Partitions induced on 2-subsets of a set Consider the set $[n] = \{1,\dots,n\}$, and let $\binom{[n]}{2} =\{\{i,j\} : i < j\}$ be the set of 2-subsets of $[n]$. Every partition $P$ of $[n]$ into two subsets induces a partition of $\binom{[n]}{2}$ based on whether both elements of $\{i,j\}$ belong to the same class of $P$ or not. How many such induced partitions are there? Is there a link between this problem and well-known problems in combinatorics?
EDIT: The number of these might seem trivial (?). I am actually more interested in any links to other known concepts, keywords, etc. A better question might be this: Given a 2-partition of $\binom{[n]}{2}$, how can one determine whether it is an induced partition of $[n]$ or not?

Let $2^X/2$ denote the collection of partitions of a set $X$ into two parts. You have defined a map $2^X/2\to2^{\binom{X}{2}}/2$ and ask for the size of its image as a function of $|X|$. I will show the map is injective if we assume $|X|>4$, which afterwards reduces the problem to merely $|X|=2,3,4$.
Suppose $\{U,V\}\in 2^X/2$ is a $2$-part partition of $X$ inducing $\Pi=\{A,B\}\in2^\binom{X}{2}/2$. We want to recover $U$ and $V$ from $\Pi$. The first line of business is to determine which element of $\Pi$ functions as the collection of "bridges" between $U$ and $V$. Since one of $U$ or $V$ has size greater than $2$, the set (say $A$ wlog) which is not the set of bridges will contain a 'triangle' (subset $\{\{i,j\},\{j,k\},\{k,i\}\}$) whereas the bridge set $B$ will not, so we can distinguish $A,B$. Now, pick a bridge $\{i,j\}\in B$, then pick an element $i\in\{i,j\}$. Form the set $U(i)=\{i'\in X:\{i,i'\}\in A\}$ and similarly for $U(j)$, so that we end up with $\{U,V\}=\{U(i),U(j)\}$. Thus we can recover $\{U,V\}$ from $\Pi=\{A,B\}$.
As the map $\nu:2^X/2\to2^\binom{X}{2}/2$ is injective, the number of induces partitions is $|2^X/2|=2^{|X|}/2$.
A partition $\Pi\in2^\binom{X}{2}/2$ is seen to be induced iff only one of its parts contains triangles and the procedure outlined above outputs a partition of $X$ (as opposed to just any $2$-subset of $2^X$).
