The cycle containing $1$ of a uniform permutation has length which is uniformly distributed. I was wondering if the converse is true:



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> Suppose $\sigma$ is a permutation on $\{1,\dots,n\}$ and let $u \in \{1,\dots,n\}$ be a uniformly chosen random variable, independent of $\sigma$. Let $V$ be the size of the cycle of $\sigma$ containing $u$. If $V$ is uniformly distributed on $\{1,\dots,n\}$ then is it true that $\sigma$ must be a uniform permutation? 
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Uniformly permutation and the length of a size biased cycle
Observe that the distribution of $V|\sigma$ depends only on the lengths of the cycles in $\sigma$. Therefore, the distribution of $V$ does not change if we replace the random variable $\sigma$ with $f(\sigma)$, where $f:S\_n\to S\_n$ is any function that maintains the lengths of the cycles.


Example: let $\sigma$ be the identity permutation with probability $1/6$, $(12)$ with probability $3/6$, and $(123)$ with probability $2/6$.


