6.5.26  Listing all compositions of an integer into k parts: icomp

A composition of a positive integer n is an ordered set of non-negative integers which sum to n. For example, three compositions of 4 are

These compositions have two, two and three elements, respectively. The icomp command finds all compositions of an integer with a given number of elements.

• icomp accepts two mandatory arguments and one optional argument:
  • n, a positive integer.
  • k, a positive integer not larger than n.
  • Optionally, either zeros=true or zeros=false.
• icomp(n,k ⟨,zeros=bool⟩) returns the list of all compositions of n into k parts, where a part can be 0. This is equivalent to the optional argument with bool equal to true. With bool equal to false, icomp(n,k,zeros=false) returns the list of all compositions of n into k parts, where each part is nonzero (positive).

Examples.

• Input:
  icomp(4,2)
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  4 0         3 1         2 2         1 3         0 4

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

• Input:
  icomp(6,3,zeros=false)
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  4 1 1         3 2 1         2 3 1         1 4 1         3 1 2         2 2 2         1 3 2         2 1 3         1 2 3         1 1 4

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦