A386296 Array read by descending antidiagonals: T(n,k) is the number of ways to partition n X n X n cube into k noncongruent cuboids.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 4, 3, 2, 1, 0, 0, 2, 12, 8, 2, 1, 0, 0, 1, 31, 47, 11, 3, 1, 0, 0, 0, 70, 209, 85, 19, 3, 1, 0, 0, 0, 115, 846, 560, 183, 23, 4, 1, 0, 0, 0, 97, 3131, 3508, 1561, 266, 35, 4, 1, 0, 0, 0, 40, 9533, 21699, 12960

Offset: 1

Author

Janaka Rodrigo, Jul 17 2025

Keywords

• tabl
• nonn

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 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     12     31
  1      2      8     47    209
  1      2     11     85    560
  1      3     19    183   1561
  1      3     23    266   2852
  1      4     35    466   5894
  1      4     40    613   9093
		

Links

• Sean A. Irvine, Java program (github)
• Sean A. Irvine, The 31 possible partitions of a 4 X 4 X 4 cube into 5 distinct cuboids, 2025.
• Sean A. Irvine, The 47 possible partitions of a 5 X 5 X 5 cube into 4 distinct cuboids, 2025.

Crossrefs

Cf. A333296 (index of maximum nonzero term on each row).

Columns: A004526 (k=2), A381847 (k=3), A384311 (k=4), A384479 (k=5).

Formula

T(n,1) = 1.

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