Algorithm, software to calculate combination lexicographical order, rank, index

Algorithm, algorithms, formula, formulas, formulae, math, mathematics to calculate combination index, order, index, lexicographic, lexicographical.

Algorithm, software to calculate combination lexicographical order, rank, index

By Ion Saliu, Algorithmically At-Large

Two algorithms to calculate the combination lexicographical order, or rank, or index; generate the combination for a given lexicographic order or rank. Also known as the problem of the lotto combination sequence number. The first by B.P. Buckles and M. Lybanon; the second by Ion Saliu.

Written by Ion Saliu on July 23, 2001.
• When you gotta do it, you'll do it. For your mind won't find peace until a goal is reached.
I presented earlier this month of July 2001 an algorithm to calculate the index, or rank, or lexicographical order, or combination sequence number (CSN). The algorithm was 100% precise. It had one problem, however. The speed of execution left much to be desired for monster-odds games, such as Keno (80/20, 70/20, even 70/10). The speed was acceptable for many other lotto games. Even a big game such as the 49/5/42 PowerBall required less than half a minute to determine the CSN. Nevertheless, running the algorithm for Keno games took an eternity!
I came back to the original “B.P. Buckles and M. Lybanon” algorithm (algorithm “ACM #515”). I was able to add to it the reverse task: calculate the CSN for any lotto game, but the PowerBall. Later, the PowerBall will be added to the procedure.
I recompiled NTHINDEX.EXE to SEQUENCE.EXE, the program that uses the new algorithm. The program is freeware at the software download site.
Here is the main algorithm. Listed is also my function that calculates the number of combinations 'N taken M at a time': Combine## (N, M). The program SEQUENCE.EXE was compiled with PowerBasic for DOS.

' *** the new 'combination index (rank)' algorithm ' V = the biggest number in the lotto game (e.g. 49, or 80, etc.) ' K = numbers per combination (e.g. 6, 10, 20, etc.) REDIM C(K) REDIM Num(K) 'the numbers in the combination LI## = 0 CSN## = 0 P1 = K - 1 FOR I = 1 TO P1 C(I) = 0 If I <> 1 THEN C(I) = C(I - 1) 2001 C(I) = C(I) + 1 R## = Combine##(V - C(I), K - I) 'call the combination calculator LI## = LI## + R## IF C(I) < Num(I) THEN GOTO 2001 ' the key command LI## = LI## - R## NEXT I CSN## = LI## + Num(K) - Num(P1) 'combination sequence number FUNCTION Combine##(N, M) REM *** This function calculates the total number of combinations *** REM *** 'N taken M' at a time FA## = 1 FOR i = 1 TO M: FA## = FA## * i: NEXT i va## = 1 FOR j = (N - M + 1) TO N: va## = va## * j: NEXT j Combine## = va## / FA## END FUNCTION

I verified the algorithm with many cases. I believe the procedure is 100% accurate. But, if you think you discovered a miscalculation, please let me know. Document the error as detailed and as clearly as possible. I believe I can fix the errors – if any.
Ion Saliu

This is a comprehensive list of my writings on the topic of lexicographical order or indexing, including algorithms and software.

Lexicographic, lexicographical order, index, rank of permutations, exponential sets, combinations.
This is the most intuitive, if not the best, introduction to the apparently difficult concept of lexicographic ordering (or indexing).

Lexicographical Order: Lotto, Powerball, Mega Millions, Euromillions.
This is the most comprehensive and intuitive presentation of the concept of lexicographic ordering (or indexing), including the superior software to tackle the task.

Combination Lexicographic Order, Rank, Index: The Comprehensive and Fast Algorithm.
I expanded the lexicographical index concept to two-in-one games, particularly the U.S. lotto game of Powerball (applicable to Mega Millions as well).

Calculate the combination lexicographic order of lotto data files.
The introduction of DrawIndex.EXE, the software that calculates the lexicographic indexes for every combination (draw) in a lotto results file.
The following are older and somehow outdated materials I wrote on lexicographical order subjects. Most of the older software is outdated. Nothing (in the whole world) can beat LexicographicSets.EXE or DrawIndex.EXE. Sentimentalism is good only to a close extent.

Index, rank lexicographic order lotto combination.
Software combination algorithm formula index ordering.
Combination sequence number or Lexicographic Order Debates.
Of course, everybody loves to feast on great software, especially when it is free! My combinatorics software (and other categories) is absolutely free to run, for an unlimited period of time. However, only the registered members have a right to download the software. Membership requires a nominal fee — the most reasonable there is to connect to the greatest and most useful software ever created. No kidding! Read the conditions to becoming a registered member: Download Great Free Software: Paid Membership Required.

Download the free software from the download site:
LexicographicSets.EXE, the universal permutations, arrangements and combinations lexicographic indexing (ranking);
DrawIndex.EXE - highly automated lexicographical indexing for lotto draw files.
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;
PermuteCombine.exe, the universal permutations, arrangements and combinations generator for any numbers and words;
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.

Subject:

Comments:
: : Combination sequence number or lexicographical (lexicographic) order, index - the fast algorithm. The only problem I have with your algorithm is that it's kind of slow on my HP48 calculator. Just running a program which is a simple FOR loop to count from 1 to 14000000 takes a few hours. At least the other algorithms ran in a reasonable time. For going from set to CSN I wrote a program that does a binary search in the CSN (combination sequence number, index, rank, lexicograhic order) set until it hits the right one. It's not fast, about twenty seconds, but it does work. Why do you run it on a calculator? What are the PCs for? As of your binary algorithm to calculate the CSN (combination sequence number) and vice versa. This is the best analysis of "Algorithm, algorithms, formula, formulas, formulae, math, mathematics to calculate combination index, order, index, lexicographic, lexicographical. Such an algorithm is incorrect IF it doesn't take into account the largest number in the lotto game. Such an algorithm leads to correct results ONLY in the very beginning of lotto sequences. Study Algorithm, algorithms, formula, formulas, formulae, math, mathematics to calculate combination index, order, index, lexicographic, lexicographical. Let's think of a simple combination, such as 1-2-3-4-5-8. What is its CSN? An absurd result, for there is no C(0,1), C(1,2), C(2,3), C(3,4), C(4,5); there can be only C(7,6). Thus, in a lotto 49/6 game, the CSN for combination 6-7-16-20-28-47 is 6,991,908 (exactly the mid-point in the combination set). The same combination 6-7-16-20-28-47 has an index (CSN) of 45,148,858 in a lotto 69/6 game (PA lottery). Your formulae do not lead to these results, the only correct results. Although not a general-purpose formula, the algorithm I presented a few days ago represents the easiest way to calculate the indexes of any lotto combinations. The algorithm is for the BASIC language, but it can be converted to any programming language. I compiled NTHINDEX.EXE with PowerBasic, especially because of increased speed of execution. Read Algorithm, algorithms, formula, formulas, formulae, math, mathematics to calculate combination index, order, index, lexicographic, lexicographical - at www.saliu.com. The number of sets of 1 number when you have 0 is 0, the number of sets of 2 numbers when you have 1 is 0, the number of sets of 3 numbers when you only have 2 is again 0, the number of sets of 4 numbers with only 3 available is 0, the number of sets of five numbers with only 4 on the table is also zero. There are kingdoms in this world where the pupils are taught the combinatorial mathematics. The number of combinations of any N elements taken 0 at a time is 1. On the other hand, everybody has the right to set new knowledge, including new mathematics. You have a right to design a combinatorial algorithm with the following premise: The speed of execution left much to be desired for monster-odds games, such as Keno (80/20, 70/20, even 70/10). The speed was acceptable for many other lotto games. Even a big game such as the 49/5/42 PowerBall required less than half a minute to determine the CSN. Nevertheless, running the algorithm for Keno games took an eternity! I came back to the original "B.P. Buckles and M. Lybanon" algorithm (algorithm "ACM #515"). I was able to add to it the reverse task: calculate the CSN for any lotto game, but the PowerBall. Later, the PowerBall will be added to the procedure. I recompiled NTHINDEX.EXE to SEQUENCE.EXE, the program that uses the new algorithm. The program is freeware at the download site: Here is the main algorithm. Listed is also my function that calculates the number of combinations 'N taken M at a time': Combine## (N, M). I verified the algorithm with many cases. I believe the procedure is 100% accurate. All about algorithm formula math calculate combination index order. Title: Mathematics, combinatorics: algorithm, formula, software to generate combinations and calculate combination lexicographical order or index, rank. All about combination, combinations, combinatorics, formula, formulas, formulae, algorithm, algorithms. This is the best analysis of lexicographic, lexicographical order, lex, combination sequence number, mathematics, math, maths, lotto, lottery. Master the knowledge of odds, calculate odds, generate combinations, generator, function, probability, software, freeware, program, source code, code. Title: "Algorithm, software to calculate combination lexicographical order, rank, index". Two algorithms to calculate the combination lexicographical order, or rank, or index; generate the combination for a given lexicographic order or rank . By B.P. Buckles - M. Lybanon; by Ion Saliu.

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