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List of partitions of natural numbers

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This is a list of partitions of natural numbers up to 7. The partitions are written with the terms in decreasing order, grouped by the number of terms required. Note that any natural number can be written as a "trivial sum" of one term: the integer itself. Including this trivial partition, the function giving the number of unique partitions of each natural number is called the partition function.

  • 2 = 1 + 1
  • 3 = 2 + 1
    = 1 + 1 + 1
  • 4 = 3 + 1 = 2 + 2
    = 2 + 1 + 1
    = 1 + 1 + 1 + 1
  • 5 = 4 + 1 = 3 + 2
    = 3 + 1 + 1 = 2 + 2 + 1
    = 2 + 1 + 1 + 1
    = 1 + 1 + 1 + 1 + 1
  • 6 = 5 + 1 = 4 + 2 = 3 + 3
    = 4 + 1 + 1 = 3 + 2 + 1 = 2 + 2 + 2
    = 3 + 1 + 1 + 1 = 2 + 2 + 1 + 1
    = 2 + 1 + 1 + 1 + 1
    = 1 + 1 + 1 + 1 + 1 + 1
  • 7 = 6 + 1 = 5 + 2 = 4 + 3
    = 5 + 1 + 1 = 4 + 2 + 1 = 3 + 3 + 1 = 3 + 2 + 2
    = 4 + 1 + 1 + 1 = 3 + 2 + 1 + 1 = 2 + 2 + 2 + 1
    = 3 + 1 + 1 + 1 + 1 = 2 + 2 + 1 + 1 + 1
    = 2 + 1 + 1 + 1 + 1 + 1
    = 1 + 1 + 1 + 1 + 1 + 1 + 1

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