Cycle notation

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In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles.

[edit] Definition

Let $S$ be a finite set, and

$a_1,\ldots,a_k,\quad k\geq 2$

be distinct elements of $S$ . The expression

$(a_1\ \ldots\ a_k)$

denotes the cycle σ whose action is

$a_1\mapsto a_2\mapsto a_3\ldots a_k \mapsto a_1.$

For each index i,

σ(a i) = a i + 1,

where $a k + 1$ is taken to mean $a 1$ .

There are $k$ different expressions for the same cycle; the following all represent the same cycle:

$(a_1\ a_2\ a_3\ \ldots\ a_k) = (a_2\ a_3\ \ldots\ a_k\ a_1) = \cdots = (a_k\ a_1\ a_2\ \ldots\ a_{k-1}).\,$

A 1-element cycle is the same thing as the identity permutation and is omitted. It is customary to express the identity permutation simply as $()\,$ .

[edit] Permutation as product of cycles

Let $π$ be a permutation of $S$ , and let

$S_1,\ldots, S_k\subset S,\quad k\in\mathbb{N}$

be the orbits of $π$ with more than 1 element. For each $j=1,\ldots,k$ let $n j$ denote the cardinality of $S j$ . Also, choose an $a_{1,j}\in S_j$ , and define