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Cycle type of a permutation - Groupprops

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Let be a finite set and be a permutation. The cycle type of is the data of how many cycles of each length are present in the cycle decomposition of . There are two typical ways of specifying the cycle type. The definition also applies for infinite sets; here, we also need to include cycles of infinite length, i.e., the chains. The cycle type of a permutation is defined as the unordered list of the sizes of the cycles in the cycle decomposition of . For instance, consider the permutation with cycle decomposition: This permutation has cycle type . Since this is an unordered list, this can also be written as or . Note that the sum of all the cycle sizes must equal the size of the whole set . Thus, the cycle type of a permutation is an unordered integer partition of the size of the set. This decsribes the cycle type as an ordered sequence where is the number of cycles of size (length) . Thus, the permutation: has and . Note that give the cycle type in either form, it can be converte

Cycle type of a permutation - Groupprops Jump to content Want site search autocompletion? See here Encountering 429 Too Many Requests errors when browsing the site? See here From Groupprops Definition Let S {\displaystyle S} be a finite set and σ : S → S {\displaystyle \sigma :S\to S} be a permutation. The cycle type of σ {\displaystyle \sigma } is the data of how many cycles of each length are present in the cycle decomposition of σ {\displaystyle \sigma } . There are two typical ways of specifying the cycle type. The definition also applies for infinite sets; here

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