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Let $C_n$ be the cyclic group of order $n$ acting on a finite set $X$ and let $Z_C(p_1,p_2,\dots)$ 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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fixed points and the cycle indexLet $C_n$ be the cyclic group of order $n$ acting on a finite set $X$ and let $Z_C(p_1,p_2,\dots)$ 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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