A permutation can be represented in both linear and cyclic notations. The functions described in this section can be used to convert between the two forms.
The linear notation is an index mapping, and has already been described above. The cyclic notation represents a permutation as a series of circular rearrangements of groups of elements, or cycles.
Any permutation can be decomposed into a combination of cycles. For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element which is unchanged by the permutation and is referred to as a singleton.
The cyclic notation for a permutation is not unique, but can be rearranged into a unique canonical form by a reordering of elements. The library uses the canonical form defined in Knuth's Art of Computer Programming (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.
The procedure for obtaining the canonical form given by Knuth is,
For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
The important property of the canonical form is that it can be reconstructed from the contents of each cycle without the brackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation. In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping between linear permutations defined by the canonical form has many important uses in the theory of permutations.