# Counting subsets of bipartite graph part which admit an induced perfect matching

Given a bipartite graph $$G=(U \sqcup V, E)$$, count $$U^\prime \subseteq U$$ for which $$\exists V^\prime \subseteq V$$ such that the induced subgraph $$G[U^\prime \sqcup V^\prime]$$ contains a perfect matching.

I believe that this problem is at least #P-hard, but I can't find anything that states such.

I've found a similar question on cstheory which states that counting subsets of vertices on a general graph for which there is an induced perfect matching is #P-Complete. The difference between our questions is I consider $$U^\prime$$ unique whereas they consider $$U^\prime \sqcup V^\prime$$ unique.

An alternative expression of my question could be: count $$U^\prime \subseteq U$$ for which the induced subgraph $$G[U^\prime \sqcup V]$$ contains a (maximum, potentially imperfect) matching $$M$$ where $$\lvert M \rvert = \lvert U^\prime \rvert$$, i.e. every $$u^\prime \in U^\prime$$ is adjacent to some $$m \in M$$.