Cycle index
= Cycle index
{title2=$Z_G(t_1,t_2,\ldots)=|G|^{-1}\sum_{g\in G}\prod_rt_r^{c_r(g)}$}
For a <permutation group>, $c_r(g)$ is the number of length-$r$ cycles of $g$. The cycle index records the average cycle structure. On coloring a finite underlying set, a coloring fixed by $g$ is constant on each cycle. Replacing $t_r$ by the <power-sum symmetric polynomial> $p_r(x)$ therefore gives the <pattern inventory> by the <weighted Burnside lemma>.