A14907 | 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 $n+1 \le L \le R \le 2n$ .

The first line contains one integer $n$ ( $2 \le n \le 1500$ ) — the number of vertices in each part.

Then $2n$ lines follow, each denoting an edge of the graph. The $i$ -th line contains two integers $x_i$ and $y_i$ ( $1 \le x_i \le n$ ; $n + 1 \le y_i \le 2n$ ) — the endpoints of the $i$ -th edge.

There are no multiple edges in the given graph, and each vertex has exactly two incident edges.

Print one integer — 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 $n+1 \le L \le R \le 2n$ .