A386779 Array read by descending antidiagonals: T(n,k) is the number of ways to partition an n X n X n cube into k noncongruent cuboids excluding cube-shaped parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 2, 0, 0, 0, 1, 10, 8, 2, 0, 0, 0, 0, 21, 43, 11, 3, 0, 0, 0, 0, 37, 179, 81, 19, 3, 0, 0, 0, 0, 38, 644, 513, 177, 23, 4, 0, 0, 0, 0, 15, 2068, 3024, 1471, 260, 35, 4, 0, 0, 0, 0, 4, 4995, 17489, 11776, 2736, 458, 40, 5

Offset: 1

Author

Janaka Rodrigo, Aug 02 2025

Keywords

• tabl
• nonn

Comments

The partition here must be valid packing 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 volume x*y*z excluding x=y=z.

Examples

			Array begins
  0    0    0    0     0
  0    0    0    0     0
  0    1    2    2     1
  0    1    3   10    21
  0    2    8   43   179
  0    2   11   81   513
  0    3   19  177  1471
  0    3   23  260  2736
  0    4   35  458  5713
  0    4   40  605  8881
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..97

Crossrefs

Cf. columns: A004526 (k=2), A381847 (k=3), A386756 (k=4), A386757 (k=5).

Formula

T(n,1) = 0

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

T(n,k) = A381847(n) for k = 3.

Extensions

More terms from Sean A. Irvine, Aug 03 2025