#include <bits/stdc++.h>
#include <sys/types.h>
#include <unistd.h>

#define _overload(_1,_2,_3,name,...) name
#define _rep(i,n) _range(i,0,n)
#define _range(i,a,b) for(int i=int(a);i<int(b);++i)
#define rep(...) _overload(__VA_ARGS__,_range,_rep,)(__VA_ARGS__)

#define _rrep(i,n) _rrange(i,n,0)
#define _rrange(i,a,b) for(int i=int(a)-1;i>=int(b);--i)
#define rrep(...) _overload(__VA_ARGS__,_rrange,_rrep,)(__VA_ARGS__)

#define _all(arg) begin(arg),end(arg)
#define uniq(arg) sort(_all(arg)),(arg).erase(unique(_all(arg)),end(arg))
#define getidx(ary,key) lower_bound(_all(ary),key)-begin(ary)
#define clr(a,b) memset((a),(b),sizeof(a))
#define bit(n) (1LL<<(n))
#define popcount(n) (__builtin_popcountll(n))

using namespace std;

template<class T>bool chmax(T &a, const T &b) { return (a<b)?(a=b,1):0;}
template<class T>bool chmin(T &a, const T &b) { return (b<a)?(a=b,1):0;}

using ll=long long;

class ModInt {
public:
	static unsigned MOD;
	ModInt(): x(0) {}
	ModInt(signed y) : x(y >= 0 ? y % MOD : MOD - (-y) % MOD) {}
	ModInt(signed long long y) : x(y >= 0 ? y % MOD : MOD - (-y) % MOD) {}


	// Arithmetic Oprators
	ModInt &operator+=(ModInt that) {
		if ((x += that.x) >= MOD) x -= MOD; 
		return *this;
	}

	ModInt &operator-=(ModInt that) {
		if ((x += MOD - that.x) >= MOD) x -= MOD; 
		return *this;
	}
	ModInt &operator*=(ModInt that) {
		x = 1LL * x * that.x % MOD;
		return *this;
	}

	ModInt &operator+=(const int that) { return *this += ModInt(that);}
	ModInt &operator-=(const int that) { return *this -= ModInt(that);}
	ModInt &operator*=(const int that) { return *this *= ModInt(that);}

	// Comparators
	bool operator <(ModInt that) { return x < that.x; }
	bool operator >(ModInt that) { return x > that.x; }
	bool operator<=(ModInt that) { return x <= that.x; }
	bool operator>=(ModInt that) { return x >= that.x; }
	bool operator!=(ModInt that) { return x != that.x; }
	bool operator==(ModInt that) { return x == that.x; }

	// Utilities
	unsigned getval() const { return x;}
	ModInt operator+(ModInt that) const { return ModInt(*this) += that;}
	ModInt operator-(ModInt that) const { return ModInt(*this) -= that;}
	ModInt operator*(ModInt that) const { return ModInt(*this) *= that;}
	ModInt operator+(const int that) const { return ModInt(*this) += that;}
	ModInt operator-(const int that) const { return ModInt(*this) -= that;}
	ModInt operator*(const int that) const { return ModInt(*this) *= that;}
	ModInt operator=(const int that) { return *this = ModInt(that);}
	friend istream &operator>>(istream& is, ModInt &that) {ll tmp; is >> tmp; that = ModInt(tmp);return is;}
	friend ostream &operator<<(ostream& os, const ModInt &that) { return os << that.x; }
	
	ModInt power(ll n) const {
		ll b = 1LL, a = x;
		while(n){
			if(n & 1) b = b * a % MOD;
			a = a * a % MOD;
			n >>= 1;
		}
		return ModInt(b);
	}

	ModInt inv() const {
		ll n = MOD - 2, b = 1LL, a = x;
		while(n){
			if(n & 1) b = b * a % MOD;
			a = a * a % MOD;
			n >>= 1;
		}
		return ModInt(b);
	}
private:
	unsigned x;
};

unsigned ModInt::MOD = 1000000007;
using mint = ModInt;
const mint ZERO = mint(0);
const mint ONE = mint(1);
const mint TWO = mint(2);

template <int depth> struct Binary_indexed_tree {
	using T = mint;

	const static int h = depth;
	const static int n = 1 << h;
	T data[n];

	void init() {
		fill_n(data, n, ZERO);
	}

	void update(int i, T x) {
		for (i++; i < n; i += i & -i) {
			data[i] += x;
		}
	}

	T query(int i) {
		T ret = 0;
		for (i++; i > 0; i -= i & -i) ret += data[i];
		return ret;
	}
};


int main(void){
	int n;
	cin >> n;
	
	vector<int> a(n);
	vector<int> c(n + 1, 0);
	
	rep(i, n) {
		cin >> a[i];
		c[a[i]]++;
	}

	rrep(i, n, 1) {
		c[i] += c[i + 1];
	}

	rep(i, 1, n + 1) {
		c[i] -= n - i;
	}

	mint s = ONE;

	rep(i, 1, n + 1) {
		if (c[i] <= 0) {
			puts("0");
			exit(0);
		}

		s *= mint(c[i]);
	}


	mint ans = ZERO;

	vector<int> z(n + 1, 0);
	vector<mint> d(n + 1, ZERO);
	Binary_indexed_tree<18> dsum, dinvsum, anum;

	d[0] = ONE;

	rep(i, 1, n + 1) {
		z[i] = z[i - 1];	
		d[i] = d[i - 1];

		if (c[i] == 1) {
			z[i]++;
		} else {
			d[i] *= mint(c[i] - 1) * (mint(c[i]).inv());
		}
	}


	rep(i, n) {
		const int from = lower_bound(begin(z) + 1, end(z), z[a[i]]) - begin(z);
		ans += s * d[a[i]] * (dinvsum.query(a[i]) - dinvsum.query(from)) * TWO.inv();
		ans += s * (mint(i) - anum.query(a[i]));
		const int to = lower_bound(begin(z) + 1, end(z), z[a[i]] + 1) - begin(z) - 1;
		// cerr << from << " " << a[i] << " " << to << endl;
		ans -= s * (dsum.query(to) - dsum.query(a[i])) * d[a[i]].inv() * TWO.inv();
		
		anum.update(a[i], 1);
		dsum.update(a[i], d[a[i]]);
		dinvsum.update(a[i], d[a[i]].inv());
	}

	cout << ans << endl;

	return 0;
}
### Prompt

Generate a CPP solution to the following problem:

Snuke has an integer sequence A whose length is N. He likes permutations of (1, 2, ..., N), P, that satisfy the following condition:

* P_i \leq A_i for all i ( 1 \leq i \leq N ).



Snuke is interested in the inversion numbers of such permutations. Find the sum of the inversion numbers over all permutations that satisfy the condition. Since this can be extremely large, compute the sum modulo 10^9+7.

Constraints

* 1 \leq N \leq 2 \times 10^5
* 1 \leq A_i \leq N ( 1 \leq i \leq N )
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N
A_1 A_2 ... A_N


Output

Print the sum of the inversion numbers over all permutations that satisfy the condition.

Examples

Input

3
2 3 3


Output

4


Input

6
4 2 5 1 6 3


Output

7


Input

5
4 4 4 4 4


Output

0


Input

30
22 30 15 20 10 29 11 29 28 11 26 10 18 28 22 5 29 16 24 24 27 10 21 30 29 19 28 27 18 23


Output

848414012

### Response

```cpp
#include <bits/stdc++.h>
#include <sys/types.h>
#include <unistd.h>

#define _overload(_1,_2,_3,name,...) name
#define _rep(i,n) _range(i,0,n)
#define _range(i,a,b) for(int i=int(a);i<int(b);++i)
#define rep(...) _overload(__VA_ARGS__,_range,_rep,)(__VA_ARGS__)

#define _rrep(i,n) _rrange(i,n,0)
#define _rrange(i,a,b) for(int i=int(a)-1;i>=int(b);--i)
#define rrep(...) _overload(__VA_ARGS__,_rrange,_rrep,)(__VA_ARGS__)

#define _all(arg) begin(arg),end(arg)
#define uniq(arg) sort(_all(arg)),(arg).erase(unique(_all(arg)),end(arg))
#define getidx(ary,key) lower_bound(_all(ary),key)-begin(ary)
#define clr(a,b) memset((a),(b),sizeof(a))
#define bit(n) (1LL<<(n))
#define popcount(n) (__builtin_popcountll(n))

using namespace std;

template<class T>bool chmax(T &a, const T &b) { return (a<b)?(a=b,1):0;}
template<class T>bool chmin(T &a, const T &b) { return (b<a)?(a=b,1):0;}

using ll=long long;

class ModInt {
public:
	static unsigned MOD;
	ModInt(): x(0) {}
	ModInt(signed y) : x(y >= 0 ? y % MOD : MOD - (-y) % MOD) {}
	ModInt(signed long long y) : x(y >= 0 ? y % MOD : MOD - (-y) % MOD) {}


	// Arithmetic Oprators
	ModInt &operator+=(ModInt that) {
		if ((x += that.x) >= MOD) x -= MOD; 
		return *this;
	}

	ModInt &operator-=(ModInt that) {
		if ((x += MOD - that.x) >= MOD) x -= MOD; 
		return *this;
	}
	ModInt &operator*=(ModInt that) {
		x = 1LL * x * that.x % MOD;
		return *this;
	}

	ModInt &operator+=(const int that) { return *this += ModInt(that);}
	ModInt &operator-=(const int that) { return *this -= ModInt(that);}
	ModInt &operator*=(const int that) { return *this *= ModInt(that);}

	// Comparators
	bool operator <(ModInt that) { return x < that.x; }
	bool operator >(ModInt that) { return x > that.x; }
	bool operator<=(ModInt that) { return x <= that.x; }
	bool operator>=(ModInt that) { return x >= that.x; }
	bool operator!=(ModInt that) { return x != that.x; }
	bool operator==(ModInt that) { return x == that.x; }

	// Utilities
	unsigned getval() const { return x;}
	ModInt operator+(ModInt that) const { return ModInt(*this) += that;}
	ModInt operator-(ModInt that) const { return ModInt(*this) -= that;}
	ModInt operator*(ModInt that) const { return ModInt(*this) *= that;}
	ModInt operator+(const int that) const { return ModInt(*this) += that;}
	ModInt operator-(const int that) const { return ModInt(*this) -= that;}
	ModInt operator*(const int that) const { return ModInt(*this) *= that;}
	ModInt operator=(const int that) { return *this = ModInt(that);}
	friend istream &operator>>(istream& is, ModInt &that) {ll tmp; is >> tmp; that = ModInt(tmp);return is;}
	friend ostream &operator<<(ostream& os, const ModInt &that) { return os << that.x; }
	
	ModInt power(ll n) const {
		ll b = 1LL, a = x;
		while(n){
			if(n & 1) b = b * a % MOD;
			a = a * a % MOD;
			n >>= 1;
		}
		return ModInt(b);
	}

	ModInt inv() const {
		ll n = MOD - 2, b = 1LL, a = x;
		while(n){
			if(n & 1) b = b * a % MOD;
			a = a * a % MOD;
			n >>= 1;
		}
		return ModInt(b);
	}
private:
	unsigned x;
};

unsigned ModInt::MOD = 1000000007;
using mint = ModInt;
const mint ZERO = mint(0);
const mint ONE = mint(1);
const mint TWO = mint(2);

template <int depth> struct Binary_indexed_tree {
	using T = mint;

	const static int h = depth;
	const static int n = 1 << h;
	T data[n];

	void init() {
		fill_n(data, n, ZERO);
	}

	void update(int i, T x) {
		for (i++; i < n; i += i & -i) {
			data[i] += x;
		}
	}

	T query(int i) {
		T ret = 0;
		for (i++; i > 0; i -= i & -i) ret += data[i];
		return ret;
	}
};


int main(void){
	int n;
	cin >> n;
	
	vector<int> a(n);
	vector<int> c(n + 1, 0);
	
	rep(i, n) {
		cin >> a[i];
		c[a[i]]++;
	}

	rrep(i, n, 1) {
		c[i] += c[i + 1];
	}

	rep(i, 1, n + 1) {
		c[i] -= n - i;
	}

	mint s = ONE;

	rep(i, 1, n + 1) {
		if (c[i] <= 0) {
			puts("0");
			exit(0);
		}

		s *= mint(c[i]);
	}


	mint ans = ZERO;

	vector<int> z(n + 1, 0);
	vector<mint> d(n + 1, ZERO);
	Binary_indexed_tree<18> dsum, dinvsum, anum;

	d[0] = ONE;

	rep(i, 1, n + 1) {
		z[i] = z[i - 1];	
		d[i] = d[i - 1];

		if (c[i] == 1) {
			z[i]++;
		} else {
			d[i] *= mint(c[i] - 1) * (mint(c[i]).inv());
		}
	}


	rep(i, n) {
		const int from = lower_bound(begin(z) + 1, end(z), z[a[i]]) - begin(z);
		ans += s * d[a[i]] * (dinvsum.query(a[i]) - dinvsum.query(from)) * TWO.inv();
		ans += s * (mint(i) - anum.query(a[i]));
		const int to = lower_bound(begin(z) + 1, end(z), z[a[i]] + 1) - begin(z) - 1;
		// cerr << from << " " << a[i] << " " << to << endl;
		ans -= s * (dsum.query(to) - dsum.query(a[i])) * d[a[i]].inv() * TWO.inv();
		
		anum.update(a[i], 1);
		dsum.update(a[i], d[a[i]]);
		dinvsum.update(a[i], d[a[i]].inv());
	}

	cout << ans << endl;

	return 0;
}
```