# Decomposing a permuation into a product of disjoint cycles: permu2cycles

### 5.9.3 Decomposing a permuation into a product of disjoint cycles: permu2cycles

Any permutation can be decomposed as a sequence of cycles which have no elements in common. For example, the permutation [1,3,4,0,2] can be written as a combination of the cycles [0,1,3] and [2,4].

The permu2cycles command decomposes a permutation into a combination of cycles.

- permu2cycles takes one argument:
  p, a permutation.
- permu2cycles(p) returns the decomposition of p as a product of disjoint cycles. A cycle is represented by a list, a cyclic decomposition is represented by a list of lists.

#### Examples.

- Input:
  permu2cycles([1,3,4,5,2,0])
  Output:

  | ⎡ ⎣ | ⎡ ⎣ | 0,1,3,5 | ⎤ ⎦ | , | ⎡ ⎣ | 2,4 | ⎤ ⎦ | ⎤ ⎦

  In the answer the cycles of size 1 are omitted, except if n−1 is a fixed point of the permutation (this is required to find the value of n from the cycle decomposition).
- Input:
  permu2cycles([0,1,2,4,3,5])
  Output:

  | ⎡ ⎣ | ⎡ ⎣ | 5 | ⎤ ⎦ | , | ⎡ ⎣ | 3,4 | ⎤ ⎦ | ⎤ ⎦

- Input:
  permu2cycles([0,1,2,3,5,4])
  Output:

  | ⎡ ⎣ | ⎡ ⎣ | 4,5 | ⎤ ⎦ | ⎤ ⎦