1,4

T(n, k) = A362176(k)*A362208(n, k).

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, (Sqrt[8n+1]-1)/2}]//Flatten

Cf. A000290, A003056 (row lengths), A345132, A362176, A362208.

allocated

nonn,tabf

Stefano Spezia, Apr 11 2023

approved

editing