cp's OEIS Frontend

Partitioning n X n X n cube is done by decomposing the triplet (n,n,n) into five unordered integer triplets of distinct element composition in three different stages and applying the inclusion-exclusion principle to obtain all geometrically feasible triplets without repetitions.

First stage:

Generating the sequence of sets A(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).

The number of sets in each term describes the number of ways a cube can be partitioned into five cuboids by cutting one of the cuboids in the previous stage into two cuboids. The algorithm explained in A381847 (partitioning into three cuboids) and A384311 (partitioning into four cuboids). As an example: A(3) = {{(3,3,1), (3,2,2), (3,1,1), (2,1,1), (1,1,1)}, {(3,2,2), (3,2,1), (2,2,1), (3,1,1), (2,1,1)}}.

Therefore the number of sets in A(3) is |A(3)| = 2.

Second stage:

Generating the sequence of sets B(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).

The number of sets in each term represents number of distinct five-cuboid combinations filling the n X n X n cube with three full-length axial spanning sharing only two cube corners each.

Sets of five distinct triplets of the n-th term are defined by {(n,n-x,y), (n,n-y,z), (n,n-z,x), (x,y,z), (n-x,n-y,n-z)}, where 0 < x,y,z < n.

The triplet (3,3,3) can be decomposed by the rule only in one way giving, B(3) = {{(1,3,1), (3,1,2), (2,2,3), (2,1,1), (1,2,2)}}. Therefore |B(3)| = 1.

Third stage:

Generating the sequence of sets C(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).

The number of sets in each term represents number of different ways to partition n X n X n cube into five distinct cuboids such that all five cuboids going across the cube parallel to each other avoiding any cut-plane that cuts through the whole pile.

Sets of five distinct triplets of the n-th term are defined by {(n,x,y), (n,p,y+q), (n,n-p,q), (n,n-p-x,n-q), (n,p+x,n-y-q)}, where p+x < n,q+y < n and 0 < x,y,p,q < n.

Triplets (1,1,1), (2,2,2) and (3,3,3) cannot be decomposed by this rule and the triplet (4,4,4) has only one way of decomposing, C(4) = {{(1,1,4), (1,2,4), (1,3,4), (2,2,4), (2,3,4)}}. Therefore |C(4)| = 1.

Since there are no intersections between B(n) and C(n), the number of ways to partition n X n X n cube is given by: a(n) = |A(n) union B(n) union C(n)| = |A(n)| + |B(n)| + |C(n)| - |A(n) intersection B(n)| - |A(n) intersection C(n)|.

Alternatively, a(n) is the number of ways to decompose the triplet (n,n,n) into 3 distinct unordered triplets.

Initial stage: One triplet of the form (n,n,n).

Second stage: Decompose the original triplet into two distinct triplets by splitting one of the elements of (x,y,z) into two parts at a time according to the following rule; (x,y,z) is replaced by (x,y,z-r), (x,y,r), where 0 < r <= z/2. Each resulting triplet must be distinct in element composition when comparing with the rest of the triplets in the set. Sets including the same element composition including the case r = z/2 are named as duplicates and set aside to reconsider when forming the next term.

Third stage: Apply the same rule to one of the two triplets of the second term at a time to create another two distinct triplets. At this stage consider the duplicates of the second term and apply the same rule to decompose one of the two identical triplets into two triplets of distinct element composition.

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.

A386296 is the main sequence for this topic.

Alternatively a(n) is the number of distinct six-cuboid combinations filling an n X n X n cube.

Alternatively, a(n) is the number of ways to decompose (n,n,n) triplet into geometrically feasible four distinct unordered triplets of the form (x,y,z) with no pairs of triplets having equal value for the product x*y*z.

Alternatively a(n) is the number of distinct five-triplet sets produced by A(n)-D(n); that is, a(n) = |A(n)-D(n)|, where the sequences of sets A(n), B(n) and C(n) are introduced in A384479 and D(n) = B(n) U C(n).

Alternatively a(n) is the number of distinct five-triplet sets of the terms produced by D(n)-A(n); that is, a(n) = |D(n)-A(n)|, where A(n), B(n) and C(n) are introduced in A384479 and D(n) = B(n) U C(n).

Developed as the three-dimensional extension of the Mondrian Art Problem.

Alternatively, a(n) is the minimum defect when an n X n X n cube is partitioning into four cuboids of different dimensions.

Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set S of four cuboids.

Let V(x,y,z) = x*y*z be the volume and for a given element of S. Define min(S) = min{V(x,y,z): (x,y,z) in S}, max(S) = max{V(x,y,z): (x,y,z) in S}, and defect(S) = max(S)-min(S).

a(n) is the smallest value of the defect(S) across all possible partitions of the n X n X n cuboid into four cuboids of different dimensions.

Alternatively, a(n) is the number of ways to decompose a (n,n,n) triplet into sets of four unordered triplets of the form (x,y,z) without having x=y=z in any of the triplets.

A strict cuboid is a cuboid with all three dimensions different to each other.

Alternatively a(n) is number of ways to decompose (n,n,n) triplet into sets of distinct unordered geometrically feasible four triplets of the form (x,y,z) excluding x != y != z in any of the triplets.