A387121 Array read by antidiagonals: T(n,k) is the number of sets of k integer sided cuboids with distinct volumes that fill an n X n X n cube.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 4, 3, 2, 1, 0, 0, 2, 11, 8, 2, 1, 0, 0, 1, 26, 47, 11, 3, 1, 0, 0, 0, 55, 206, 77, 19, 3, 1, 0, 0, 0, 48, 793, 442, 183, 23, 4, 1, 0, 0, 0, 23, 2653, 2451, 1531, 259, 35, 4, 1, 0, 0, 0, 0, 6706, 13022, 12178

Offset: 1

Author

Janaka Rodrigo, Aug 16 2025

Keywords

• tabl
• nonn
• new

Comments

The partitions here must be valid packings of the n X n X n cube, hence T(n,k) is generally less than the number of partitions of n^3 into distinct cuboids (x,y,z) with 1 <= x,y,z <= n and no pair of triplets having equal volume x*y*z.

Examples

			Array begins:
  1     0     0     0     0
  1     0     0     0     0
  1     1     2     4     2
  1     1     3    11    26
  1     2     8    47   206
  1     2    11    77   442
  1     3    19   183  1531
  1     3    23   259  2661
  1     4    35   457  5574
  1     4    40   599  8514
  ...
		

Crossrefs

Columns are A004526 (k=2), A381847 (k=3), A385580 (k=4), A387040 (k=5).

Formula

T(n,1) = 1, T(n,k) = 0 for k > n^3.

Extensions

More terms from Sean A. Irvine, Aug 25 2025