Cycles

A permutation written in disjoint-cycle notation: $(i_1\,i_2\,\dots\,i_k)$ sends $i_1 \to i_2 \to \dots \to i_k \to i_1$ and fixes everything else. A carrier, like PermutationGroup — it holds the cycles rather than computing anything from them.

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Cycles({{i1, ..., ik}, ...})the permutation that cycles each listed tuple and fixes every other point

Cycles({{i1, i2, ...}, ...})A permutation written in disjoint-cycle notation:

(i_1\,i_2\,\dots\,i_k)

sends

i_1 \to i_2 \to \dots \to i_k \to i_1

and fixes everything else. A carrier, like PermutationGroup — it holds the cycles rather than computing anything from them.

Typed signatures

Type	Package	Applies
`((list<list<integer>>) -> expression<Cycles>) & ((cycle_decomposition) -> expression<Cycles>)`	`combinatorics`replaces `groupalgebra`	always
`(list<list<integer>>) -> expression<Cycles>`	`groupalgebra`	always

Domain: Permutations

Details

Fixed points (singleton cycles) are dropped on construction — $\mathrm{Cycles}(\{\{1\},\{2,3\}\})$ and $\mathrm{Cycles}(\{\{2,3\}\})$ are the same value
The cycles and their internal order are kept exactly as given otherwise — nothing is sorted or rotated to a canonical start
PermutationCycles builds one from a one-line word; Permute applies one to a list; InversePermutation reverses one
Written from a CycleDecomposition, it drops that value's fixed points. CycleDecomposition(cycles, n) is the way back, given the size