Number of independent subcubes of a hypercube

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There are 2^n hypercubes at the next step.In summary, the maximum number of independent subcubes of a hypercube with m vertices is determined by the number of vertices on each (n-r)D face, with 2^n hypercubes at the next step. The positions of the m vertices are crucial in determining this maximum number.
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Number of "independent" subcubes of a hypercube

Hello, I am trying to solve this problem: I have an n-dimensional hypercube and m of its vertices. Now I want to compute the maximum number of subcubes of the entire hypercube such that:
- each subcube from the set may contain only those m vertices
- no subcube from the set is part of another subcube from the set

Does anybody have any idea?

Thank you very much.


Standa
 
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I take the 'subcubes' to be hypercubes of lower dimension.
The answer depends on the positions of the m vertices critically. For each (n-1)dimensional face , find the number of vertices on it. If two adjacent faces contain 2 or more vertices, subtract the number of hypercubes on the common 'edge'. Inductively, calculate the number of hypercubes on each (n-r) D face (r=1,2,...).
 

Related to Number of independent subcubes of a hypercube

1. How is the number of independent subcubes of a hypercube calculated?

The number of independent subcubes of a hypercube can be calculated using the formula 2^n - 1, where n is the dimension of the hypercube. This formula takes into account all possible combinations of subcubes within the hypercube.

2. What is the significance of the number of independent subcubes in a hypercube?

The number of independent subcubes in a hypercube is significant because it represents the level of complexity and diversity within the hypercube. It also plays a role in various mathematical and scientific applications, such as graph theory and network analysis.

3. Can the number of independent subcubes in a hypercube be larger than the total number of subcubes?

No, the number of independent subcubes in a hypercube cannot be larger than the total number of subcubes. This is because the formula 2^n - 1 takes into account all possible subcubes, including overlapping and redundant ones.

4. Are there any patterns or relationships between the number of independent subcubes and the dimension of a hypercube?

Yes, there is a relationship between the number of independent subcubes and the dimension of a hypercube. As the dimension increases, the number of independent subcubes also increases exponentially.

5. How does the number of independent subcubes in a hypercube relate to its symmetry?

The number of independent subcubes in a hypercube is closely related to its symmetry. A hypercube with a higher number of independent subcubes has more symmetry and is considered to be more regular and uniform in its structure.

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