Which of the following numbers can be orders of permutations \alpha of 11 symbols such that it does not fix any symbol?
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A permutation $\pi$ can always be written as product of disjoint cycles $$ \pi=c_1\cdot\ldots\cdot c_k $$ and then the order of $\pi$ is the lcm of the lengths of the cycles $c_1,\dots,c_k$.
Thus you are looking for partitions $$ 11=\ell_1+\dots+\ell_k $$ with $\ell_i\geq2$ for all $i$ such that $lcm(\ell_1,...,\ell_k)$ is some specified value.
For instance $11=5+6$ proves that there's a permutation as required of order 30.
Try to work out the other values by yourself!