I. Introduction To The Mathematical Sets Of Numbers: Exponents (Saliusian Sets), Permutations, Arrangements, Combinations
II. Exponents (Exponential Sets Or Saliusian Sets; Saliu Sets)
III. Permutations (Factorial Sets; Exponents Without Duplicates)
IV. Arrangements Sets (n Permutations m)
V. Combinations (Boxed Arrangements)
VI. Sets Of Letters, Words, Names
VII. Generating Words From Letters: Lexicographically And Randomly
1. Introductory Notes To The Mathematical Sets Of Numbers: Exponents (Saliusian Sets), Permutations, Arrangements, Combinations
"Let no one enter here who is ignorant of mathematics"
(The frontispiece of Plato's Academy)
• PermuteCombine.exe ~ version 6.0, 2007 ~ Freeware (for registered members).
PermuteCombine.exe is the summit of all mathematical generation. The program calculates and generates exponents, permutations, arrangements, and combinations for any numbers and words. The sets can be sequential (lexicographic order) or random.
Any finite number of elements can be put together in groups based on certain rules. Such groups are known as sets. The sets can comprise from 0 elements to infinity. There are four types of sets, from the most inclusive to the least: exponents, permutations, arrangements, and combinations. The number sets are the most important mathematically. We can substitute the numbers by alphanumerical elements, such as words, names, any strings of characters. In the case of the alphanumerical sets, mathematics works with the indices, indexes of the respective elements.
An example of exponents (N=3, M=3): 111,112,113,121,122,123,131,132, etc. (a total of 27 sets).
An example of permutations (for N = 3): 1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2, 3 2 1 (6 elements: 1* 2 * 3)).
An example of arrangements (for N = 3, M = 2): 1 2, 1 3, 2 1, 2 3, 3 1, 3 2. (6 elements in set: 3 * 2).
An example of combinations (for N = 3, M = 2): 1 2, 1 3, 2 3 (3 elements: 3 * 2 / 1 * 2).
The exponential sets had been neglected by mathematics, although they are the most important! All other sets are derived from exponents. Some number-lovers (and teasers of mine!) call the exponents the 'Ion Saliu sets'. Thanks, but I am a modest man (at least, financially). I use the in-house term of 'Saliusian sets'. The exponents are very important. They are capable of solving a wide range of probability problems. You can find revealing details in the probability articles I have written. Please follow the links at the end of this material.
2. Exponents (Exponential Sets Or Saliusian Sets; 'Ion Saliu Sets')
The pick-3 or pick-4 lottery games are the most commonly known examples of exponents. Each digit of the pick games takes values between 0 and 9 (10 values). The pick-3 game has a total of 10 to the power of 3 (10 ^ 3) = 1000 combinations.
The soccer pools, such as totocalcio, have 3 outcomes for 13 games; 3 to the power of 13 (3 ^ 13) = 1,594,323 possibilities.
The parameters for soccer pools (totocalcio, pronosport) are: 13 (items per set), 0 (lower bound), 2 (upper bound). 1 represents home victory, 2 is for win for the visitor, and 0 stands for a tie. The pick-3, pick-4 games have a lower bound of 0, and an upper bound of 9. Items per set: 3 (in pick-3) and 4 (in pick-4).
The formula of exponents is:
Exponent (N, M) = NM.
Exponents grow much faster than anything else, including permutations!
3. Permutations (Factorial Sets; Exponents Without Duplicates)
The permutations are also known as factorial, as far as calculation is concerned. Factorial of N or N! = 1 x 2 x 3 x … x N. The factorials grow extremely rapidly.
The formula of permutations is:
Permutation (N) = N! = 1 x 2 x 3 x … x N .
4. Arrangements Sets (n Permutations m)
The arrangements of N elements taken M at a time represent, in fact, a partial permutation. Older books use the notation nPm (n Permutation m) for arrangements:
nPm = n! / (n-m)!
Since n! = 1x2x3x...x(n-m)x(n-m+1)x(n-m+2)x...xn
and
(n-m)! = 1x2x3x...x(n-m)
we can simplify the fraction n! / (n-m)! and obtain:
(n-m+1)x(n-m+2)x...xn
It is the factorial of a series of elements beginning at the term (n-m+1) and ending at the term n. That's the formula of the arrangements:
Arrangement (N, M) = N x (N – 1) x (N – 2) x (N – 3) x ... x (N – M + 1) .
The exactas (top two finishers), or trifectas (top three finishers), or superfectas (top four finishers) in horse racing are some of the most common representations of the arrangements. If the famous horse race known as The Kentucky Derby has 20 horses, the total possible number of straight trifectas is:
A(20,3) = 20x19x18 = 6840. The boxed trifectas represent, in fact, the combinations of (N, M). (20x19x18) / (1x2x3) = 1140 boxed trifectas. See next.
5. Combinations (Boxed Arrangements)
The combinations are the best-known element of the four mathematical sets. The lotto drawings are some of the most common representations of the combinations. The random number generation of the arrangements is actually closer to the lotto drawings. The lotto numbers are not drawn in sequential order, but in sequences like 33, 7, 18, 44, 29, 48. The random number generation of the combinations in PermuteCombine.exe is like the lotto draws after sorting the numbers in ascending order.
The combinations formula is:
Combination (N, M) = Arrangement (N, M) / Permutation (M) = {N x (N – 1) x (N – 2) x (N – 3) x … x (N – M + 1)} / {1 x 2 x 3 x … x M} .
The combinations are the equivalent of boxed arrangements. In boxed, the order of the elements does not count.
The exponents represent the most inclusive of the four types of sets. The exponent set includes permutations; the permutation set includes arrangements; the arrangement set is inclusive of combinations.
6. Sets Of Letters, Words, Names
Generating sets of words (names, teams) is a very interesting feature of the software. The program requires that the user create first a text file. The file lists the entries that act as a generating source. The entries (items) can be single words or multiple words, including sentences. The items must be typed one per line. Each line must be terminated by pressing [Enter]. The file must not have blank lines. The list files do not require any field delimiter. But the program adds | as the field delimiter in the output files. The program is accompanied by three sample list files: TEAMS.NFL, 1X2.TXT, WORDS.TXT. TEAMS.NFL is a sorted list of the 32 teams of the National Football League. 1X2.TXT can be used as an input file to convert 0, 1, 2 to 1, X, 2 in soccer or real football pools. Here is the content of WORDS.TXT (6 items, 6 lines):
Permutations
permutation, sequential
random
lexicographic
Lexicographical, Order,
combination
Here are fragments of the output files for various word-generating options of PermuteCombine.exe:
- permutations, lexicographic order (N = 6):
Permutations | permutation, sequential | random | lexicographic | Lexicographical, Order, | combination |
Permutations | permutation, sequential | random | lexicographic | combination | Lexicographical, Order, |
...
combination | Lexicographical, Order, | lexicographic | random | Permutations | permutation, sequential |
combination | Lexicographical, Order, | lexicographic | random | permutation, sequential | Permutations |
...
- combinations, lexicographic order (N = 6, M = 5)
Permutations | permutation, sequential | random | lexicographic | Lexicographical, Order, |
Permutations | permutation, sequential | random | lexicographic | combination |
Permutations | permutation, sequential | random | Lexicographical, Order, | combination |
Permutations | permutation, sequential | lexicographic | Lexicographical, Order, | combination |
Permutations | random | lexicographic | Lexicographical, Order, | combination |
permutation, sequential | random | lexicographic | Lexicographical, Order, | combination |
- random combinations (N = 6, M = 5)
Permutations | permutation, sequential | random | lexicographic | combination |
permutation, sequential | random | lexicographic | Lexicographical, Order, | combination |
Permutations | permutation, sequential | random | lexicographic | combination |
permutation, sequential | random | lexicographic | Lexicographical, Order, | combination |
Permutations | random | lexicographic | Lexicographical, Order, | combination |
...
The word generation can be used at horse races to generate sets of horse names, instead of post numbers. This feature is useful in sports betting, too. A user can write, for example, all the NFL teams in a list file. Then generate parlays every week, using the feature: “Arrangements: Words, random”. Select, for example, 5 words (teams) per set. Check the output and disregard the sets with teams from the same game for that week.
The permutation, exponential, and combination word features can work as well. No word generation for Powerball—not of much use, in my mind.
7. Generating Words From Letters: Lexicographically And Randomly
I wrote the five letters of my name in a five-line text file, one letter per line. I got the 120 permutations of my name, from SALIU to UILAS. Obviously, I edited the output file with MDIEditor and Lotto; I replaced the | and blank spaces between the letters with nothing. I especially liked two of the permutations: Laius Usail. Looks like I was destined to become a USA guy!
Suppose we write the 26 letters of the English alphabet in a 26-line text file, one letter per line. We can generate an astronomical number of permutations (words). But the permutations have a fixed length (number of letters), whereas the natural languages have words of a variety of lengths. The exponents serve the purpose better. The words vary from one letter to some teens. The first letter of a word can be from A to Z; the second letter of a word can be from A to Z; the third letter of a word can be from A to Z; etc. The words can have repeat letters. For example, this Indonesian sacred chant of initiation consists of repeated vowels only:
Aiai, ouou,
Aia oua, aia oua, aia oua, ai.
I wrote another computer program that generates random words in a manner close to the natural languages. The words can have repeat letters; the words are of variable length; a word must contain at least one vowel. The free software is named WRITER.EXE.
Please be advised that the permutations can grow astronomically! A low number such as 10 has a factorial of 3,628,800 (total permutations)! Factorial of 100 (100!): Don't even try! It's mind blowing!
Please be advised that the arrangements, too, can grow astronomically!
Try first random numbers to see how many sets there are. The default number of sets to generate randomly is 100. The user, of course, can choose different amounts to generate.
Said Tabaki Paravicius, netizen extraordinaire:
You ain't seen nothing quite like it, Kulai! PermuteCombine.exe is the mother of all generations.
Resources in Theory of Probability, Mathematics, Combinatorics, Permutations, Software.
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