Random Mapping

$\sigma$ of a set $X = \{1,2,\ldots,n\}$ into itself

A random variable taking values in the set $\Sigma_n$ of all single-valued mappings of $X$ into itself. The random mappings $\sigma$ for which the probability $\mathsf{P}\{\sigma=s\}$ is positive only for one-to-one mappings $s$ are called random permutations of degree (order) $n$. The most thoroughly studied random mappings are those for which $\mathsf{P}\{\sigma=s\} = n^{-n}$ for all $s \in\Sigma_n$. A realization of such a random mapping is the result of a simple random selection from $\Sigma_n$.

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