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edited Oct 5, 2014 at 0:53

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?

By connected,"connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we choose all the "1" sites and all the "0" sites are not chosen. Then

is connected and

is not connected.

A more difficult question would be, how many ways could we choose sites. Such that there are at most 2 connected "island"(or component):

For example:

has 2 connected component and

has 3 connected component.

The ultimate question is, given a cube size of n by m by k, how many ways could we choose sites, such that there are at most j connected components?

I face this question in my research in designing algorithm for material science. Thank you:)

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?

By connected, we will illustrate by example, say, we choose all the "1" sites and all the "0" sites are not chosen. Then

is connected and

is not connected.

A more difficult question would be, how many ways could we choose sites. Such that there are at most 2 connected "island"(or component):

For example:

has 2 connected component and

has 3 connected component.

The ultimate question is, given a cube size of n by m by k, how many ways could we choose sites, such that there are at most j connected components?

I face this question in my research in designing algorithm for material science. Thank you:)

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?

By "connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we choose all the "1" sites and all the "0" sites are not chosen. Then

is connected and

is not connected.

A more difficult question would be, how many ways could we choose sites. Such that there are at most 2 connected "island"(or component):

For example:

has 2 connected component and

has 3 connected component.

The ultimate question is, given a cube size of n by m by k, how many ways could we choose sites, such that there are at most j connected components?

I face this question in my research in designing algorithm for material science. Thank you:)

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asked Oct 4, 2014 at 20:41

user40780

enumeration of connected blocks in finite size square

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?

By connected, we will illustrate by example, say, we choose all the "1" sites and all the "0" sites are not chosen. Then

is connected and

is not connected.

A more difficult question would be, how many ways could we choose sites. Such that there are at most 2 connected "island"(or component):

For example:

has 2 connected component and

has 3 connected component.

The ultimate question is, given a cube size of n by m by k, how many ways could we choose sites, such that there are at most j connected components?

I face this question in my research in designing algorithm for material science. Thank you:)