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Cycle Notation of Permutations

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Permutations can also be written in cycle notation. A cycle of a permutation consists of an element and its images we obtain by successively applying of this permutation until the starting element is reached again. By means of the elements not occurring in the first cycle further cycles are constructed until each element occurs in (exactly) one of the cycles found by the above construction. The cycles of a permutation are written down in parentheses one after another in descending order by the number of elements occurring in the cycle. Cycles of length 1 are usually omitted.

For example

$\displaystyle \pi = \left( \begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6 \\ 4 & 3 & 2 & 6 & 5 & 1 \end{array}\right) \equiv (1 \, 4 \, 6) \ (2 \, 3) \ (5)$    respectively $\displaystyle \pi = (1 \, 4 \, 6) \ (2 \, 3) \; . $

Example:

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  automatically generated 3/31/2005