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This operator, together with the set operator <math>\mathfrak{P}</math>, and their restrictions to specific degrees are used to compute [[random permutation statistics]]. There are two useful restrictions of this operator, namely to even and odd cycles.
The labelled even cycle operator <math>\mathfrak{C}_\{\operatorname{even}}</math> is
:<math>C_2 + C_4 + C_6 + \cdots\,</math>
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\frac{1}{2} \log \frac{1}{1-g(z)^2}.</math>
This implies that the labelled odd cycle operator <math>\mathfrak{C}_\{\operatorname{odd}}</math>
:<math>C_1 + C_3 + C_5 + \cdots</math>
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