Combinatorial Fractalization
C(n, r) = [from z = 1 to z = n - r + 1] å 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] y {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] y {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] y {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] y {C(n - z - 3, r - 2)}
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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] y {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] y {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] y {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. Along side the combinatorial set are a few subsets of combinations. Table 1 shows only one leg of the fractal path. The fractals are in blue and transformed to the right. The symbol Y is the superset fractal and extends to infinity.
Table 1
Fractal Path is Y ® y(8,6) ® y(7,5) ® y(6,4) ® y(5,3) ® y(4,2) ® y(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 | | | | | | | | | | | | | | | | | | | | |