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{{Other uses|Composition (disambiguation){{!}}Composition}}
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In [[  mathematics]], a '''composition''' of an [[integer]] ''n'' is a way of writing ''n'' as the [[sum]] of a sequence of (strictly) [[positive integer]]s. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same [[partition (number theory)|partition]] of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer ''n'' has 2<sup>''n''−1</sup> distinct compositions.
[[File:Binary and compositions 4.svg|thumb|center|600px|[[Bijection]] between 3 bit [[binary numeral system|binary numbers]] and compositions of 4]]
A '''weak composition''' of an integer ''n'' is similar to a composition of ''n'', but allowing terms of the sequence to be zero: it is a way of writing ''n'' as the sum of a sequence of [[non-negative integer]]s. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the ''end'' of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are assumed to be implicitly extended indefinitely by terms&nbsp;0.


To further generalize, an ''' ''A''-restricted composition''' of an integer ''n'', for a subset ''A'' of the (nonnegative or positive) integers, is an ordered collection of one or more elements in ''A'' whose sum is ''n''.<ref>
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{{cite journal
| last1=Heubach | first1=Silvia
| last2=Mansour | first2=Toufik
| year=2004
| title=Compositions of n with parts in a set
| journal=[[Congressus Numerantium]] | volume=168 | pages=33–51
| url=http://web.calstatela.edu/faculty/sheubac/papers/compositions%20with%20parts%20in%20A.pdf
}}</ref>
 
== Examples ==
[[File:Compositions of 6.svg|thumb|The 32 compositions of 6<br><br>1 + 1 + 1 + 1 + 1 + 1<br>2 + 1 + 1 + 1 + 1<br>1 + 2 + 1 + 1 + 1<br>. . .<br>1 + 5<br>6]]
[[File:Partitions of 6.svg|thumb|The 11 partitions of 6<br><br>1 + 1 + 1 + 1 + 1 + 1<br>2 + 1 + 1 + 1 + 1<br>3 + 1 + 1 + 1<br>. . .<br>3 + 3<br>6]]
 
The sixteen compositions of 5 are:
*5
*4 + 1
*3 + 2
*3 + 1 + 1
*2 + 3
*2 + 2 + 1
*2 + 1 + 2
*2 + 1 + 1 + 1
*1 + 4
*1 + 3 + 1
*1 + 2 + 2
*1 + 2 + 1 + 1
*1 + 1 + 3
*1 + 1 + 2 + 1
*1 + 1 + 1 + 2
*1 + 1 + 1 + 1 + 1.
 
Compare this with the seven partitions of 5:
*5
*4 + 1
*3 + 2
*3 + 1 + 1
*2 + 2 + 1
*2 + 1 + 1 + 1
*1 + 1 + 1 + 1 + 1.
 
It is possible to put constraints on the parts of the compositions.  For example the five compositions of 5 into distinct terms are:
*5
*4 + 1
*3 + 2
*2 + 3
*1 + 4.
 
Compare this with the three partitions of 5 into distinct terms:
*5
*4 + 1
*3 + 2.
 
== Number of compositions ==
 
Conventionally the empty composition is counted as the sole composition of 0, and there are no compositions of negative integers.
There are 2<sup>''n''−1</sup> compositions of ''n''&nbsp;&ge;&nbsp;1; here is a proof:
 
Placing either a plus sign or a comma in each of the ''n''&nbsp;&minus;&nbsp;1 boxes of the array
:<math>
    \big(\,
      \overbrace{1\, \square\, 1\, \square\, \ldots\, \square\, 1\,
      \square\, 1}^n\,
    \big)
</math>
 
produces a unique composition of ''n''. Conversely, every composition of ''n'' determines an assignment of pluses and commas. Since there are ''n''&nbsp;&minus;&nbsp;1 binary choices, the result follows. The same argument shows that the number of compositions of ''n'' into exactly ''k'' parts is given by the [[binomial coefficient]] <math>{n-1\choose k-1}</math>.  Note that by summing over all possible number of parts we recover 2<sup>''n''−1</sup> as the total number of compositions of ''n'':
 
: <math> \sum_{k=1}^n {n-1 \choose k-1} = 2^{n-1}.</math>
 
For weak compositions, the number is <math>{n+k-1\choose k-1}</math>, since each ''k''-composition of ''n''&nbsp;+&nbsp;''k'' corresponds to a weak one of&nbsp;''n'' by the rule [''a''&nbsp;+&nbsp;''b''&nbsp;+&nbsp;...&nbsp;+&nbsp;''c''&nbsp;=&nbsp;''n''&nbsp;+&nbsp;''k'']&nbsp;&rarr;&nbsp;[(''a''&nbsp;&minus;&nbsp;1)&nbsp;+&nbsp;(''b''&nbsp;&minus;&nbsp;1)&nbsp;+&nbsp;...&nbsp;+&nbsp;(''c''&nbsp;&minus;&nbsp;1)&nbsp;=&nbsp;''n''].
 
For ''A''-restricted compositions, the number of compositions of ''n'' into exactly ''k'' parts is given by the extended binomial (or polynomial) coefficient <math>\binom{k}{n}_{(1)_{a\in A}}=[x^n](\sum_{a\in A} x^a)^k</math>.<ref>
{{cite journal
| last1=Eger | first1=Steffen
| year=2013
| title=Restricted weighted integer compositions and extended binomial coefficients
| journal=[[Journal of Integer Sequences]] | volume=16
| url=https://cs.uwaterloo.ca/journals/JIS/VOL16/Eger/eger6.pdf
}}</ref>
 
== See also ==
[[Stars and bars (combinatorics)]]
 
== References ==
{{reflist}}
* {{ cite book | title=Combinatorics of Compositions and Words | first1=Silvia | last1=Heubach | first2=Toufik | last2=Mansour | series=Discrete Mathematics and its Applications | location=Boca Raton, FL | publisher=CRC Press | year=2009 | isbn=978-1-4200-7267-9 | zbl=1184.68373 }}
 
== External links ==
* [http://www.se16.info/js/partitions.htm Partition and composition calculator]
 
[[Category:Number theory]]
[[Category:Combinatorics]]

Latest revision as of 23:59, 4 December 2014

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