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Home › Blogs › JADELottery's Blog › Combinatorial Fractalization reformated

Combinatorial Fractalization reformated

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A while back we post the Combinatorial Fractalization.

Unfortunately, we used a format that could only be seen correctly in Internet Explorer.

We reformatted it for other browsers.

 

Combinatorial Fractalization

    C(n, r) =  [from z = 1 to z = n - r + 1]  Σ C(n - z, r - 1)

    C(n, r) = C(n - 1, r - 1) + C(n - 2, r - 1) + C(n - 3, r - 1) + ... + C(r + 3, r - 1) + C(r + 2, r - 1) + C(r + 1, r - 1) + C(r, r - 1) + C(r - 1, r - 1)

    Fractals of C(n, r) -  {C(n - 1, r - 1), C(n - 2, r - 1), C(n - 3, r - 1), ... , C(r + 3, r - 1), C(r + 2, r - 1), C(r + 1, r - 1), C(r, r - 1), C(r - 1, r - 1)}
        Fractals of C(n, r) - [from z = 1 to z = n - r + 1] ψ {C(n - z, r - 1)}

    Iteration of C(n, r) Fractals
          Fractals of C(n - 1, r - 1) - {C(n - 2, r - 2), C(n - 3, r - 2), C(n - 4, r - 2), ... , C(r + 2, r - 2), C(r + 1, r - 2), C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
          Fractals of C(n - 1, r - 1) - [from z = 1 to z = n - r + 1] ψ {C(n - z - 1, r - 2)}


      Fractals of C(n - 2, r - 1) - {C(n - 3, r - 2), C(n - 4, r - 2), C(n - 5, r - 2), ... , C(r + 2, r - 2), C(r + 1, r - 2), C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(n - 2, r - 1) - [from z = 1 to z = n - r] ψ {C(n - z - 2, r - 2)}


      Fractals of C(n - 3, r - 1) - {C(n - 4, r - 2), C(n - 5, r - 2), C(n - 6, r - 2), ... , C(r + 2, r - 2), C(r + 1, r - 2), C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(n - 3, r - 1) - [from z = 1 to z = n - r -1] ψ {C(n - z - 3, r - 2)}
      .
          .
          .

          Fractals of C(r + 1, r - 1) - {C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(r + 1, r - 1) - [from z = 1 to z = 3] ψ {C(r - z + 1, r - 2)}


      Fractals of C(r, r - 1) - {C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(r, r - 1) - [from z = 1 to z = 2] ψ {C(r - z, r - 2)}


      Fractals of C(r - 1, r - 1) - {C(r - 2, r - 2)}
      Fractals of C(r - 1, r - 1) - [from z = 1 to z = 1] ψ {C(r - z - 1, r - 2)}

Within every combinatorial set there are subsets of combinations that are similar in characteristics to the whole combination. Example, below is a sample combinatorial set of 8 numbers taken 6 at a time. Alongside the combinatorial set are a few subsets of combinations. Table 1 shows only one leg of the fractal path. The fractals are in red and transformed to the right. The symbol Ψ is the superset fractal and extends to infinity.

 

Table 1
Fractal Path is Ψ → ψ(8,6) → ψ(7,5) → ψ(6,4) → ψ(5,3) → ψ(4,2) → ψ(3,1)

1

2

3

4

5

6

→

1

2

3

4

5

→

1

2

3

4

→

1

2

3

→

1

2

→

1

1

2

3

4

5

7

→

1

2

3

4

6

→

1

2

3

5

→

1

2

4

→

1

3

→

2

1

2

3

4

5

8

→

1

2

3

4

7

→

1

2

3

6

→

1

2

5

→

1

4

→

3

1

2

3

4

6

7

→

1

2

3

5

6

→

1

2

4

5

→

1

3

4

→

2

3

 

 

1

2

3

4

6

8

→

1

2

3

5

7

→

1

2

4

6

→

1

3

5

→

2

4

 

 

1

2

3

4

7

8

→

1

2

3

6

7

→

1

2

5

6

→

1

4

5

→

3

4

 

 

1

2

3

5

6

7

→

1

2

4

5

6

→

1

3

4

5

→

2

3

4

 

 

 

 

 

1

2

3

5

6

8

→

1

2

4

5

7

→

1

3

4

6

→

2

3

5

 

 

 

 

 

1

2

3

5

7

8

→

1

2

4

6

7

→

1

3

5

6

→

2

4

5

 

 

 

 

 

1

2

3

6

7

8

→

1

2

5

6

7

→

1

4

5

6

→

3

4

5

 

 

 

 

 

1

2

4

5

6

7

→

1

3

4

5

6

→

2

3

4

5

 

 

 

 

 

 

 

 

 

1

2

4

5

6

8

→

1

3

4

5

7

→

2

3

4

6

 

 

 

 

 

 

 

 

 

1

2

4

5

7

8

→

1

3

4

6

7

→

2

3

5

6

 

 

 

 

 

 

 

 

 

1

2

4

6

7

8

→

1

3

5

6

7

→

2

4

5

6

 

 

 

 

 

 

 

 

 

1

2

5

6

7

8

→

1

4

5

6

7

→

3

4

5

6

 

 

 

 

 

 

 

 

 

1

3

4

5

6

7

→

2

3

4

5

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

5

6

8

→

2

3

4

5

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

5

7

8

→

2

3

4

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

6

7

8

→

2

3

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

5

6

7

8

→

2

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

4

5

6

7

8

→

3

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

6

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

4

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

4

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here's another fractal path shown in Table 2.

 

Table 2
Fractal Path is Ψ → ψ(8,6) → ψ(7,5) → ψ(5,4) → ψ(3,3) → ψ(2,2) → ψ(1,1)

1

2

3

4

5

6

→

1

2

3

4

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

4

5

7

→

1

2

3

4

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

4

5

8

→

1

2

3

4

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

4

6

7

→

1

2

3

5

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

4

6

8

→

1

2

3

5

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

4

7

8

→

1

2

3

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

5

6

7

→

1

2

4

5

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

5

6

8

→

1

2

4

5

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

5

7

8

→

1

2

4

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

3

6

7

8

→

1

2

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

4

5

6

7

→

1

3

4

5

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

4

5

6

8

→

1

3

4

5

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

4

5

7

8

→

1

3

4

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

4

6

7

8

→

1

3

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

5

6

7

8

→

1

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

5

6

7

→

2

3

4

5

6

→

1

2

3

4

 

 

 

 

 

 

 

 

 

1

3

4

5

6

8

→

2

3

4

5

7

→

1

2

3

5

 

 

 

 

 

 

 

 

 

1

3

4

5

7

8

→

2

3

4

6

7

→

1

2

4

5

 

 

 

 

 

 

 

 

 

1

3

4

6

7

8

→

2

3

5

6

7

→

1

3

4

5

 

 

 

 

 

 

 

 

 

1

3

5

6

7

8

→

2

4

5

6

7

→

2

3

4

5

→

1

2

3

→

1

2

→

1

1

4

5

6

7

8

→

3

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

6

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

4

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

4

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
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