Codeforces #663 (Div. 2) C. Cyclic Permutations

C. Cyclic Permutations

原题链接：https://codeforces.com/contest/1391/problem/C

time limit per test：1 second memory limit per test：256 megabytes input：standard input output：standard output

A permutation of length $n$ is an array consisting of n distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4] $is a permutation, but $[1,2,2]$ is not a permutation $( 2$ appears twice in the array $)$ and $[1,3,4]$ is also not a permutation $(n=3$ but there is $4$ in the array $)$.

Consider a permutation $p$ of length $n$, we build a graph of size $n$ using it as follows:

For every $1≤i≤n$, find the largest $j$ such that $1≤j<i$ and $p_j>p_i$, and add an undirected edge between node $i$ and node $j$
For every $1≤i≤n$, find the smallest $j$ such that $i<j≤n$ and $p_j>p_i$, and add an undirected edge between node $i$ and node $j$
In cases where no such $j$ exists, we make no edges. Also, note that we make edges between the corresponding indices, not the values at those indices.

For clarity, consider as an example $n=4$, and $p=[3,1,4,2]$; here, the edges of the graph are $(1,3),(2,1),(2,3),(4,3)$.

A permutation $p$ is cyclic if the graph built using $p$ has at least one simple cycle.

Given $n$, find the number of cyclic permutations of length $n$. Since the number may be very large, output it modulo $10^9+7$.

Please refer to the Notes section for the formal definition of a simple cycle

Input
The first and only line contains a single integer $n (3≤n≤10^6)$.

Output
Output a single integer $0≤x<10^9+7$, the number of cyclic permutations of length $n$ modulo $10^9+7$.
Examples
input
4
output
16
input
583291
output
135712853

Note
There are $16$ cyclic permutations for $n=4.$ $[4,2,1,3]$ is one such permutation, having a cycle of length four $: 4→3→2→1→4$.

Nodes $v_1, v_2, …, v_k$ form a simple cycle if the following conditions hold:

• k≥3.
• $v_i≠v_j$ for any pair of indices $i$ and $j$. $(1≤i<j≤k)$
• $v_i$ and $v_{i+1}$ share an edge for all $i (1≤i<k)$, and $v_1$ and $v_k$ share an edge.

(先上代码，题解稍后送上)

#include<iostream>
#include<cstdio>
#include<string>
#include<cstring>
#include<cmath>
#include<algorithm>
#define ll long long
const ll mod = 1e9 + 7;
using namespace std;
ll fun(ll n)// n!
{
	ll ans = 1;
	for (ll i = 1; i <= n; i++)
		ans = ans * i % mod;
	return ans;
}
ll quick_pow(ll a, ll b, ll c) //2^(n-1)快速幂
{
	ll ans = 1;
	a = a % c;
	while (b)
	{
		if (b & 1)
			ans = (ans * a) % c;
		a = (a * a) % c;
		b = b / 2;
	}
	return ans;
}

int main()
{
	ll n;
	cin >> n;
	ll res;
	res = fun(n) - pow2(2, n - 1, mod) + mod;
	printf("%d\n", res % mod);
	return 0;
}