Let $N \subset \mathbb N$ and $\Pi$ the set of partitions of $N$. I want to define a permutation such that the partition $\pi \in \Pi$ is identical with $\pi' \in \Pi$ up to a permutation $p : N \to N$ of the index $i \in N$.
Exmaple Consider $N = \{1,2,3\}$ and $\pi = \{\{1\},\{2,3\}\}$. The permutation $p(N) = \{2,3,1\}$ shall result in $\pi' = \{\{2\},\{3,1\}\}$. Does there exist a common definition or how would you define $\pi'$ from $p$?
Edit With respect to the comment one may defines $\rho : \Pi \to \Pi$ by $\rho(\pi) := \{\{p(i)\}_{i \in S} \mid S \in \pi\}$. Reconsdering the example $\pi = \{\{1\},\{2,3\}\}$ and $p(\{1,2,3\}) = \{2,3,1\}$ yields \begin{align} \rho(\pi)& = \{\{p(i)\}_{i \in \{1\}}, \{p(i)\}_{i \in \{2,3\}}\}\\ & = \{\{p(1)\}, \{p(2),p(3)\}\\ & = \{\{2\},\{3,1\}\}\\ & = \pi'. \end{align}
Is the definition sound?