#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;
template<typename T> using Tree = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
typedef long long int ll;
typedef long double ld;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
#define fastio ios_base::sync_with_stdio(false); cin.tie(NULL)
#define pb push_back
#define endl '\n'
#define sz(a) (int)a.size()
#define setbits(x) __builtin_popcountll(x)
#define ff first
#define ss second
#define conts continue
#define ceil2(x,y) ((x+y-1)/(y))
#define all(a) a.begin(), a.end()
#define rall(a) a.rbegin(), a.rend()
#define yes cout << "Yes" << endl
#define no cout << "No" << endl
#define rep(i,n) for(int i = 0; i < n; ++i)
#define rep1(i,n) for(int i = 1; i <= n; ++i)
#define rev(i,s,e) for(int i = s; i >= e; --i)
#define trav(i,a) for(auto &i : a)
template<typename T>
void amin(T &a, T b) {
a = min(a,b);
}
template<typename T>
void amax(T &a, T b) {
a = max(a,b);
}
#ifdef LOCAL
#include "debug.h"
#else
#define debug(x) 42
#endif
/*
read some solutions long back, remember some ideas from there
*/
const int MOD = 1e9 + 7;
const int N = 1e5 + 5;
const int inf1 = int(1e9) + 5;
const ll inf2 = ll(1e18) + 5;
#include "rect.h"
template<typename T>
struct fenwick {
int siz;
vector<T> tree;
fenwick() {
}
fenwick(int n) {
siz = n;
tree = vector<T>(n + 1);
}
int lsb(int x) {
return x & -x;
}
void build(vector<T> &a, int n) {
for (int i = 1; i <= n; ++i) {
int par = i + lsb(i);
tree[i] += a[i];
if (par <= siz) {
tree[par] += tree[i];
}
}
}
void pupd(int i, T v) {
i++;
while (i <= siz) {
tree[i] += v;
i += lsb(i);
}
}
T sum(int i) {
i++;
T res = 0;
while (i) {
res += tree[i];
i -= lsb(i);
}
return res;
}
T query(int l, int r) {
if (l > r) return 0;
T res = sum(r) - sum(l - 1);
return res;
}
};
long long count_rectangles(std::vector<std::vector<int> > a) {
int n = sz(a), m = sz(a[0]);
if(n <= 2 or m <= 2) return 0;
vector<pii> up[n][m];
int ptr = 0;
rep(j,m){
stack<int> st;
rep(i,n){
bool eq = false;
while(!st.empty() and a[i][j] >= a[st.top()][j]){
int k = st.top();
st.pop();
if(a[i][j] == a[k][j]){
eq = true;
}
if(i-k >= 2){
up[i][j].pb({k,ptr++});
}
}
if(!st.empty() and !eq){
int k = st.top();
if(i-k >= 2){
up[i][j].pb({k,ptr++});
}
}
st.push(i);
}
}
vector<int> farthest(ptr);
int seg_ptr[n][n];
memset(seg_ptr,-1,sizeof seg_ptr);
rev(j,m-1,0){
if(j+2 < m){
rep(i,n){
for(auto [k,ptr] : up[i][j+2]){
seg_ptr[k][i] = -1;
}
}
}
if(j+1 < m){
rep(i,n){
for(auto [k,ptr] : up[i][j+1]){
seg_ptr[k][i] = ptr;
}
}
}
rep(i,n){
for(auto [k,ptr] : up[i][j]){
if(seg_ptr[k][i] != -1){
farthest[ptr] = farthest[seg_ptr[k][i]];
}
else{
farthest[ptr] = j;
}
}
}
}
int prev_occ[m][m], first_occ[m][m];
memset(prev_occ,-1,sizeof prev_occ);
memset(first_occ,-1,sizeof first_occ);
ll ans = 0;
vector<int> queries[ptr];
rep1(i,n-2){
stack<int> st;
vector<pii> good;
rep(j,m){
bool eq = false;
while(!st.empty() and a[i][j] >= a[i][st.top()]){
int k = st.top();
st.pop();
if(a[i][j] == a[i][k]){
eq = true;
}
if(j-k >= 2){
good.pb({k,j});
}
}
// cout << j << endl;
// cout << eq << endl;
// auto st_copy = st;
// while(!st_copy.empty()){
// cout << st_copy.top() << " ";
// st_copy.pop();
// }
// cout << endl;
if(!st.empty() and !eq){
int k = st.top();
if(j-k >= 2){
// cout << k << " " << j << endl;
good.pb({k,j});
}
}
st.push(j);
}
for(auto [l,r] : good){
if(prev_occ[l][r] != i-1){
first_occ[l][r] = i;
}
prev_occ[l][r] = i;
int lo = 0, hi = sz(up[i+1][l+1])-1;
int pos = -1;
while(lo <= hi){
int mid = (lo+hi) >> 1;
if(up[i+1][l+1][mid].ff >= first_occ[l][r]-1){
pos = mid;
lo = mid+1;
}
else{
hi = mid-1;
}
}
if(pos != -1){
int ind = up[i+1][l+1][pos].ss;
queries[ind].pb(r);
}
/*
for(auto [k,ptr] : up[i+1][l+1]){
if(k >= first_occ[l][r]-1){
if(farthest[ptr] >= r-1){
ans++;
}
}
else{
break;
}
}
*/
}
}
fenwick<int> fenw(m);
rep(i,n){
rep(j,m){
for(auto [k,ptr] : up[i][j]){
fenw.pupd(farthest[ptr],1);
trav(r,queries[ptr]){
ans += fenw.query(r-1,m-1);
}
}
for(auto [k,ptr] : up[i][j]){
fenw.pupd(farthest[ptr],-1);
}
}
}
/*
int oans = 0;
rep1(i1,n-2){
rep1(j1,m-2){
for(int i2 = i1; i2 < n-1; ++i2){
for(int j2 = j1; j2 < m-1; ++j2){
bool ok = true;
for(int r = i1; r <= i2; ++r){
for(int c = j1; c <= j2; ++c){
if(a[r][c] >= min({a[i1-1][c],a[i2+1][c],a[r][j1-1],a[r][j2+1]})){
ok = false;
break;
}
}
if(!ok) break;
}
if(ok){
oans++;
cout << i1 << " " << j1 << " " << i2 << " " << j2 << endl;
}
}
}
}
}
*/
// cout << ans << endl;
// cout << oans << endl;
// cout << endl;
// assert(ans == oans);
return ans;
}
