Example of the $Ω$−partite
A permutation on a set is a bijection from the set to itself; thus $\sigma$ is a (bijective) map from $P$ to itself, not from $p(P)$ to itself (though it does induce a map from $p(P)$ to itself in a natural manner).

Also, if $f$ is a map from $A$ to $B$ and $X$ is a subset of $A$, then $f(X) = \{f(x), x \in X\}$. This is completely standard notation.

For a collection $\Lambda$ of subsets of $P$, $\sigma(\Lambda) = \{\sigma(A), A \in \Lambda\}$ is the collection of subsets of $P$ which is obtained by applying $\sigma$ to all the elements of $\Lambda$.

Example: $P = \{1,2,\dots,8\}$, $\Omega = \{\{1,2\},\dots,\{7,8\}\}$. Then $\Lambda = \{\{1,2,3,4\},\{5,6,7,8\}\}$ is $\Omega$-partite since clearly every permutation of $P$ that fixes each $P_i$ also fixes each element of $\Lambda$, so in fact we even have $\sigma(A) = A$ for all $A$. On the other hand, $\Lambda' = \{\{1,2,3\},\{4,5,6\},\{7,8\}\}$ is not; for instance $\sigma = (34)$ yields $\sigma(\Lambda') = \{\{1,2,4\},\{3,5,6\},\{7,8\}\} \ne \Lambda'$.