# Count permutations of first N natural numbers having sum of adjacent elements equal to a perfect square

Given a positive integer **N**, the task is to find the number of unique permutations of first **N** natural numbers having sum of the adjacent elements equal to a perfect square.

**Examples:**

Input:N = 17Output:2Explanation:

Following permutations have sum of adjacent elements equal to a perfect square:

- {17, 8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16}
- {16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8, 17}
Therefore, count of such permutations is 2.

Input:N = 13Output:0

**Approach:** The given problem can be solved by using the concepts of Graph. Follow the steps below to solve the problem:

- List all the perfect square numbers up to
**(2*N – 1)**that can be obtained by adding any two positive integers. - Represent the graph as the adjacency list representation such that if the sum of two numbers
**X**and**Y**is a perfect square, then add an edge from node**X**to node**Y**. - Count the number of nodes in the graph whose in-degree is 1 and store it in a variable
**X**. - Now, the number of permutation can be calculated as per the following conditions:
- If the value of
**X**is**0**, then a total of**N**permutations are possible. Hence, print**N**as the result. - If the value of
**X**is**1**or**2**, then a total of**2**permutations are possible. Hence, print**2**as the result. - Otherwise, no such permutations exist satisfying the given criteria. Hence, print
**0**as the result.

- If the value of

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to count total number of` `// permutation of the first N natural` `// number having the sum of adjacent` `// elements as perfect square` `int` `countPermutations(` `int` `N)` `{` ` ` `// Create an adjacency matrix` ` ` `vector<vector<` `int` `> > adj(105);` ` ` `// Count elements whose indegree` ` ` `// is 1` ` ` `int` `indeg = 0;` ` ` `// Generate adjacency matrix` ` ` `for` `(` `int` `i = 1; i <= N; i++) {` ` ` `for` `(` `int` `j = 1; j <= N; j++) {` ` ` `if` `(i == j)` ` ` `continue` `;` ` ` `// Find the sum of i and j` ` ` `int` `sum = i + j;` ` ` `// If sum is perfect square.` ` ` `// then move from i to j` ` ` `if` `(` `ceil` `(` `sqrt` `(sum))` ` ` `== ` `floor` `(` `sqrt` `(sum))) {` ` ` `// Add it in adjacency` ` ` `// list of i` ` ` `adj[i].push_back(j);` ` ` `}` ` ` `}` ` ` `// If any list is of size 1,` ` ` `// then the indegree is 1` ` ` `if` `(adj[i].size() == 1)` ` ` `indeg++;` ` ` `}` ` ` `// If there is no element whose` ` ` `// indegree is 1, then N such` ` ` `// permutations are possible` ` ` `if` `(indeg == 0)` ` ` `return` `N;` ` ` `// If there is 1 or 2 elements` ` ` `// whose indegree is 1, then 2` ` ` `// permutations are possible` ` ` `else` `if` `(indeg <= 2)` ` ` `return` `2;` ` ` `// If there are more than 2` ` ` `// elements whose indegree is` ` ` `// 1, then return 0` ` ` `else` ` ` `return` `0;` `}` `// Driver Code` `int` `main()` `{` ` ` `int` `N = 17;` ` ` `cout << countPermutations(N);` ` ` `return` `0;` `}` |

## Java

`// Java program for the above approach` `import` `java.io.*;` `import` `java.util.*;` `import` `java.lang.*;` `class` `GFG{` ` ` `// Function to count total number of` `// permutation of the first N natural` `// number having the sum of adjacent` `// elements as perfect square` `static` `int` `countPermutations(` `int` `N)` `{` ` ` ` ` `// Create an adjacency matrix` ` ` `ArrayList<` ` ` `ArrayList<Integer>> adj = ` `new` `ArrayList<` ` ` `ArrayList<Integer>>(` `105` `);` ` ` ` ` `for` `(` `int` `i = ` `0` `; i < ` `105` `; i++)` ` ` `adj.add(` `new` `ArrayList<Integer>());` ` ` `// Count elements whose indegree` ` ` `// is 1` ` ` `int` `indeg = ` `0` `;` ` ` `// Generate adjacency matrix` ` ` `for` `(` `int` `i = ` `1` `; i <= N; i++)` ` ` `{` ` ` `for` `(` `int` `j = ` `1` `; j <= N; j++)` ` ` `{` ` ` `if` `(i == j)` ` ` `continue` `;` ` ` `// Find the sum of i and j` ` ` `int` `sum = i + j;` ` ` `// If sum is perfect square.` ` ` `// then move from i to j` ` ` `if` `(Math.ceil(Math.sqrt(sum)) == ` ` ` `Math.floor(Math.sqrt(sum)))` ` ` `{` ` ` ` ` `// Add it in adjacency` ` ` `// list of i` ` ` `adj.get(i).add(j);` ` ` `}` ` ` `}` ` ` `// If any list is of size 1,` ` ` `// then the indegree is 1` ` ` `if` `(adj.get(i).size() == ` `1` `)` ` ` `indeg++;` ` ` `}` ` ` `// If there is no element whose` ` ` `// indegree is 1, then N such` ` ` `// permutations are possible` ` ` `if` `(indeg == ` `0` `)` ` ` `return` `N;` ` ` `// If there is 1 or 2 elements` ` ` `// whose indegree is 1, then 2` ` ` `// permutations are possible` ` ` `else` `if` `(indeg <= ` `2` `)` ` ` `return` `2` `;` ` ` `// If there are more than 2` ` ` `// elements whose indegree is` ` ` `// 1, then return 0` ` ` `else` ` ` `return` `0` `;` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `N = ` `17` `;` ` ` ` ` `System.out.println(countPermutations(N));` `}` `}` `// This code is contributed by Dharanendra L V.` |

## Python3

`# python program for the above approach` `from` `math ` `import` `sqrt,floor,ceil` `# Function to count total number of` `# permutation of the first N natural` `# number having the sum of adjacent` `# elements as perfect square` `def` `countPermutations(N):` ` ` `# Create an adjacency matrix` ` ` `adj ` `=` `[[] ` `for` `i ` `in` `range` `(` `105` `)]` ` ` `# bCount elements whose indegree` ` ` `# bis 1` ` ` `indeg ` `=` `0` ` ` `# bGenerate adjacency matrix` ` ` `for` `i ` `in` `range` `(` `1` `, N ` `+` `1` `):` ` ` `for` `j ` `in` `range` `(` `1` `, N ` `+` `1` `):` ` ` `if` `(i ` `=` `=` `j):` ` ` `continue` ` ` `# Find the sum of i and j` ` ` `sum` `=` `i ` `+` `j` ` ` `# If sum is perfect square.` ` ` `# then move from i to j` ` ` `if` `(ceil(sqrt(` `sum` `)) ` `=` `=` `floor(sqrt(` `sum` `))):` ` ` `# Add it in adjacency` ` ` `# list of i` ` ` `adj[i].append(j)` ` ` `# If any list is of size 1,` ` ` `# then the indegree is 1` ` ` `if` `(` `len` `(adj[i]) ` `=` `=` `1` `):` ` ` `indeg ` `+` `=` `1` ` ` `# If there is no element whose` ` ` `# indegree is 1, then N such` ` ` `# permutations are possible` ` ` `if` `(indeg ` `=` `=` `0` `):` ` ` `return` `N` ` ` `# If there is 1 or 2 elements` ` ` `# whose indegree is 1, then 2` ` ` `# permutations are possible` ` ` `elif` `(indeg <` `=` `2` `):` ` ` `return` `2` ` ` `# If there are more than 2` ` ` `# elements whose indegree is` ` ` `# 1, then return 0` ` ` `else` `:` ` ` `return` `0` `# Driver Code` `if` `__name__ ` `=` `=` `'__main__'` `:` ` ` `N ` `=` `17` ` ` `print` `(countPermutations(N))` `# This code is contributed by mohit kumar 29.` |

## C#

`// C# program for the above approach` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG{` ` ` `// Function to count total number of` `// permutation of the first N natural` `// number having the sum of adjacent` `// elements as perfect square` `static` `int` `countPermutations(` `int` `N)` `{` ` ` ` ` `// Create an adjacency matrix` ` ` `List<List<` `int` `>> adj = ` `new` `List<List<` `int` `>>(105);` ` ` ` ` `for` `(` `int` `i = 0; i < 105; i++)` ` ` `adj.Add(` `new` `List<` `int` `>());` ` ` `// Count elements whose indegree` ` ` `// is 1` ` ` `int` `indeg = 0;` ` ` `// Generate adjacency matrix` ` ` `for` `(` `int` `i = 1; i <= N; i++)` ` ` `{` ` ` `for` `(` `int` `j = 1; j <= N; j++)` ` ` `{` ` ` `if` `(i == j)` ` ` `continue` `;` ` ` `// Find the sum of i and j` ` ` `int` `sum = i + j;` ` ` `// If sum is perfect square.` ` ` `// then move from i to j` ` ` `if` `(Math.Ceiling(Math.Sqrt(sum)) == ` ` ` `Math.Floor(Math.Sqrt(sum)))` ` ` `{` ` ` ` ` `// Add it in adjacency` ` ` `// list of i` ` ` `adj[i].Add(j);` ` ` `}` ` ` `}` ` ` `// If any list is of size 1,` ` ` `// then the indegree is 1` ` ` `if` `(adj[i].Count == 1)` ` ` `indeg++;` ` ` `}` ` ` `// If there is no element whose` ` ` `// indegree is 1, then N such` ` ` `// permutations are possible` ` ` `if` `(indeg == 0)` ` ` `return` `N;` ` ` `// If there is 1 or 2 elements` ` ` `// whose indegree is 1, then 2` ` ` `// permutations are possible` ` ` `else` `if` `(indeg <= 2)` ` ` `return` `2;` ` ` `// If there are more than 2` ` ` `// elements whose indegree is` ` ` `// 1, then return 0` ` ` `else` ` ` `return` `0;` `}` `// Driver Code` `public` `static` `void` `Main()` `{` ` ` `int` `N = 17;` ` ` ` ` `Console.WriteLine(countPermutations(N));` `}` `}` `// This code is contributed by SoumikMondal` |

## Javascript

`<script>` `// JavaScript program for the above approach` `// Function to count total number of` `// permutation of the first N natural` `// number having the sum of adjacent` `// elements as perfect square` `function` `countPermutations(N)` `{` ` ` `// Create an adjacency matrix` ` ` `let adj = [];` ` ` ` ` `for` `(let i = 0; i < 105; i++)` ` ` `adj.push([]);` ` ` ` ` `// Count elements whose indegree` ` ` `// is 1` ` ` `let indeg = 0;` ` ` ` ` `// Generate adjacency matrix` ` ` `for` `(let i = 1; i <= N; i++)` ` ` `{` ` ` `for` `(let j = 1; j <= N; j++)` ` ` `{` ` ` `if` `(i == j)` ` ` `continue` `;` ` ` ` ` `// Find the sum of i and j` ` ` `let sum = i + j;` ` ` ` ` `// If sum is perfect square.` ` ` `// then move from i to j` ` ` `if` `(Math.ceil(Math.sqrt(sum)) ==` ` ` `Math.floor(Math.sqrt(sum)))` ` ` `{` ` ` ` ` `// Add it in adjacency` ` ` `// list of i` ` ` `adj[i].push(j);` ` ` `}` ` ` `}` ` ` ` ` `// If any list is of size 1,` ` ` `// then the indegree is 1` ` ` `if` `(adj[i].length == 1)` ` ` `indeg++;` ` ` `}` ` ` ` ` `// If there is no element whose` ` ` `// indegree is 1, then N such` ` ` `// permutations are possible` ` ` `if` `(indeg == 0)` ` ` `return` `N;` ` ` ` ` `// If there is 1 or 2 elements` ` ` `// whose indegree is 1, then 2` ` ` `// permutations are possible` ` ` `else` `if` `(indeg <= 2)` ` ` `return` `2;` ` ` ` ` `// If there are more than 2` ` ` `// elements whose indegree is` ` ` `// 1, then return 0` ` ` `else` ` ` `return` `0;` `}` `// Driver Code` `let N = 17;` `document.write(countPermutations(N));` `// This code is contributed by patel2127` `</script>` |

**Output:**

2

**Time Complexity:** O(N^{2})**Auxiliary Space:** O(N^{2})

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