A386296 - OEIS

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

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OFFSET

1,13

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.

LINKS

Table of n, a(n) for n=1..73.

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.

FORMULA

T(n,1) = 1.

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

EXAMPLE

Array begins:

1 0 0 0 0

1 1 2 4 2