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Count number of Special Set
• Difficulty Level : Hard
• Last Updated : 16 Oct, 2019

An ordered set of integers is said to be a special set if for every element of the set X, the set does not contain the element X + 1. Given an integer N, the task is to find the number of special sets whose largest element is not greater than N. Since, the number of special sets can be very large, print the answer modulo 109 + 7.

Example:

Input: N = 3
Output: 5
{1}, {2}, {3}, {1, 3} and {3, 1} are the
only special sets possible.

Input: N = 4
Output: 10

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: This problem can be solved using dynamic programming. Create an array dp[][] where dp[i][j] stores the number of special sets of length i ending with j. Now, the recurrence relation will be:

dp[i][j] = dp[i – 1] + dp[i – 1] + … + dp[i – 1][j – 2]
dp[i][j] can be computed in O(1) by taking the prefix sum of the previous dp[i – 1] once.

Now the total special sets of size i can be calculated by multiplying dp[i][n] with factorial(i).

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;`` ` `typedef` `long` `long` `ll;`` ` `const` `int` `MAX = 2 * 1000 + 10;``const` `int` `MOD = 1e9 + 7;`` ` `// To store the states of the dp``ll dp[MAX][MAX];`` ` `// Function to return (a + b) % MOD``ll sum(ll a, ll b)``{``    ``return` `((a % MOD) + (b % MOD)) % MOD;``}`` ` `// Function to return (a * b) % MOD``ll mul(ll a, ll b)``{``    ``return` `((a % MOD) * (b % MOD)) % MOD;``}`` ` `// Function to return the count``// of special sets``int` `cntSpecialSet(``int` `n)``{`` ` `    ``// Fill the dp[][] array with the answer``    ``// for the special sets of size 1``    ``for` `(``int` `i = 1; i <= n; i++) {``        ``dp[i] = 1;`` ` `        ``// Take prefix sum of the current row which``        ``// will be used to fill the next row``        ``dp[i] += dp[i - 1];``    ``}`` ` `    ``// Fill the rest of the dp[][] array``    ``for` `(``int` `i = 2; i <= n; i++) {`` ` `        ``// Recurrence relation``        ``for` `(``int` `j = 2; j <= n; j++) {``            ``dp[i][j] = dp[i - 1][j - 2];``        ``}`` ` `        ``// Calculate the prefix sum``        ``for` `(``int` `j = 1; j <= n; j++) {``            ``dp[i][j] = sum(dp[i][j], dp[i][j - 1]);``        ``}``    ``}`` ` `    ``ll ways(1), ans(0);`` ` `    ``for` `(``int` `i = 1; i <= n; i++) {`` ` `        ``// To find special set of size i``        ``ways = mul(ways, i);`` ` `        ``// Addition of special sets of all sizes``        ``ans = sum(ans, mul(ways, dp[i][n]));``    ``}`` ` `    ``return` `ans;``}`` ` `// Driver code``int` `main()``{``    ``int` `n = 3;`` ` `    ``cout << cntSpecialSet(n);`` ` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;`` ` `class` `GFG ``{``static` `int` `MAX = ``2` `* ``1000` `+ ``10``;``static` `int` `MOD = (``int``) (1e9 + ``7``);`` ` `// To store the states of the dp``static` `long` `[][]dp = ``new` `long``[MAX][MAX];`` ` `// Function to return (a + b) % MOD``static` `long` `sum(``long` `a, ``long` `b)``{``    ``return` `((a % MOD) + (b % MOD)) % MOD;``}`` ` `// Function to return (a * b) % MOD``static` `long` `mul(``long` `a, ``long` `b)``{``    ``return` `((a % MOD) * (b % MOD)) % MOD;``}`` ` `// Function to return the count``// of special sets``static` `long` `cntSpecialSet(``int` `n)``{`` ` `    ``// Fill the dp[][] array with the answer``    ``// for the special sets of size 1``    ``for` `(``int` `i = ``1``; i <= n; i++) ``    ``{``        ``dp[``1``][i] = ``1``;`` ` `        ``// Take prefix sum of the current row which``        ``// will be used to fill the next row``        ``dp[``1``][i] += dp[``1``][i - ``1``];``    ``}`` ` `    ``// Fill the rest of the dp[][] array``    ``for` `(``int` `i = ``2``; i <= n; i++) ``    ``{`` ` `        ``// Recurrence relation``        ``for` `(``int` `j = ``2``; j <= n; j++) ``        ``{``            ``dp[i][j] = dp[i - ``1``][j - ``2``];``        ``}`` ` `        ``// Calculate the prefix sum``        ``for` `(``int` `j = ``1``; j <= n; j++) ``        ``{``            ``dp[i][j] = sum(dp[i][j], dp[i][j - ``1``]);``        ``}``    ``}`` ` `    ``long` `ways = ``1``, ans = ``0``;``  ` `    ``for` `(``int` `i = ``1``; i <= n; i++) ``    ``{`` ` `        ``// To find special set of size i``        ``ways = mul(ways, i);`` ` `        ``// Addition of special sets of all sizes``        ``ans = sum(ans, mul(ways, dp[i][n]));``    ``}`` ` `    ``return` `ans;``}`` ` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``3``;`` ` `    ``System.out.println(cntSpecialSet(n));``}``}`` ` `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 implementation of the approach`` ` `# Function to print the nodes having ``# maximum and minimum degree ``def` `minMax(edges, leng, n) : `` ` `    ``# Map to store the degrees of every node ``    ``m ``=` `{};``     ` `    ``for` `i ``in` `range``(leng) :``        ``m[edges[i][``0``]] ``=` `0``;``        ``m[edges[i][``1``]] ``=` `0``;``         ` `    ``for` `i ``in` `range``(leng) :``         ` `        ``# Storing the degree for each node``        ``m[edges[i][``0``]] ``+``=` `1``;``        ``m[edges[i][``1``]] ``+``=` `1``; `` ` `    ``# maxi and mini variables to store ``    ``# the maximum and minimum degree ``    ``maxi ``=` `0``; ``    ``mini ``=` `n; `` ` `    ``for` `i ``in` `range``(``1``, n ``+` `1``) :``        ``maxi ``=` `max``(maxi, m[i]); ``        ``mini ``=` `min``(mini, m[i]); `` ` `    ``# Printing all the nodes ``    ``# with maximum degree ``    ``print``(``"Nodes with maximum degree : "``, ``                                ``end ``=` `"")``     ` `    ``for` `i ``in` `range``(``1``, n ``+` `1``) :``        ``if` `(m[i] ``=``=` `maxi) :``            ``print``(i, end ``=` `" "``); `` ` `    ``print``()`` ` `    ``# Printing all the nodes ``    ``# with minimum degree ``    ``print``(``"Nodes with minimum degree : "``, ``                                ``end ``=` `"")``     ` `    ``for` `i ``in` `range``(``1``, n ``+` `1``) :``        ``if` `(m[i] ``=``=` `mini) :``            ``print``(i, end ``=` `" "``); `` ` `# Driver code ``if` `__name__ ``=``=` `"__main__"` `: `` ` `    ``# Count of nodes and edges ``    ``n ``=` `4``; m ``=` `6``; `` ` `    ``# The edge list ``    ``edges ``=` `[[ ``1``, ``2` `], [ ``1``, ``3` `], ``             ``[ ``1``, ``4` `], [ ``2``, ``3` `], ``             ``[ ``2``, ``4` `], [ ``3``, ``4` `]]; `` ` `    ``minMax(edges, m, ``4``); `` ` `# This code is contributed by AnkitRai01`

## C#

 `// C# implementation of the approach``using` `System;``     ` `class` `GFG ``{``     ` `static` `int` `MAX = 2 * 1000 + 10;``static` `int` `MOD = (``int``) (1e9 + 7);`` ` `// To store the states of the dp``static` `long` `[,]dp = ``new` `long``[MAX, MAX];`` ` `// Function to return (a + b) % MOD``static` `long` `sum(``long` `a, ``long` `b)``{``    ``return` `((a % MOD) + (b % MOD)) % MOD;``}`` ` `// Function to return (a * b) % MOD``static` `long` `mul(``long` `a, ``long` `b)``{``    ``return` `((a % MOD) * (b % MOD)) % MOD;``}`` ` `// Function to return the count``// of special sets``static` `long` `cntSpecialSet(``int` `n)``{`` ` `    ``// Fill the dp[,] array with the answer``    ``// for the special sets of size 1``    ``for` `(``int` `i = 1; i <= n; i++) ``    ``{``        ``dp[1, i] = 1;`` ` `        ``// Take prefix sum of the current row which``        ``// will be used to fill the next row``        ``dp[1, i] += dp[1, i - 1];``    ``}`` ` `    ``// Fill the rest of the dp[,] array``    ``for` `(``int` `i = 2; i <= n; i++) ``    ``{`` ` `        ``// Recurrence relation``        ``for` `(``int` `j = 2; j <= n; j++) ``        ``{``            ``dp[i, j] = dp[i - 1, j - 2];``        ``}`` ` `        ``// Calculate the prefix sum``        ``for` `(``int` `j = 1; j <= n; j++) ``        ``{``            ``dp[i, j] = sum(dp[i, j], dp[i, j - 1]);``        ``} ``    ``}`` ` `    ``long` `ways = 1, ans = 0;`` ` `    ``for` `(``int` `i = 1; i <= n; i++) ``    ``{`` ` `        ``// To find special set of size i``        ``ways = mul(ways, i);`` ` `        ``// Addition of special sets of all sizes``        ``ans = sum(ans, mul(ways, dp[i, n]));``    ``}`` ` `    ``return` `ans;``}`` ` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `n = 3;`` ` `    ``Console.WriteLine(cntSpecialSet(n));``}``}`` ` `// This code is contributed by Princi Singh`
Output:
```5
```

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