Lexicographic lexicographical order, index permutations sets

Lexicographic, lexicographical order, index, ranking, sets: permutations, exponential sets, combinations.

Lexicographic, Lexicographical Order, Index, Rank of Permutations, Combinations, Sets

By Ion Saliu, Rank-And-File At-Large

I. The Best Software To Find The Lexicographic (Or Lexicographical) Index
II. Types Of Sets And Their Lexicographic (Or Lexicographical) Indexing, Order
III. Analysis Of Lexicographic (Or Lexicographical) Order And Indexing
IV. Essential Resources In Lexicographic (Or Lexicographical) Order And Indexing
Written by Ion Saliu on August 23, 2003 (3 WE).
• LexicographicSets.EXE version 3.0, Augustus 24, 2003 (3 WE) - Freeware.
1. The Best Software To Find The Lexicographic (Or Lexicographical) Index — and vice versa
LexicographicSets.EXE is the summit of all lexicographic . . . make it lexicographical ordering. The program finds (calculates) the index, or order, or rank, or numeral of all types of sets: exponents, permutations, arrangements, and combinations, including Powerball. Conversely, the program finds (constructs) the set for a given index, or order, or rank, or numeral. The sets are considered in sequential, or lexicographic (lexicographical), or dictionary order.
An example of exponents (N=3, M=3): 111,112,113,121,122,123,131,132, etc. (a total of 27 sets).
The index of set (1,2,2) = 5.
The set for index = 4 is: (1,2,1).
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.
The index of set (3,1,2) = 5.
The set for index = 4 is: (2,3,1).
An example of arrangements (for N = 3, M = 2): 1 2, 1 3, 2 1, 2 3, 3 1, 3 2.
The index of set (2,3) = 4.
The set for index = 5 is: (3,1).
An example of combinations (for N = 3, M = 2): 1 2, 1 3, 2 3. The index of set (2,3) = 3.
The set for index = 2 is: (1,3).
2. Types Of Sets And Their Lexicographic (Or Lexicographical) Indexing
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) 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).
Exponents grow much faster than anything else, including permutations!
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 arrangements of N elements taken M at a time are calculated as N x (N-1) x (N-2) 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.
The combination formula is: Combinations (N, M) = Arrangements (N, M) / Permutations (M).
The combinations are the best-known element of the three mathematical entities. The lotto draws are some of the most common representations of the combinations. The software also covers Powerball-type of combinations; i.e. the last number (the 'power ball') can be equal to any of the previous numbers in the combination.

3. Analysis Of Lexicographic (Or Lexicographical) Order And Indexing
I wrote previously about the lexicographical order. I wrote also software to calculate (find) indexes (ranks) and corresponding combinations. This page shows at the end related pages at this web site and previous software I wrote (freeware). I realize now how much interest there is out there in lexicographic(al) order. I thought I had received way too many emails on the subject. But that amounts to almost nothing compared to what's on the Internet! I noticed that the keyword 'lexicographical' directs quite a few visitors to my site. Checking at Google, Yahoo, and MSN, the keyword ranks my site pretty high. Yet, the very same keyword (from a human perspective), spelled 'lexicographic' does not rank my web site in the first, who knows how many hundred pages!
Searching on lexicographical, lexicographic, sets, permutations, combinations, etc. etc. yields probably over 100,000 unique search hits! It's a huge mess! Tons of algorithms and source code in every imaginable programming language! I couldn't go too far. I saw newsgroups postings from the early 1980's. It didn't look to me that the issue was ever solved. I mean, I couldn't find the 'mother of all sets generating'; or the 'mother of all lexicographical indexes'! That finding corroborates with the requests I received to write specific lexicographical indexing and generating software. I was unable to find comprehensive software to tackle sets generating and lexicographic indexing. I know, the Internet is of gigantic proportions. I assume, however that the two programs I wrote represent the most comprehensive answer to sets generating and lexicographic indexing. The accuracy is also a very important issue. From what I was able to read, there has been a lot of bickering in the newsgroups. It appears that many attempts to tackle the job resulted in faulty algorithms and/or software. I can guarantee that my (comprehensive) software is fault-free to a very high degree. I haven't been able to find a wrong result in quite a few tries.
Speed of execution is also a very important issue. The combinations functions are as fast as it gets. The exponential functions are also close to 'as fast as it gets'. The arrangements functions are slower. The permutations functions are the slowest. I have multiple methods at my disposition. I chose the most accurate ones. Given time, I may come back and insert faster methods. For now, accuracy comes first and second. Speed comes in the third place.
The saying goes that the universities make public the algorithms and source code. Therefore I should make my stuff public, too. Well, the universities are funded. I am not a university and am not funded. Nevertheless, I offer a lot of free software of my own, probably more freeware than most universities.
Said Caraola Breechew, the front lady of netizenship:
You ain't seen nothing quite like it, Tabaki! LexicographicSets.EXE is the dictionary of all the words in your mother tongue.
4. Essential Resources In Lexicographic (Or Lexicographical) Order And Indexing
Download the freeware from the FTP download site:

• PermuteCombine.exe, the universal permutations, arrangements and combinations generator for any numbers and words;
• LexicographicSets.EXE, the universal permutations, arrangements and combinations lexicographic indexing (ranking);
• Combinations.exe, the universal combinations generator for any lotto, Keno, Powerball game: N numbers taken M at a time, in K steps;
• WRITER.EXE, random generator of letters to words, passwords, sentences;
• NthIndex.EXE ~ lexicographic indexing superseded by LexicographicSets.EXE;
• SEQUENCE.EXE ~ lexicographic indexing superseded by LexicographicSets.EXE;
• COLORDER.EXE - lexicographical indexing for lotto and Powerball only;
• DrawIndex.EXE - highly automated lexicographical indexing for lotto draw files.

See also: Online Odds, Probability Calculator & Random Numbers Combinations Generator: Lotto, Powerball, Lottery, Horse Races, Roulette, Sports Betting, Soccer, Pools, 1X2; The best that ever was!

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: : : Lexicographic, lexicographical order, index, ranking, sets. LexicographicSets.EXE is the summit of all lexicographic, er, make it lexicographical ordering. The program finds (calculates) the index, or order, or rank, or numeral of all types of sets: exponents, permutations, arrangements, and combinations, including Powerball/Mega millions. Conversely, the program finds (constructs) the set for a given index, or order, or rank, or numeral. The sets are considered in sequential, or lexicographic (lexicographical), or dictionary order. 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. An example of combinations (for N = 3, M = 2): 1 2, 1 3, 2 3. The index of set (2,3) = 3. The set for index = 2 is: (1,3). 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 exponents (Ion Saliu sets or Saliusian sets) grow much faster than anything else, including permutations! 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 arrangements of N elements taken M at a time are calculated as N x (N-1) x (N-2) 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. The formula of permutations is: Permutation (N) = N! = 1 x 2 x 3 x … x N. The formula of arrangements is: Arrangement (N, M) = {(N – M + 1)!} = N x (N – 1) x (N – 2) x (N – 3) x … x (N – M + 1). 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 best-known element of the three mathematical entities. The lotto draws are some of the most common representations of the combinations. The software also covers Powerball-type (Mega Millions) of combinations; i.e. the last number (the 'power ball') can be equal to any of the previous numbers in the combination. I wrote previously about the lexicographical order. I wrote also software to calculate alphanumeric indexes and corresponding combinations. This page shows at the end related pages at this web site and previous software I wrote. I realize now how much interest there is out there in lexicographic(al) order. I noticed that the keyword 'lexicographical' alphanumerical index directs quite a few visitors to my site. Spelled 'lexicographic' does not rank my web site in the first, who knows, thousand pages! Searching on lexicographical, lexicographic, sets, permutations, combinations yields probably over 100,000 unique web pages! Study "Lexicographic, lexicographical order, index, ranking, sets: permutations, exponential sets, combinations". Tons of algorithms and source code in every imaginable programming language! I couldn't go too far. I couldn't find a 'mother of all sets generation'; or 'mother of all lexicographical indexes'! That corroborates with the requests I received to write specific software to find the rank index of combinations or sets of numbers. I was unable to find comprehensive software to tackle sets generating and lexicographic indexing. I assume that the two programs I wrote represent the most comprehensive answer to sets generating and lexicographic indexing, rank, ranking. The saying goes that the universities make public the algorithms and source code. Studies on Lexicographic, lexicographical order, index, ranking, sets: permutations, exponential sets, combinations www.saliu.com. PermuteCombine.exe, the universal permutations, arrangements and combinations generator for any numbers and words; LexicographicSets.EXE , the universal permutations, arrangements and combinations lexicographic indexing (ranking). Combinations.exe, the universal combinations generator for any lotto, Keno, Powerball game: N numbers taken M at a time, in K steps; NthIndex.EXE ~ lexicographic indexing superseded by LexicographicSets.EXE. SEQUENCE.EXE ~ lexicographic indexing superseded by LexicographicSets.EXE; COLORDER.EXE - lexicographical indexing, rank for lotto and Powerball only; DrawIndex.EXE - highly automated lexicographical indexing for lotto draw files. Calculate, generate exponents, permutations (factorial), sets, arrangements, combinations (including Powerball) for any numbers and word. Lexicographic, lexicographical order, index, ranking, sets: permutations, exponential sets, combination. Calculate the combination lexicographic order of lotto data files. Combination Lexicographic Order, Rank, Index: THE Comprehensive and Fast Algorithm. Index rank lexicographic order lotto combination. Software combination algorithm formula index ordering. Combination sequence number or Lexicographic Order Debates. Combination sequence number or Lexicographic Order Fast Algorithm. Combination Lexicographica Order, Rank, Index: THE Comprehensive and Fast Algorithm. Index rank lexicographical order lotto combination. Software combination algorithm formula index ordering. Combination sequence number or Lexicographical Order Debates. Combination sequence number or Lexicographical Order Fast Algorithm. Title: Lexicographic, lexicographical order, index, rank, indexing, ranking. Offering software to find the index, order, rank, numeral of sets: exponents, permutations, arrangements, combinations; find set from index (rank). On mathematics, math of lexicographic, lexicographical, order, index, rank, ranking, numeral, dictionary. Software is applied to permutations, permutation, factorial, sets, exponents (Saliusian sets, Ion Saliu sets), arrangements. Get free combinations, combination, number, numbers, mathematical, mathematics, math, maths. On lexicographic, lexicographical, order, index, rank, ranking, numeral, dictionary, permutations, permutation. Factorial, set, sets, sub-sets, exponents, exponential, arrangements, combinations, combination, number, numbers, mathematical, mathematics, math, maths.

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