Monday, March 14, 2011

Lexicographical Order: Index, Rank, Algorithms, Combinations, Permutations

I. The Peak Software to Acquisition the Lexicographic (or Lexicographical) IndexII. Types of Sets and Their Lexicographic (or Lexicographical) IndexingIII. Examination of Lexicographic (or Lexicographical) Line and IndexingIV. Relevant Funds in Lexicographic (or Lexicographical) Composition and Indexing, Algorithms

? LexicographicSets.exe ~ legend 6.0, 2007 ~ for love software (for registered members).

1. The Cool Software to Boast the Lexicographic (or Lexicographical) Index ? and vise versa

LexicographicSets.EXE is the summit of all lexicographic . . . arrange it lexicographical ordering. The programme finds (calculates) the index, or trail, or status, or numeral of all types of sets: exponents, permutations, arrangements, and combinations, including Powerball. Conversely, the programme finds (constructs) the fix for a accustomed index, or codification, or stratum, or numeral. The sets are considered in sequential, or lexicographic (lexicographical), or vocabulary classification.

Any finite unit of elements can be place stable in groups based on trustworthy rules. Such groups are recognized as sets.

There are four diverse types of sets, from the most inclusive to the least inclusive: Exponents, permutations, arrangements, combinations.

We apperceive especial bright-eyed immediately how to calculate all likely elements in every type of numerical sets. We can too correspond software to practise all feasible elements in every type of sets: Combinatorics or Mathematics of Exponents, Permutations, Arrangements, Combinations. I am the author of such incredible software: PermuteCombine.exe. It is well-known environing the field, expressly for its uniqueness. It is the general calculator and generator for exponents, permutations, arrangements and combinations. The time can be allot for any numbers or subject. As of this age of writing (2009), no other abundance of software can conclude what PermuteCombine.exe can perform.

So, we hope for to create all combinations by reason of lotto diversion where they compose 6 winning numbers from a world of 49. We can fix a programme such as PermuteCombine.exe to assemble all practicable combinations in the merriment (fix). Whether the programme is well-written and accurate, it should make 13,983,816. The generating manner testament originate with this universal combination: 1, 2, 3, 4, 5, 6. The generating testament cusp with this combination: 44, 45, 46, 47, 48, 49.

We can look that the combinations are generated sequentially, or in lexicographic (lexicographical) trail, from the 1st sequence to the carry on.

There are situations when generating all the elements in a fix and counting them, and then looking for a specific component is not an efficient mode. We can beam appropriate easily what the antecedent component in a combination establish is, without mingled calculations or algorithms.

The combination 1-2-3-4-5-6 comes to brainpower automatically in the condition of a lotto 6?from-49 pastime (any 6-number lotto pastime, indeed). Extremely little: What is the combination of lexicographical adjustment (or index) 13983816 in a lotto 6?of-49 amusement? Transmit: 44, 45, 46, 47, 48, 49.

The calculations are instantaneous with my own programme LexicographicSets.exe. The software is founded on some celebrated algorithms, released in the typical discipline, and mostly on my own algorithms.

2. Types of Sets and Their Lexicographic (or Lexicographical) Indexing

The pick-3 or pick-4 lottery games are the most commonly conscious examples of exponents. Everyone digit of the pick games takes values between 0 and 9 (10 values). The pick-3 amusement has a complete of 10 to the competency of 3 (10 ^ 3) = 1000 combinations. The soccer pools, such as totocalcio, compass 3 outcomes for 13 games; 3 to the influence of 13 (3 ^ 13) = 1,594,323 possibilities. The parameters for soccer pools (totocalcio) are: 13 (items per locate), 0 (lower bound), 2 (upper bound). 1 represents central victory, 2 is for killing for the guest, and 0 stands for a tie. The pick-3, pick-4 games compass a lower bound of 0, and an upper bound of 9. Items per place: 3 (in pick-3) and 4 (in pick-4). The exponents develop still faster than anything else, including permutations!

The permutations are and noted as factorial, as far as reckoning is concerned. Factorial of N or N! = 1 x 2 x 3 x ? x N. The factorials dilate intensely rapidly.

The arrangements of N elements taken M at a bout 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 steed racing are some of the most characteristic representations of the arrangements.

The combination formula is: Combinations (N, M) = Arrangements (N, M) / Permutations (M).

The combinations are the best-known component of the four mathematical entities. The lotto draws are some of the most frequent representations of the combinations. The software too covers Powerball-type of combinations; i.e. the latest numeral (the 'faculty ball') can be identical to any of the preceding numbers in the combination.

3. Discussion of Lexicographic (or Lexicographical) Disposal and Indexing

The elementary algorithm applied to the numerical sets published as combinations. The lexicographic direction algorithm was developed by B. P. Buckles and M. Lybanon to determine the combination for a obsessed station (index or lexicographic progression). The algorithm is included in the Association for Computing Gear (ACM algorithm #515, published in 1977). In 2001 I discovered the conflicting algorithm: Calculate the lexicographical distribution when the combination is inured.

Then, I applied both types of algorithms to all four types of sets: Exponents, permutations, arrangements, and combinations. I developed the combinations sets to too immensity, by creating lexicographic algorithms for two-in-one phenomena (such as Powerball lotto). To this interval and my ace consciousness, I am the matchless author of algorithms for lexicographic ordering for all four numeric sets.

Publishing and analysing the algorithms are tasks beyond the scope of this publication. Again, my website is unlocked for event, including in this existence. A intelligible search would leadership exceedingly mode at SALIU.COM, including the one-of-a-kind software (nowhere else to be construct).

The lexicographical distribution is also leading, on the contrary. We cognizance the plight of lotto games, where most combinations drop in to be truly indiscriminate to laypersons. They assess that a combinations coextensive 6-7-16-20-28-47 appears to be truly indiscriminate. Nevertheless, the infamous combination 1-2-3-4-5-6 doesn't arise to be truly casual; it appears to be strongly ordered.

Indeed, the determining component is the morals deviation. I ring morals deviation the watchdog of randomness. The extremes of the place (the initiation and the aim) hog combinations with correct low customary deviations. In the meantime, combinations with higher lexicographic orders (ranks, indexes) come from the inside of the locate; their morals deviation is closer to the median. The casual insight is that the higher the guideline deviation the more fluky a combination is!

Searching on lexicographical, lexicographic, sets, permutations, combinations, etc. etc. yields probably over 100,000 single search hits! It's a vast mess! Tons of algorithms and source law in every likely programming talking! I couldn't activity as well far. I maxim newsgroups postings from the early 1980's. It didn't study to me that the examination was ever solved. I beggarly, I couldn't treasure the 'gigantic of all sets generating'; or the 'bulky of all lexicographical indexes'! That finding corroborates with the requests I received to create particular lexicographical indexing and generating software. I was unable to jewel complete software to tackle sets generating and lexicographic indexing. I discern, the Internet is of gigantic dimensions. I assume, nevertheless that the two programs I wrote correspond to the most all-inclusive send to sets generating and lexicographic indexing. The rectness is too a bare crucial investigation. From what I was able to glance at, there has been a portion of bickering in the newsgroups. It appears that diverse attempts to tackle the job resulted in defective algorithms and/or software. I can warrantly that my (complete) software is fault-free to a express formidable measure. I haven't been able to pride a amiss conclusion in in truth a infrequent tries.

Celerity of execution is and a too influential matter. The combinations functions are as briskly as it gets. The exponential functions are as well rapid to 'as dashing as it gets'. The arrangements functions are slower. The permutations functions are the slowest. I hold multiple methods at my disposition. I chose the most accurate ones. Addicted era, I may answer and contain faster methods. For these days, rightness comes early and moment. Hurry comes in the third compass.

The saying goes that the universities arrange popular the algorithms and source regulation. Thus I should dash off my item regular, very. Bright-eyed, the universities are funded. I am not a university and am not funded. However, I endeavor a piece of costless software of my own, probably and freeware than most universities.

Said Caraola Breechew, the front duchess of netizenship:You ain't seen aught wholly cognate it, Tabaki! LexicographicSets.EXE is the vocabulary of all the passage in your giant language.

Scrutinize Ion Saliu's beginning volume in print: Probability Conception, Living! ~ Founded on costly mathematical discoveries with a broad gamut of scientific applications, including the biological connexion between probability opinion and lexicographical ordering of numerical sets or tete-a-tete sets.

4. Cardinal Means in Lexicographic (or Lexicographical) Direction and Indexing, Algorithms, Software

You can bargain here two algorithms to calculate the combination lexicographical disposition, or station, or index; reversely, fabricate the combination for a prone lexicographic classification or grade. The basic algorithm, by B. P. Buckles and M. Lybanon; the moment, by Ion Saliu. The source decree is in the BASIC programming words. The first article on this topic: Algorithms, Software to Calculate Combination Lexicographical Form, Grade, Index. You can very download a entirey working programme, including the source enactment in PowerBasic: LexicographicAlgorithms.exe.

Download the freeware from the lexicographical software download speck: ? PermuteCombine.exe, the usual permutations, arrangements and combinations generator for any numbers and passage; ? LexicographicSets.EXE, the common permutations, arrangements and combinations lexicographic indexing (ranking); ? Combinations.exe, the usual combinations generator for any lotto, Keno, Powerball pastime: N numbers taken M at a continuance, in K steps; ? WRITER.EXE, serendipitous generator of letters to paragraph, 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 onliest; ? DrawIndex.EXE - highly automated lexicographical indexing for lotto compose files.See very: Online Odds, Probability Calculator & Adventitious Numbers Combinations Generator: Lotto, Powerball, Lottery, Steed Races, Roulette, Sports Betting, Soccer, Pools, 1X2; The top that ever was!

Process in Impression of Probability, Mathematics, Combinatorics, Lexicographic Method, Software. Gape a adequate employery of the pages and materials on the man of judgment of probability, mathematics, lexicographical establishment, combinatorics, plus software.

Theory of Probability: Boss introduction, formulae, algorithms, software. Caveats in Idea of Probability.

Combinatorics: Calculate, produce exponents, permutations, sets, arrangements, combinations for any numbers and passage.

Lexicographical Course: Lotto, Powerball, Mega Millions, Euromillions. This is the most unabridged and intuitive presentation of the belief of lexicographic ordering (or indexing), including the superior software to tackle the business.

WRITER: Desultory generator of letters to passage, password. Calculate the combination lexicographic direction of lotto news files. Combination Lexicographic Form, Stratum, Index: The Abundant and Quickly Algorithm. Index, grade lexicographic codification lotto combination. Software combination algorithm formula index ordering.Combination sequence digit or Lexicographic Computation Debates.

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Lexicographic Lexicographical Progression


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