#include "rect.h"
#include <bits/stdc++.h>
using namespace std;

#define pb push_back
#define sz(a) (int)a.size()
#define all(a) begin(a),end(a)

using ll = long long;
using vi = vector<int>;
using ar2 = array<int,2>;

const int mxN = 2502;

int n, m;
bool grid[mxN][mxN];
vector<int> row[mxN][mxN], col[mxN][mxN];
set<pair<int,int>> found[mxN][mxN];
deque<ar2> lines[mxN];
vector<int> grid2[mxN];

struct Fenwick2D{
	int fen[mxN][mxN];
	void init(int n, int m){
		for(int i = 1; i <= n; i++)
			for(int j = 1; j <= m; j++)
				fen[i][j]=0;
	}
	void upd(int _x, int _y, int v){
		_x++, _y++;
		for(int x = _x; x<mxN; x+=x&-x)
			for(int y = _y; y < mxN; y+=y&-y)
				fen[x][y]+=v;
	}
	int sum(int _x, int _y){
		int s = 0; _x++, _y++;
		for(int x = _x; x>0; x-=x&-x)
			for(int y = _y; y>0; y-=y&-y)
				s+=fen[x][y];
		return s;
	}
	int sum(int x1, int y1, int x2, int y2){
		return sum(x2,y2)-sum(x1-1,y2)-sum(x2,y1-1)+sum(x1-1,y1-1);
	}
} fen2d;

ll count_rectangles(vector<vi> a) {
	n = sz(a); m = sz(a[0]); ll ans = 0;
	for(int i = 0; i < n; i++){
		stack<int> S = stack<int>();
		for(int j = m-1; j >= 0; j--){
			while(sz(S) and a[i][S.top()]<a[i][j]) S.pop();
			if(sz(S) and abs(S.top()-j)>1)
				row[i][j].pb(S.top());
			S.push(j);
		}
		S = stack<int>();
		for(int j = 0; j < m; j++){
			while(sz(S) and a[i][S.top()]<a[i][j]) S.pop();
			if(sz(S) and abs(S.top()-j)>1)
				if(a[i][S.top()]!=a[i][j]) 
					row[i][S.top()].pb(j);
			S.push(j);
		}
	}
	for(int j = 0; j < m; j++){
		stack<int> S = stack<int>();
		for(int i = n-1; i >= 0; i--){
			while(sz(S) and a[S.top()][j]<a[i][j]) S.pop();
			if(sz(S) and abs(S.top()-i)>1)
				col[i][j].pb(S.top());
			S.push(i);
		}
		S = stack<int>();
		for(int i = 0; i < n; i++){
			while(sz(S) and a[S.top()][j]<a[i][j]) S.pop();
			if(sz(S) and abs(S.top()-i)>1)
				if(a[S.top()][j]!=a[i][j])
					col[S.top()][j].pb(i);
			S.push(i);
		}
	}
	for(int i = 0; i < n; i++)
		for(int j = 0; j < m; j++)
			sort(all(row[i][j])), sort(all(col[i][j]));
	fen2d.init(n,m);
	for(int i = 1; i < n-1; i++){
		for(int j = 0; j < m; j++)
			for(auto u : col[i-1][j]) 
				grid[u][j]=1, grid2[u].pb(j);
		for(int k = 0; k < n; k++){
			for(auto u : grid2[k]){
				int y = u;
				if(!y or !grid[k][y-1]){
					while(y<m-1 and grid[k][y+1]) y++;
					lines[u].pb({k,y});
				}
			}
		}
		for(auto [x,y] : lines[0]) fen2d.upd(x,y,1);
		for(int j = 0; j < m; j++){
			for(auto [x,y] : lines[j+1]) fen2d.upd(x,y,1);
			for(auto k : row[i][j]){
				int R = i+1;
				if(i>1 and binary_search(all(row[i-1][j]),k)){
					R = found[i-1][j].lower_bound({k,-1})->second;
				}
				else{
					while(R<n and binary_search(all(row[R][j]),k)) R++;
				}
				found[i][j].insert({k,R});
				int xd = fen2d.sum(i+1,k-1,R,mxN-2); ans+=xd;
				
			}
			for(auto u : col[i-1][j]) grid[u][j]=0;
		}
		for(int j = 0; j < m; j++){
			for(auto [x,y] : lines[j])
				fen2d.upd(x,y,-1);
			lines[j].clear(); lines[j].shrink_to_fit();
		}
		for(int k = 0; k < n; k++) grid2[k].clear();
	}
	return ans;
}