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In mathematics, a combinadic is an ordered integer partition, or composition. Combinadics provide a lexicographical index for combinations. Applications for combinadics include software testing, sampling, quality control, and the analysis of gambling games such as Canada's national 6/49 lottery. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...
In mathematics, a composition of a positive integer n is a way of writing n as a sum of positive integers. ...
In mathematics, the lexicographical order, or dictionary order, is a natural order structure of the cartesian product of two ordered sets. ...
In mathematics, an index is a superscript or subscript to a symbol. ...
It has been suggested that Permutations and combinations be merged into this article or section. ...
Software testing is the process used to help identify the correctness, completeness, security, and quality of developed computer software. ...
Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ...
In engineering and manufacturing, quality control and quality engineering are involved in developing systems to ensure products or services are designed and produced to meet or exceed customer requirements. ...
Gambling or Gaming [1] has had many different meanings depending on the cultural and historical context in which it is used. ...
Lotto 6/49 is one of Canadas national lottery games, along with Lotto Super 7. ...
For definiteness, we will consider the k-combinations on the set S = {0, 1, ..., n − 1} of the first n integers starting from 0. Recall that there are C(n,k) = n! / ( k! (n − k)! ) of these. The index we are looking for is the mapping associating the numbers i = 0, 1, ..., C(n,k) − 1 with the list of k-combinations having their elements written in ascending order, and themselves ordered lexicographically. By abuse of notation we denote as C(n,k)(i) the ith k-combination taken from the set of n elements. Then the index for the ten 3-combinations of the set of five elements can be written out as follows. It has been suggested that Permutations and combinations be merged into this article or section. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition (while being unlikely to introduce errors or cause confusion). ...
ith 3-combination Index i C(5,3)(i) 0 {0, 1, 2} 1 {0, 1, 3} 2 {0, 1, 4} 3 {0, 2, 3} 4 {0, 2, 4} 5 {0, 3, 4} 6 {1, 2, 3} 7 {1, 2, 4} 8 {1, 3, 4} 9 {2, 3, 4} The definition we want for an ith combinadic M(n,k)(i) on the C(n,k) k-combinations of the set of n elements is most directly achieved by first considering the list of all possible words n letters long consisting of k 1s and n − k 0s. We designate the letter positions, or places, in the word as n − 1, n − 2, ..., 1, 0 in descending order, lowest letter position on the right. Then the ith combinadic in this system is defined to be the ordered n-tuple consisting of the letter positions for the 1s in the nth word in lexicographical order, reading from left to right. In computer programming and some branches of mathematics, strings are sequences of various simple objects. ...
In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...
In mathematics, the lexicographical order, or dictionary order, is a natural order structure of the cartesian product of two ordered sets. ...
We can now write out the combinadics for the ten 3-combinations of the set of five elements in the following table.
ith word (places) ith combinadic Index i (43210) M(5,3)(i) ||||| 0 00111 (2, 1, 0) 1 01011 (3, 1, 0) 2 01101 (3, 2, 0) 3 01110 (3, 2, 1) 4 10011 (4, 1, 0) 5 10101 (4, 2, 0) 6 10110 (4, 2, 1) 7 11001 (4, 3, 0) 8 11010 (4, 3, 1) 9 11100 (4, 3, 2) Clearly the combinadics also list all the k-combinations from the set of n elements in lexicographical order, but here the elements are listed in decreasing, not increasing order. In this MSDN article, James D. McCaffrey shows that the conventional combinations index is easily obtained from the combinadics. James D. McCaffrey (1950 -) is a software engineer and author specializing in software test automation and mathematical combinatorics. ...
The more useful composition-based definition for a combinadic can be illustrated using an example. Consider the combinadic with index 72 in the system of 5-combinations from the set of 10 elements. This is M(10,5)(72) = (8, 6, 3, 1, 0). By studying the seventy-three words consisting of five 1s and five 0s that lead up to this combinadic, it becomes clear that there is a simple relationship between the numbers in the combinadic code and an ordered partition of the index number. In mathematics, a composition of a positive integer n is a way of writing n as a sum of positive integers. ...
R ith word e Index Breakdown of Index (places) ith combinadic f i (9876543210) M(10,5)(i) |||||||||| A) 0 0000011111 (4,3,2,1,0) 1 0000101111 (5,3,2,1,0) 2 0000110111 (5,4,2,1,0) 3 0000111011 (5,4,3,1,0) 4 0000111101 (5,4,3,2,0) 5 0000111110 (5,4,3,2,1) 6 0001001111 (6,3,2,1,0) 7 0001010111 (6,4,2,1,0) ... ... ... ... 53 0011110010 (7,6,5,4,1) 54 0011110100 (7,6,5,4,2) B) 55 C(8,5) − 1 0011111000 (7,6,5,4,3) C) 56 C(8,5) 0100001111 (8,3,2,1,0) ^ ^ 57 0100010111 (8,4,2,1,0) 58 0100011011 (8,4,3,1,0) 59 0100011101 (8,4,3,2,0) 60 0100011110 (8,4,3,2,1) 61 0100100111 (8,5,2,1,0) ... ... ... ... 67 0100110110 (8,5,4,2,1) 68 0100111001 (8,5,4,3,0) 69 0100111010 (8,5,4,3,1) D) 70 C(8,5) + C(6,4) − 1 0100111100 (8,5,4,3,2) E) 71 C(8,5) + C(6,4) 0101000111 (8,6,2,1,0) ^ ^ ^ ^ F) 72 C(8,5) + C(6,4) + C(3,3) + C(1,2) + C(0,1) 0101001011 (8,6,3,1,0) ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ Notice first of all that between reference A and reference B, the five 1s run through all the possible patterns among the eight rightmost places 0, 1, 2, ..., 7 in the word. There are C(8, 5) = 56 of these, so since reference A is at index 0, reference B has to be at index C(8,5) − 1 = 55. At the next index, reference C, the leftmost 1 achieves its final resting place of position 8 on index C(8,5) = 56, while the remaining four 1s are reset to the far right of the word. Next notice that between reference C and reference D, the remaining four 1s run through all the possible patterns in the six rightmost places 0, 1, 2, ..., 5 in the word. There are C(6,4) = 15 of these, so reference D lies at index C(8,5) + C(6,4) − 1 = 70, and the next index, reference E, where the second leftmost 1 achieves its final resting place of position 6 and the remaining three 1s are reset to the right, is index C(8,5) + C(6,4) = 71. The three rightmost 1s run through the single (C(3,3) = 1) pattern on the three rightmost places before the third rightmost 1 takes up its final station at place 3 at reference F. The two rightmost 1s run through no (C(1,2) = 0) patterns before the second rightmost 1 doesn't move; similarly the one rightmost 1 runs through no (C(0,1) = 0) patterns before it doesn't move.
The relationship between combinadics and ordered integer partitions is formalized in the following.
- Definition
- (after McCaffrey)
- combinadic
- In the context of the system of C(n, k) k-combinations on the set of n objects, and for each non-negative integer i less than C(n,k), the ith combinadic M(n,k)(i) is the unique strictly decreasing sequence (mk−1, mk−2, ..., m0) of k integers lying between n − 1 and 0 so that C(mk−1,k) + C(mk−2,k − 1) + ... + C(m0,1) = i.
Clearly the index value i for any combinadic can be immediately calculated from its elements. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
To get the elements of the combinadic M(n,k)(i) from its index i, we first find the largest first element mk−1 so that C(mk−1, k) is less than or equal to i. The second element is found by repeating the procedure in the reduced combinadic system where i is reduced by the previous C(mk−1,k), n is set to the previous mk−1, and k is reduced by one. This is continued until k is reduced to zero. As McCaffrey points out in his MSN article, efficiently finding the largest mk−1 is the key to calculating a k-combination from its index value (see his discussion of the helper function LargestV()). It has been suggested that Permutations and combinations be merged into this article or section. ...
Partial plot of a function f. ...
Combinadics, unlike factoradics, does not appear to be a numeral system, although it has very much the look and feel of one. In particular, it is different from a mixed radix system. The factorial based radix or factoradic is a factorial based mixed radix numeral scheme: radix: 5! 4! 3! 2! 1! decimal: 120 24 6 2 1 In this numbering system, the rightmost digit may be 0 or 1, the next 0, 1, or 2, and so on. ...
A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ...
Look and feel refers to design aspects of a graphical user interface - in terms of both colours, shapes, layout, typefaces, etc (the look); and, the behaviour of dynamic elements such as buttons, boxes, and menus (the feel). It is used in reference to both software and websites. ...
Mixed radix numeral systems are more general than the usual ones in that the numerical base may vary from position to position. ...
External links
- "Generating the mth Lexicographical Element of a Mathematical Combination", by James McCaffrey. Volt Information Sciences, Inc. July 2004 (a Visual C# .NET article from the MSDN Library) Implementation of combinadics in C# source code
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ...
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