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Let $C_n$ be the cyclic group of order $n$ acting on a finite set $X$ and let $Z(C_n, X; p_1,p_2,\dots)$ be the cycle index of the corresponding permutation group.

I wonder whether the knowledge of the cycle index alone is enough to determine the number of fixed points of the action of a given element $g\in C_n$ on $X$?

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Yes; fixed points are cycles of length 1. –  Qiaochu Yuan Feb 16 '11 at 14:12
    
Thank's, I missed the obvious. –  Martin Rubey Feb 16 '11 at 14:57
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