# Complexity of counting the number of Good-perfect matching in the bipartite graph

Let's $G=(U, V, E)$ be a balanced bipartite graph which $|U|=|V|=n$ and $|E|=n*(n-1)$; All nodes in $U$ are connected to all nodes in $V$ except $u_i$ to $v_i$ for $1\leq i \leq n$.

Definition1: Cross edges are two edges in $E$, one with two end points $u_i$, $v_j$ and the other with $u_j$, $v_i$.
Definition2: Good-perfect matching is a perfect matching with no cross edges.

What is the complexity of counting the number of Good-perfect matching in $G$?