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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.

%I A386296 #40 Jul 28 2025 18:36:44
%S A386296 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,
%T A386296 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,
%U A386296 3131,3508,1561,266,35,4,1,0,0,0,40,9533,21699,12960
%N 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.
%C A386296 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.
%H A386296 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a386/A386296.java">Java program</a> (github)
%H A386296 Sean A. Irvine, <a href="/A386296/a386296.txt">The 31 possible partitions of a 4 X 4 X 4 cube into 5 distinct cuboids</a>, 2025.
%H A386296 Sean A. Irvine, <a href="/A386296/a386296_1.txt">The 47 possible partitions of a 5 X 5 X 5 cube into 4 distinct cuboids</a>, 2025.
%F A386296 T(n,1) = 1.
%F A386296 T(n,k) = 0 for k > n^3.
%e A386296 Array begins:
%e A386296   1      0      0      0      0
%e A386296   1      0      0      0      0
%e A386296   1      1      2      4      2
%e A386296   1      1      3     12     31
%e A386296   1      2      8     47    209
%e A386296   1      2     11     85    560
%e A386296   1      3     19    183   1561
%e A386296   1      3     23    266   2852
%e A386296   1      4     35    466   5894
%e A386296   1      4     40    613   9093
%Y A386296 Cf. A333296 (index of maximum nonzero term on each row).
%Y A386296 Columns: A004526 (k=2), A381847 (k=3), A384311 (k=4), A384479 (k=5).
%K A386296 tabl,nonn
%O A386296 1,13
%A A386296 _Janaka Rodrigo_, Jul 17 2025