Cyclic Permutations
Description
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 $1 \leq i \leq n$, find the largest $j$ such that $1 \leq j < i$ and $p_j > p_i$, and add an undirected edge between node $i$ and node $j$
- For every $1 \leq i \leq n$, find the smallest $j$ such that $i < j \leq n$ 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
The first and only line contains a single integer $n$ ($3 \le n \le 10^6$).
Output a single integer $0 \leq x < 10^9+7$, the number of cyclic permutations of length $n$ modulo $10^9+7$.
Input
The first and only line contains a single integer $n$ ($3 \le n \le 10^6$).
Output
Output a single integer $0 \leq x < 10^9+7$, the number of cyclic permutations of length $n$ modulo $10^9+7$.
Samples
4
16
583291
135712853
Note
There are $16$ cyclic permutations for $n = 4$. $[4,2,1,3]$ is one such permutation, having a cycle of length four: $4 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 4$.
Nodes $v_1$, $v_2$, $\ldots$, $v_k$ form a simple cycle if the following conditions hold:
- $k \geq 3$.
- $v_i \neq v_j$ for any pair of indices $i$ and $j$. ($1 \leq i < j \leq k$)
- $v_i$ and $v_{i+1}$ share an edge for all $i$ ($1 \leq i < k$), and $v_1$ and $v_k$ share an edge.