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A386848 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 excluding strict cuboids.

%I A386848 #13 Aug 06 2025 01:00:39
%S A386848 1,0,1,0,0,1,0,0,1,1,0,0,0,1,1,0,0,2,0,2,1,0,0,1,3,0,2,1,0,0,1,3,4,1,
%T A386848 3,1,0,0,0,1,10,6,1,3,1,0,0,0,6,9,19,6,2,4,1,0,0,0,5,34,24,30,9,3,4,1,
%U A386848 0,0,0,0,78,37,47,44,8,4,5,1,0,0,0,0,93
%N A386848 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 excluding strict cuboids.
%C A386848 A strict cuboid is a cuboid with all dimensions different to each other.
%C A386848 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 excluding x != y != z.
%H A386848 Sean A. Irvine, <a href="/A386848/b386848.txt">Table of n, a(n) for n = 1..140</a>
%F A386848 T(n,1) = 1,
%F A386848 T(n,k) = 0 for k > n^3.
%e A386848 1    0    0    0    0
%e A386848 1    0    0    0    0
%e A386848 1    1    0    2    1
%e A386848 1    1    0    3    3
%e A386848 1    2    0    4   10
%e A386848 1    2    1    6   19
%e A386848 1    3    1    6   30
%e A386848 1    3    2    9   44
%e A386848 1    4    3    8   64
%e A386848 1    4    4    13  84
%Y A386848 Cf. A386296.
%Y A386848 Columns: A004526(k=2), A211540(k=3), A386846(k=4), A386847(k=5).
%K A386848 nonn,tabl
%O A386848 1,18
%A A386848 _Janaka Rodrigo_, Aug 05 2025
%E A386848 More terms from _Sean A. Irvine_, Aug 05 2025