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In [[mathematics]], and in particular in [[combinatorics]], the '''combinatorial number system''' of degree ''k'' (for some positive integer ''k''), also referred to as '''combinadics''', is a correspondence between [[natural number]]s (taken to include&nbsp;0) ''N'' and ''k''-[[combination]]s, represented as [[strictly decreasing]] sequences ''c''<sub>''k''</sub>&nbsp;&gt;&nbsp;...&nbsp;&gt;&nbsp;''c''<sub>2</sub>&nbsp;&gt;&nbsp;''c''<sub>1</sub>&nbsp;≥&nbsp;0. Since the latter are strings of numbers, one can view this as a kind of [[numeral system]] for representing ''N'', although the main utility is representing a ''k''-combination by ''N'' rather than the other way around. Distinct numbers correspond to distinct ''k''-combinations, and produce them in [[lexicographic order]]; moreover the numbers less than <math>\tbinom nk</math> correspond to all ''k''-combinations of {{nowrap|{ 0, 1, ..., ''n'' − 1}}}. The correspondence does not depend on the size&nbsp;''n'' of the set that the ''k''-combinations are taken from, so it can be interpreted as a map from '''N''' to the ''k''-combinations taken from '''N'''; in this view the correspondence is a [[bijective map|bijection]].
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The number ''N'' corresponding to (''c''<sub>''k''</sub>,...,''c''<sub>2</sub>,''c''<sub>1</sub>) is given by
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:<math>N=\binom{c_k}k+\cdots+\binom{c_2}2+\binom{c_1}1.</math>
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The fact that a unique sequence so corresponds to any number ''N'' was observed by [[Derrick Henry Lehmer|D.H. Lehmer]].<ref>''Applied Combinatorial Mathematics'', Ed. E. F. Beckenbach (1964), pp.27−30.</ref> Indeed a [[greedy algorithm]] finds the ''k''-combination corresponding to ''N'': take ''c''<sub>''k''</sub> maximal with <math>\tbinom{c_k}k\leq N</math>, then take ''c''<sub>''k'' − 1</sub> maximal with <math>\tbinom{c_{k-1}}{k-1}\leq N - \tbinom{c_k}k</math>, and so forth. Finding the number ''N'', using the formula above, from the ''k''-combination (''c''<sub>''k''</sub>,...,''c''<sub>2</sub>,''c''<sub>1</sub>) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; these operations are known by those names in most [[Computer algebra system]]s, and in computational mathematics.<ref>http://www.site.uottawa.ca/~lucia/courses/5165-09/GenCombObj.pdf</ref><ref>http://www.sagemath.org/doc/reference/sage/combinat/subset.html</ref>
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The originally used term "combinatorial representation of integers" is shortened to "combinatorial number system" by [[Donald Ervin Knuth|Knuth]],<ref>{{citation
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| last=Knuth | first = D. E. | author-link = Donald Ervin Knuth
| title = The Art of Computer Programming
| volume =4, Fascicle 3
| contribution = Generating All Combinations and Partitions | publisher = Addison-Wesley | year = 2005
| isbn = 0-201-85394-9
| pages = 5−6}}.</ref>
who also gives a much older reference;<ref>{{citation
| last = Pascal | first = Ernesto | title=Giornale di Matematiche | volume = 25 | year = 1887 | pages = 45−49}}</ref>
the term "combinadic" is introduced by James McCaffrey <ref>{{citation
| last = McCaffrey | first = James | author-link = James D. McCaffrey
  | publisher = Microsoft Developer Network
| title = Generating the mth Lexicographical Element of a Mathematical Combination
| url = http://msdn.microsoft.com/en-us/library/aa289166%28VS.71%29.aspx
| year = 2004}}</ref> (without reference to previous terminology or work).


Unlike the [[factorial number system]], the combinatorial number system of degree ''k'' is not a [[mixed radix]] system: the part <math>\tbinom{c_i}i</math> of the number ''N'' represented by a "digit" ''c''<sub>''i''</sub> is not obtained from it by simply multiplying by a place value.
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The main application of the combinatorial number system is that it allows rapid computation of the ''k''-combination that is at a given position in the lexicographic ordering, without having to explicitly list the ''k''-combinations preceding it; this allows for instance random generation of ''k''-combinations of a given set. Enumeration of ''k''-combinations has many applications, among which [[software testing]], [[sampling (statistics)|sampling]], [[quality control]], and the analysis of [[lottery]] games.
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A ''k''-combination of a set ''S'' is a subset of ''S'' with ''k'' (distinct) elements. The main purpose of the combinatorial number system is to provide a representation, each by a single number, of all <math>\tbinom nk</math> possible ''k''-combinations of a set ''S'' of ''n'' elements. Choosing, for any ''n'', {{nowrap|{0, 1, ..., ''n''&nbsp;&minus;&nbsp;1}}} as such a set, it can be arranged that the representation of a given ''k''-combination ''C'' is independent of the value of ''n'' (although ''n'' must of course be sufficiently large); in other words considering ''C'' as a subset of a larger set by increasing ''n'' will not change the number that represents&nbsp;''C''. Thus for the combinatorial number system one just considers ''C'' as a ''k''-combination of the set '''N''' of all natural numbers, without explicitly mentioning ''n''.
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In order to ensure that the numbers representing the ''k''-combinations of {{nowrap|{0, 1, ..., ''n''&nbsp;&minus;&nbsp;1}}} are less than those representing ''k''-combinations not contained in {{nowrap|{0, 1, ..., ''n''&nbsp;&minus;&nbsp;1}}}, the ''k''-combinations must be ordered in such a way that their largest elements are compared first. The most natural ordering that has this property is [[lexicographic order]]ing of the ''decreasing'' sequence of their elements. So comparing the 5-combinations ''C''&nbsp;=&nbsp;{0,3,4,6,9} and ''C''<nowiki>'</nowiki>&nbsp;=&nbsp;{0,1,3,7,9}, one has that ''C'' comes before ''C''<nowiki>'</nowiki>, since they have the same largest part&nbsp;9, but the next largest part&nbsp;6 of ''C'' is less than the next largest part&nbsp;7 of ''C''<nowiki>'</nowiki>; the sequences compared lexicographically are (9,6,4,3,0) and (9,7,3,1,0). Another way to describe this ordering is view combinations as describing the ''k'' raised bits in the binary representation of a number, so that ''C''&nbsp;=&nbsp;{''c''<sub>1</sub>,...,''c''<sub>''k''</sub>} describes the number
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:<math>2^{c_1}+2^{c_2}+\cdots+2^{c_k}</math>
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(this associates distinct numbers to ''all'' finite sets of natural numbers); then comparison of ''k''-combinations can be done by comparing the associated binary numbers. In the example ''C'' and ''C''<nowiki>'</nowiki> correspond to numbers 1001011001<sub>2</sub>&nbsp;=&nbsp;601<sub>10</sub> and 1010001011<sub>2</sub>&nbsp;=&nbsp;651<sub>10</sub>, which again shows that ''C'' comes before ''C''<nowiki>'</nowiki>. This number is not however the one one wants to represent the ''k''-combination with, since many binary numbers have a number of raised bits different from ''k''; one wants to find the relative position of ''C'' in the ordered list of (only) ''k''-combinations.
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== Place of a combination in the ordering ==
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The number associated in the combinatorial number system of degree ''k'' to a ''k''-combination ''C'' is the number of ''k''-combinations strictly less than ''C'' in the given ordering. This number can be computed from ''C''&nbsp;=&nbsp;{ ''c''<sub>''k''</sub>, ..., ''c''<sub>2</sub>, ''c''<sub>1</sub>&nbsp;} with ''c''<sub>''k''</sub>&nbsp;&gt;&nbsp;...&nbsp;&gt; ''c''<sub>2</sub> &gt; ''c''<sub>1</sub> as follows. From the definition of the ordering it follows that for each ''k''-combination ''S'' strictly less than&nbsp;''C'', there is a unique index&nbsp;''i'' such that ''c''<sub>''i''</sub> is absent from ''S'', while ''c''<sub>''k''</sub>, ..., ''c''<sub>''i''+1</sub> are present in ''S'', and no other value larger than ''c''<sub>''i''</sub> is. One can therefore group those ''k''-combinations ''S'' according to the possible values 1, 2, ..., ''k'' of ''i'', and count each group separately. For a given value of ''i'' one must include
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''c''<sub>''k''</sub>, ..., ''c''<sub>''i''+1</sub> in ''S'', and the remaining ''i'' elements of ''S'' must be chosen from the ''c''<sub>''i''</sub> non-negative integers strictly less than ''c''<sub>''i''</sub>; moreover any such choice will result in a ''k''-combinations ''S'' strictly less than&nbsp;''C''. The number of possible choices is <math>\tbinom{c_i}i</math>, which is therefore the number of combinations in group ''i''; the total number of ''k''-combinations strictly less than ''C'' then is
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:<math>\binom{c_1}1+\binom{c_2}2+\cdots+\binom{c_k}k,</math>
<ul>
and this is the index (starting from&nbsp;0) of ''C'' in the ordered list of ''k''-combinations. Obviously there is for every ''N''&nbsp;∈&nbsp;'''N''' exactly one ''k''-combination at index&nbsp;''N'' in the list (supposing ''k''&nbsp;≥&nbsp;1, since the list is then infinite), so the above argument proves that every ''N'' can be written in exactly one way as a sum of ''k'' binomial coefficients of the given form.
 
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== Finding the ''k''-combination for a given number ==
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The given formula allows finding the place in the lexicographic ordering of a given ''k''-combination immediately. The reverse process of finding the ''k''-combination at a given place ''N'' requires somewhat more work, but is straightforward nonetheless. By the definition of the lexicographic ordering, two ''k''-combinations that differ in their largest element ''c''<sub>''k''</sub> will be ordered according to the comparison of those largest elements, from which it follows that all combinations with a fixed value of their largest element are contiguous in the list. Moreover the smallest combination with ''c''<sub>''k''</sub> as largest element is <math>\tbinom{c_k}k</math>, and it has ''c''<sub>''i''</sub>&nbsp;=&nbsp;''i''&nbsp;−&nbsp;1 for all ''i''&nbsp;&lt;&nbsp;''k'' (for this combination all terms in the expression except <math>\tbinom{c_k}k</math> are zero). Therefore ''c''<sub>''k''</sub> is the largest number such that <math>\tbinom{c_k}k\leq N</math>. If ''k''&nbsp;&gt;&nbsp;1 the remaining elements of the ''k''-combination form the {{nowrap|''k'' − 1}}-combination corresponding to the number <math>N-\tbinom{c_k}k</math> in the combinatorial number system of degree {{nowrap|''k'' − 1}}, and can therefore be found by continuing in the same way for <math>N-\tbinom{c_k}k</math> and {{nowrap|''k'' − 1}} instead of ''N'' and  ''k''.
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=== Example ===
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Suppose one wants to determine the 5-combination at position 72. The successive values of <math>\tbinom n5</math> for ''n''&nbsp;=&nbsp;4, 5, 6, ... are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for ''n''&nbsp;=&nbsp;8. Therefore ''c''<sub>5</sub>&nbsp;=&nbsp;8, and the remaining elements form the 4-combination at position {{nowrap|72 − 56 {{=}} 16}}. The successive values of <math>\tbinom n4</math> for ''n''&nbsp;=&nbsp;3, 4, 5, ... are 0, 1, 5, 15, 35, ..., of which the largest one not exceeding 16 is 15, for ''n''&nbsp;=&nbsp;6, so ''c''<sub>4</sub>&nbsp;=&nbsp;6. Continuing similarly to search for a 3-combination at position {{nowrap|16 − 15 {{=}} 1}} one finds ''c''<sub>3</sub>&nbsp;=&nbsp;3, which uses up the final unit; this establishes <math>72=\tbinom85+\tbinom64+\tbinom33</math>, and the remaining values ''c''<sub>''i''</sub> will be the maximal ones with <math>\tbinom{c_i}i=0</math>, namely {{nowrap|''c''<sub>''i''</sub> {{=}} ''i'' − 1}}. Thus we have found the 5-combination {{nowrap|{8, 6, 3, 1, 0}}}.
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== Applications ==
<ul>
 
 
One could use the combinatorial number system to list or traverse all ''k''-combinations of a given finite set, but this is a very inefficient way to do that. Indeed, given some ''k''-combination it is much easier to find the next combination in lexicographic ordering directly than to convert a number to a ''k''-combination by the method indicated above. To find the next combination, find the smallest ''i''&nbsp;≥&nbsp;2 for which ''c''<sub>''i''</sub>&nbsp;≥&nbsp;''c''<sub>''i−1''</sub>+2 (if there is no such ''i'', take ''i'' = ''k''+1); then increase ''c''<sub>''i''−1</sub> by one and set all ''c''<sub>''j''</sub> with {{nowrap|''j'' &lt; ''i'' − 1}} to their minimal value {{nowrap|''j'' − 1}}. If the ''k''-combination is represented by its [[characteristic vector]], that is as a binary value with ''k'' bits&nbsp;1, then the next such value can be computed without any loop using bitwise arithmetic: the following function will advance <tt>x</tt> to that value or return <tt>false</tt>:
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<source lang="cpp">
 
// find next k-combination
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bool next_combination(unsigned long& x) // assume x has form x'01^a10^b in binary
 
{
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  unsigned long u = x & -x; // extract rightmost bit 1; u =  0'00^a10^b
    
   unsigned long v = u + x; // set last non-trailing bit 0, and clear to the right; v=x'10^a00^b
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  if (v==0) // then overflow in v, or x==0
 
    return false; // signal that next k-combination cannot be represented
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  x = v +(((v^x)/u)>>2); // v^x = 0'11^a10^b, (v^x)/u = 0'0^b1^{a+2}, and x ← x'100^b1^a
 
  return true; // successful completion
</ul>
}
</source>  
This is called [[Bill Gosper|Gosper]]'s hack;<ref>{{citation
| last=Knuth | first = D. E. | author-link = Donald Ervin Knuth
| title = The Art of Computer Programming
| volume =4, Fascicle 1
| contribution = Bitwise tricks and techniques | publisher = Addison-Wesley | year = 2009
| isbn = 0-321-58050-8
| pages = 54}}</ref>
corresponding [[assembly code]] was described as item 175 in [[HAKMEM]].
 
On the other hand the possibility to directly generate the ''k''-combination at index ''N'' has useful applications. Notably, it allows generating a random ''k''-combination of an ''n''-element set using a random integer ''N'' with <math>0\leq N<\tbinom nk</math>, simply by converting that number to the corresponding ''k''-combination. If a computer program needs to maintain a table with information about every ''k''-combination of a given finite set, the computation of the index ''N'' associated to a combination will allow the table to be accessed without searching.
 
==See also==
* [[Factorial number system]] (also called factoradics)
 
== References ==
<references/>
 
[[Category:Combinatorics]]

Latest revision as of 00:43, 11 July 2014

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