# Number of Permutations such that no Three Terms forms Increasing Subsequence

• Difficulty Level : Expert
• Last Updated : 13 May, 2021

Given a number N. The task is to find the number of permutations of 1 to N such that no three terms of the permutation form an increasing subsequence.

Examples:

```Input : N = 3
Output : 5
Valid permutations : 132, 213, 231, 312 and 321 and not 123

Input : N = 4
Output : 14 ```

The above problem is an application of Catalan numbers. So, the task is to only find the nth Catalan Number. First few Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, â€¦ (considered from 0th number)

Below is the program to find Nth Catalan Number:

## C++

 `// C++ program to find the``// nth catalan number``#include ``using` `namespace` `std;` `// Returns value of Binomial Coefficient C(n, k)``unsigned ``long` `int` `binomialCoeff(unsigned ``int` `n,``                                ``unsigned ``int` `k)``{``    ``unsigned ``long` `int` `res = 1;` `    ``// Since C(n, k) = C(n, n-k)``    ``if` `(k > n - k)``        ``k = n - k;` `    ``// Calculate value of``    ``// [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]``    ``for` `(``int` `i = 0; i < k; ++i) {``        ``res *= (n - i);``        ``res /= (i + 1);``    ``}` `    ``return` `res;``}` `// A Binomial coefficient based function``// to find nth catalan``// number in O(n) time``unsigned ``long` `int` `catalan(unsigned ``int` `n)``{``    ``// Calculate value of 2nCn``    ``unsigned ``long` `int` `c = binomialCoeff(2 * n, n);` `    ``// return 2nCn/(n+1)``    ``return` `c / (n + 1);``}` `// Driver code``int` `main()``{``    ``int` `n = 3;` `    ``cout << catalan(n) << endl;` `    ``return` `0;``}`

## Java

 `// Java program to find the``// nth catalan number``import` `java.io.*;` `class` `GFG``{` `// Returns value of Binomial``// Coefficient C(n, k)``static` `long` `binomialCoeff(``long` `n, ``long` `k)``{``    ``long` `res = ``1``;` `    ``// Since C(n, k) = C(n, n-k)``    ``if` `(k > n - k)``        ``k = n - k;` `    ``// Calculate value of``    ``// [n*(n-1)*---*(n-k+1)] /``    ``// [k*(k-1)*---*1]``    ``for` `(``int` `i = ``0``; i < k; ++i)``    ``{``        ``res *= (n - i);``        ``res /= (i + ``1``);``    ``}` `    ``return` `res;``}` `// A Binomial coefficient based``// function to find nth catalan``// number in O(n) time``static` `long` `catalan(``long` `n)``{``    ``// Calculate value of 2nCn``    ``long` `c = binomialCoeff(``2` `* n, n);` `    ``// return 2nCn/(n+1)``    ``return` `c / (n + ``1``);``}` `// Driver code``public` `static` `void` `main (String[] args)``{``    ``int` `n = ``3``;``    ` `    ``System.out.println(catalan(n));``}``}` `// This code has been contributed``// by inder_verma.`

## Python3

 `# Python program to find the``# nth catalan number` `# Returns value of Binomial``# Coefficient C(n, k)``def` `binomialCoeff(n, k):``    ``res ``=` `1``    ` `    ``# Since C(n, k) = C(n, n-k)``    ``if` `k > n ``-` `k:``        ``k``=``n``-``k``    ``# Calculate value of``    ``# [n*(n-1)*---*(n-k+1)] //``    ``# [k*(k-1)*---*1]` `    ``for` `i ``in` `range``(k):``        ``res ``=` `res ``*` `(n ``-` `i)``        ``res ``=` `res ``/``/` `(i ``+` `1``)``    ``return` `res``    ` `# A Binomial coefficient based``# function to find nth catalan``# number in O(n) time``def` `catalan(n):``    ` `    ``# Calculate value of 2nCn``    ``c ``=` `binomialCoeff(``2` `*` `n, n)``    ` `    ``# return 2nCn/(n+1)``    ``return` `c ``/``/` `(n ``+` `1``)``    ` `# Driver code``n ``=` `3``print``(catalan(n))` `# This code is contributed``# by sahil shelangia`

## C#

 `// C# program to find the``// nth catalan number``using` `System;` `class` `GFG``{` `// Returns value of Binomial``// Coefficient C(n, k)``static` `long` `binomialCoeff(``long` `n,``                          ``long` `k)``{``    ``long` `res = 1;` `    ``// Since C(n, k) = C(n, n-k)``    ``if` `(k > n - k)``        ``k = n - k;` `    ``// Calculate value of``    ``// [n*(n-1)*---*(n-k+1)] /``    ``// [k*(k-1)*---*1]``    ``for` `(``int` `i = 0; i < k; ++i)``    ``{``        ``res *= (n - i);``        ``res /= (i + 1);``    ``}` `    ``return` `res;``}` `// A Binomial coefficient based``// function to find nth catalan``// number in O(n) time``static` `long` `catalan(``long` `n)``{``    ``// Calculate value of 2nCn``    ``long` `c = binomialCoeff(2 * n, n);` `    ``// return 2nCn/(n+1)``    ``return` `c / (n + 1);``}` `// Driver code``public` `static` `void` `Main (String[] args)``{``    ``int` `n = 3;``    ` `    ``Console.WriteLine(catalan(n));``}``}` `// This code is contributed``// by Kirti_Mangal`

## PHP

 ```\$n` `- ``\$k``)``        ``\$k` `= ``\$n` `- ``\$k``;``        ` `    ``// Calculate value of``    ``// [n*(n-1)*---*(n-k+1)] //``    ``// [k*(k-1)*---*1]``    ``for` `(``\$i` `= 0; ``\$i` `< ``\$k``; ``\$i``++)``    ``{``        ``\$res` `= ``\$res` `* (``\$n` `- ``\$i``);``        ``\$res` `= ``\$res` `/ (``\$i` `+ 1);``    ``}``    ``return` `\$res``;``}` `// A Binomial coefficient based``// function to find nth catalan``// number in O(n) time``function` `catalan(``\$n``)``{``    ``// Calculate value of 2nCn``    ``\$c` `= binomialCoeff(2 * ``\$n``, ``\$n``);``    ` `    ``// return 2nCn/(n+1)``    ``return` `\$c` `/ (``\$n` `+ 1);``}` `// Driver code``\$n` `= 3;``print``(catalan(``\$n``));` `// This code is contributed``// by mits``?>`

## Javascript

 ``
Output:
`5`

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