如果所有顺时针排列都被认为是相同的，则计算排列N个不同对象的方式

给定N个不同的对象，如果所有顺时针排列都被认为是相同的，则任务是找到N个对象的不同排列的数量。

If A, B, and C are three distinct objects, then arrangements {A, B, C}, {C, A, B}, and {B, C, A} are considered the same as all the arrangements are clockwise to each other.

例子：

Input: N = 3
Output: 2
Explanation:
Permutation of 3 distinct objects can be represented as these 6 arrangements: {A, B, C}, {A, C, B}, {B, A, C}, {B, C, A}, {C, A, B} and {C, B, A}. But arrangement {A, B, C}, {C, A, B} and {B, C, A} are considered the same.
The arrangements {A, C, B}, {B, A, C} and {C, B, A} are also considered the same.
Hence, there are only 2 distinct arrangements.

Input: N = 2
Output: 1
Explanation:
There can only 2 arrangements: {A, B} and {B, A}. Both these arrangements are same considering the clockwise arrangements.
Hence, only 1 distinct arrangement exist.

天真的方法：这个想法是生成N的所有排列 不同的对象。现在，如果先前布置中不存在当前布置的顺时针旋转，则迭代所有布置并增加计数器。检查完所有布置后，将计数器打印为可能的总的独特顺时针布置。

时间复杂度： O(N * N！)
辅助空间： O(N * N！)

高效的方法：该想法是使用以下观察：

Given a permutation {A₁, A₂, …, A_N} of N distinct objects. Then, there can be N clockwise rotated arrangements denotes as follows:

{A₁, A₂, …, A_N}
{A_N, A₁, …, A_N-1}
{A_N-1, A_N, …, A_N-2}
{A_N-2, A_N-1, …, A_N-3}
and so on.

Therefore, for each permutation of length N there exist N clockwise arrangements. Also, for N distinct objects, there are N! arrangements without considering the clockwise arrangements same.

Hence, N*X = N!, where
N the clockwise arrangements of any permutation of length N.
X is the number of permutations considering clockwise arrangements the same.
N! is the total number of permutations.

Using the above equation:
X = N! / N
X = (N – 1)!

因此，考虑顺时针排列的不同排列的总数为(N – 1)！ 。因此，任务是打印阶乘(N – 1)的值。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
using namespace std;
 
// Function to return the factorial
// of given number
unsigned int factorial(unsigned int n)
{
    // Base case
    if (n == 0)
        return 1;
 
    // Recursively find the factorial
    return n * factorial(n - 1);
}
 
// Driver Code
int main()
{
    // Given N objects
    int N = 4;
 
    // Function Call
    cout << factorial(N - 1);
 
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
 
class GFG{
     
// Function to return the factorial
// of given number
static int factorial(int n)
{
     
    // Base case
    if (n == 0)
        return 1;
 
    // Recursively find the factorial
    return n * factorial(n - 1);
}
 
// Driver Code
public static void main (String[] args)
{
     
    // Given N objects
    int N = 4;
 
    // Function Call
    System.out.print(factorial(N - 1));
}
}
 
// This code is contributed by math_lover

Python3

# Python3 program for the above approach
 
# Function to return the factorial
# of a given number
def factorial(n):
 
    # Base Case
    if n == 0:
        return 1
     
    # Recursively find the factorial
    return n * factorial(n-1)
 
 
# Driver Code
 
# Given N objects
N = 4
 
# Function Call
print(factorial(N - 1))

C#

// C# program for the
// above approach
using System;
class GFG{
     
// Function to return
// the factorial
// of given number
static int factorial(int n)
{   
  // Base case
  if (n == 0)
    return 1;
 
  // Recursively find the
  // factorial
  return n * factorial(n - 1);
}
 
// Driver Code
public static void Main(String[] args)
{   
  // Given N objects
  int N = 4;
 
  // Function Call
  Console.Write(factorial(N - 1));
}
}
 
// This code is contributed by gauravrajput1

输出

6

时间复杂度： O(N)
辅助空间： O(1)