CF1651E 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 $ .