1 - Combinatorial Distribution

    Factorial - n! = n * (n -1) * (n - 2) * ... * 3 * 2 * 1 , and 0! = 1

    Permutation - P(n,r) = n! / (n - r)!

    Combination - C(n,r) = P(n,r) / r!

    Combinatorial Distribution - D(n,r,c,z) = C(z - 1, c - 1) * C(n - z, r - c)

      n - total number of items
      r - number of items in a combinatorial or permutational set
      c - column number of the distribution
      z - item number of the distribution

2 - Average Rate of Reoccurrence Distribution

    Reference - Combinatorial Distribution

    Average Rate of Reoccurrence Distribution -   m = Ar(n,r,c,z) = C(n,r) / D(n,r,c,z)

3 - Generalized Average Rate of Reoccurrence

    Reference - Combinatorial Distribution, Average Rate of Reoccurrence Distribution

    Generalized Average Rate of Reoccurrence -   m = Ag(n,r) = n / r

      n - number of items
      r - number of items in a combinatorial set

    when c = 1 and z = 1, also a! / (a - 1)! = a

    Ag = Ar(n,r,1,1)

    Ag = C(n,r) / D(n,r,1,1)

    Ag = C(n,r) / (C(1 - 1,1 - 1) * C(n - 1, r - 1))

    Ag = C(n,r) / ( 1 * C(n - 1, r - 1))

    Ag = C(n,r) / C(n - 1, r - 1)

    Ag = (n! / (r! * (n - r)!)) / ((n - 1)! / ((r - 1)! * ((n - 1) - (r - 1))!))

    Ag = (n! / (r! * (n - r)!)) / ((n - 1)! / ((r - 1)! * (n - 1 - r + 1)!))

    Ag = (n! / (r! * (n - r)!)) / ((n - 1)! / ((r - 1)! * (n - r)!))

    Ag = (n! / (r! * (n - r)!)) * ((r - 1)! * (n - r)! / (n - 1)!)

    Ag = (n! / (n - 1)!) * ((r - 1)! / r!) * ((n - r)! / (n - r)!)

    Ag = (n! / (n - 1)!) * ((r - 1)! / r!) * 1

    Ag = (n! / (n - 1)!) * (1 / (r! / (r - 1)!)

    Ag = n * (1 / r)

    Ag = n / r

4 - Relative Combinatorial Distribution

    Reference - Combinatorial Distribution, Average Rate of Reoccurrence Distribution

    Relative Combinatorial Distribution - Dr(n,r,c,z,d) = d * (D(n,r,c,z) / C(n,r)) = d / Ar(n,r,c,z)

      n - number of items (number of balls)
      r - number of items in a combinatorial set (number of picks)
      c - column number of the item (position in the pick set; when items in the set are in ascending order)
      z - item number (ball number)
      d - number of random combinatorial set selections (number of draws)

5 - Discharging Reoccurrence Distribution

    Reference - Average Rate of Reoccurrence Distribution, Generalized Average Rate of Reoccurrence

    Discharge Reoccurrence - y = (d / m² ) e ^{-(x / m )}

      d - total number of draws
      m - average rate of reoccurrence
      x - draw difference or Dd between two draw occurrences of the same number
      y - the approximate frequency of reoccurrence for a given draw difference or Dd for a given draw count d

6 - Potential Reoccurrence Probability

    Reference - Discharging Reoccurrence Distribution, Average Rate of Reoccurrence Distribution, Generalized Average Rate of Reoccurrence

    Potential Reoccurrence Probability -  y = e ^{-(x / m)}

      x - draw difference or Dd between previous occurrence and next draw number
      m - average rate of reoccurrence

      y - probability of reoccurrence relative to last occurrence

7 - Potential Occurrence Probability

    Reference - Discharging Reoccurrence Distribution, Average Rate of Reoccurrence Distribution, Generalized Average Rate of Reoccurrence, Potential Reoccurrence Probability

    Potential Occurrence Probability -  y = 1 - e ^{-(x / m)}

      x - draw difference or Dd between previous occurrence and next draw number
      m - average rate of reoccurrence

      y - probability of occurrence relative to last occurrence

8 - Half-life of Reoccurrence

    Reference - Potential Reoccurrence Probability

    Half-life of Reoccurrence -  x_l = -m ln(1 / 2)

      m - average rate of reoccurrence

      x_l - half-life of reoccurrence

9 - First and Second Order Work Equations

    Reference - Potential Reoccurrence Probability, Potential Occurrence Probability, Half-life of Reoccurrence 

    First Order Work Equation -  y = 4 (e ^{-(x / m)} - e ^{-(2 x / m)} )

    Second Order Work Equation -  y = 16 (e ^{-(x / m)} - 5 e ^{-(2x / m)} + 8 e ^{-(3x / m)} - 4 e ^{-(4x / m)} )

      x - draw difference or Dd between last occurrence and some future draw number
      m - average rate of reoccurrence

      second order region 1 -    0 £ x £ -m ln(1 / 2)
      second order region 2 -    -m ln(1 / 2) £ x £ ¥

10 - Solution for finding x in the First Order Work Equation

    low value for x -  x = -m ln((1 + Ö(1 - y)) / 2)

    high value for x -    x = -m ln((1 - Ö(1 - y)) / 2)

11 - Random Number Transforms - Normal Distribution

    Analog -      x = ±Ö-2s² ln(1 - y)

    Decimal -    x = ±(Ö-2s² ln(1 - y) - (s / 2))

    Digital -      x = ±(Ö-n ln(1 - y) - ((1 / 2) * Ön / 2))

      y - random number, 0 £ y < 1
      s - standard deviation
      n - digital deviation, | x | £ n
      x - transformed random number
       ± - values of x randomly alternate

    Analog values are not rounded to any fixed decimal or integer.

    Decimal values of x are rounded to nearest fixed decimal place or integer.
      Example: 2.591230102345664 is rounded to 2.59.

    Digital values of x are rounded to the nearest integer.
      Example: 2.591230102345664 is rounded to 3.
    Digital limits are valid y samples when | x | £ n.
    If | x | > n, then resample y.

    Relationship of n and s  -      n = 2 s²  and    s = Ön / 2

12 - Random Number Transforms - Reoccurrence Distribution

    Reference - Discharging Reoccurrence Distribution

    Random Reoccurrence Distribution -    x = -m ln(1 - y)

    m - average rate of reoccurrence
    y - random number, 0 £ y < 1

    x as analog - values of x are not rounded to any decimal place value.
    x as digital - values of x are converted to the integer part of x.
                Ex.1 -    4.129840012394543 is converted to 4
                Ex.2 -    2.781992014293572 is converted to 2
                valid digital values of x are when x ³ 1

13 - Combinatorial Symmetry - Number Symmetry

    Symmetric Number - S_n(n,z) = n - z +1

      n - total number of items
      z - item number in the set

14 - Combinatorial Symmetry - Column Symmetry

    Symmetric Column Number -    S_c(r,c) = r - c + 1

      r - items per combinatorial set (pick value)
      c - column place of the number n referred to in Combinatorial Symmetry - Number Symmetry

15 - 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)}

16 - 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)}
      .
      .
      .

      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)}