Sum of Matchings

Let's denote the size of the maximum matching in a graph $$$G$$$ as $$$\mathit{MM}(G)$$$.

You are given a bipartite graph. The vertices of the first part are numbered from $$$1$$$ to $$$n$$$, the vertices of the second part are numbered from $$$n+1$$$ to $$$2n$$$. Each vertex's degree is $$$2$$$.

For a tuple of four integers $$$(l, r, L, R)$$$, where $$$1 \le l \le r \le n$$$ and $$$n+1 \le L \le R \le 2n$$$, let's define $$$G'(l, r, L, R)$$$ as the graph which consists of all vertices of the given graph that are included in the segment $$$[l, r]$$$ or in the segment $$$[L, R]$$$, and all edges of the given graph such that each of their endpoints belongs to one of these segments. In other words, to obtain $$$G'(l, r, L, R)$$$ from the original graph, you have to remove all vertices $$$i$$$ such that $$$i \notin [l, r]$$$ and $$$i \notin [L, R]$$$, and all edges incident to these vertices.

Calculate the sum of $$$\mathit{MM}(G(l, r, L, R))$$$ over all tuples of integers $$$(l, r, L, R)$$$ having $$$1 \le l \le r \le n$$$ and $$$n+1 \le L \le R \le 2n$$$.