Math 2.0

Combinatorics Theorem

Number of compositions

Builds on Composition of an integer

Statement

Let nn be a positive integer. Then the number of compositions of nn is 2n−12^{n-1}.

Remarks

Proof: write nn as a row of nn ones. A composition is a choice of where to cut among the n−1n - 1 gaps between them. The number of compositions of nn into exactly kk parts is (n−1k−1)\binom{n-1}{k-1}. For n=0n = 0 there is one composition, the empty one.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

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Composition of an integer, definition, no evidence levelComposition of an int…definitionNumber of compositions, statement, no evidence levelNumber of compositionsstatement
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  1. Composition of an integer definition
  2. Number of compositions statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Composition_(combinatorics), Number of compositions
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Tag057H72B2MB
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Kindstatement, theorem
WritingW0 stub
MSC05A17
StatusThis copy of the site has no build data. Run m2 status 057H72B2MB.
UsesComposition of an integer

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