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Lobb Number

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In combinatorial mathematics, the Lobb number Lm, n counts the number of ways that n + m open parentheses can be arranged to form the start of a valid sequence of balanced parentheses. 
The Lobb number are parameterized by two non-negative integers m and n with n >= m >= 0. It can be obtained by: 
L_{m,n} = \frac{2\times m + 1}{m + n + 1}\binom{2\times n}{m + n}
Lobb Number is also used to count the number of ways in which n + m copies of the value +1 and n – m copies of the value -1 may be arranged into a sequence such that all of the partial sums of the sequence are non- negative.
Examples : 

Input : n = 3, m = 2
Output : 5

Input : n =5, m =3
Output :35

The idea is simple, we use a function that computes binomial coefficients for given values. Using this function and above formula, we can compute Lobb numbers. 

C++

// CPP Program to find Ln, m Lobb Number.
#include <bits/stdc++.h>
#define MAXN 109
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    int C[n + 1][k + 1];
 
    // Calculate value of Binomial Coefficient in
    // bottom up manner
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= min(i, k); j++) {
            // Base Cases
            if (j == 0 || j == i)
                C[i][j] = 1;
 
            // Calculate value using previously stored values
            else
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
        }
    }
 
    return C[n][k];
}
 
// Return the Lm, n Lobb Number.
int lobb(int n, int m)
{
    return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
}
 
// Driven Program
int main()
{
    int n = 5, m = 3;
    cout << lobb(n, m) << endl;
    return 0;
}

                    

Java

// JAVA Code For Lobb Number
import java.util.*;
 
class GFG {
     
    // Returns value of Binomial
    // Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
        int C[][] = new int[n + 1][k + 1];
      
        // Calculate value of Binomial
        // Coefficient in bottom up manner
        for (int i = 0; i <= n; i++) {
            for (int j = 0; j <= Math.min(i, k);
                                        j++) {
                // Base Cases
                if (j == 0 || j == i)
                    C[i][j] = 1;
      
                // Calculate value using
                // previously stored values
                else
                    C[i][j] = C[i - 1][j - 1] +
                              C[i - 1][j];
            }
        }
      
        return C[n][k];
    }
     
    // Return the Lm, n Lobb Number.
    static int lobb(int n, int m)
    {
        return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) /
                                             (m + n + 1);
    }
     
    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int n = 5, m = 3;
        System.out.println(lobb(n, m));
         
    }
}
 
// This code is contributed by Arnav Kr. Mandal.

                    

Python 3

# Python 3 Program to find Ln,
# m Lobb Number.
 
# Returns value of Binomial
# Coefficient C(n, k)
def binomialCoeff(n, k):
 
    C = [[0 for j in range(k + 1)]
             for i in range(n + 1)]
 
 
    # Calculate value of Binomial
    # Coefficient in bottom up manner
    for i in range(0, n + 1):
        for j in range(0, min(i, k) + 1):
            # Base Cases
            if (j == 0 or j == i):
                C[i][j] = 1
 
            # Calculate value using
            # previously stored values
            else:
                C[i][j] = (C[i - 1][j - 1]
                            + C[i - 1][j])
         
    return C[n][k]
 
# Return the Lm, n Lobb Number.
def lobb(n, m):
 
    return (((2 * m + 1) *
        binomialCoeff(2 * n, m + n))
                      / (m + n + 1))
 
# Driven Program
n = 5
m = 3
print(int(lobb(n, m)))
 
# This code is contributed by
# Smitha Dinesh Semwal

                    

C#

// C# Code For Lobb Number
using System;
 
class GFG {
 
    // Returns value of Binomial
    // Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
         
        int[, ] C = new int[n + 1, k + 1];
 
        // Calculate value of Binomial
        // Coefficient in bottom up manner
        for (int i = 0; i <= n; i++) {
            for (int j = 0; j <= Math.Min(i, k);
                j++) {
                     
                // Base Cases
                if (j == 0 || j == i)
                    C[i, j] = 1;
 
                // Calculate value using
                // previously stored values
                else
                    C[i, j] = C[i - 1, j - 1]
                                + C[i - 1, j];
            }
        }
 
        return C[n, k];
    }
 
    // Return the Lm, n Lobb Number.
    static int lobb(int n, int m)
    {
        return ((2 * m + 1) * binomialCoeff(
                 2 * n, m + n)) / (m + n + 1);
    }
 
    /* Driver program to test above function */
    public static void Main()
    {
        int n = 5, m = 3;
         
        Console.WriteLine(lobb(n, m));
    }
}
 
// This code is contributed by vt_m.

                    

PHP

<?php
// PHP Program to find Ln,
// m Lobb Number.
 
$MAXN =109;
 
// Returns value of Binomial
// Coefficient C(n, k)
function binomialCoeff($n, $k)
{
    $C= array(array());
 
    // Calculate value of Binomial
    // Coefficient in bottom up manner
    for ($i = 0; $i <= $n; $i++)
    {
        for ($j = 0; $j <= min($i, $k); $j++)
        {
            // Base Cases
            if ($j == 0 || $j == $i)
                $C[$i][$j] = 1;
 
            // Calculate value using p
            // reviously stored values
            else
                $C[$i][$j] = $C[$i - 1][$j - 1] +
                             $C[$i - 1][$j];
        }
    }
 
    return $C[$n][$k];
}
 
// Return the Lm, n Lobb Number.
function lobb($n, int $m)
{
    return ((2 * $m + 1) *
             binomialCoeff(2 * $n, $m + $n)) /
                          ($m + $n + 1);
}
 
// Driven Code
$n = 5;$m = 3;
echo lobb($n, $m);
 
// This code is contributed by anuj_67.
?>

                    

Javascript

<script>
// javascript code for Lobb Number
 
    // Returns value of Binomial
    // Coefficient C(n, k)
    function binomialCoeff(n, k)
    {
        let C = new Array(n + 1);
         
        // Loop to create 2D array using 1D array
        for (var i = 0; i < C.length; i++) {
            C[i] = new Array(2);
        }
        
        // Calculate value of Binomial
        // Coefficient in bottom up manner
        for (let i = 0; i <= n; i++) {
            for (let j = 0; j <= Math.min(i, k);
                                        j++) {
                // Base Cases
                if (j == 0 || j == i)
                    C[i][j] = 1;
        
                // Calculate value using
                // previously stored values
                else
                    C[i][j] = C[i - 1][j - 1] +
                              C[i - 1][j];
            }
        }
        
        return C[n][k];
    }
       
    // Return the Lm, n Lobb Number.
    function lobb(n, m)
    {
        return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) /
                                             (m + n + 1);
    }
       
// Driver code
 
    let n = 5, m = 3;
    document.write(lobb(n, m));
     
    // This code is contributed by sanjoy_62.
</script>

                    

Output
35

Time Complexity: O(2*n*(m+n))
Auxiliary Space: O((2*n)*(m+n))
 
Efficient approach: Space optimization

In previous approach the current value dp[i][j] is only depend upon the current and previous row values of DP. So to optimize the space complexity we use a single 1D array to store the computations.

Implementation steps:

  • Create a 1D vector C of size K+1.
  • Set a base case by initializing the values of C.
  • Now iterate over subproblems by the help of nested loop and get the current value from previous computations.
  • At last return and print the final answer stored in C[K].

Implementation:

C++

// CPP Program to find Ln, m Lobb Number.
#include <bits/stdc++.h>
#define MAXN 109
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    int C[k+1];
    memset(C, 0, sizeof(C));
    C[0] = 1;  // nC0 is 1
     
    // Calculate value of Binomial Coefficient
    for (int i = 1; i <= n; i++)
    {
        for (int j = min(i, k); j > 0; j--)
            C[j] = C[j] + C[j-1];
    }
     
    //return final answer
    return C[k];
}
 
// Return the Lm, n Lobb Number.
int lobb(int n, int m)
{
    return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
}
 
// Driven Program
int main()
{
    int n = 5, m = 3;
     
    // function call
    cout << lobb(n, m) << endl;
    return 0;
}

                    

Java

import java.util.Arrays;
 
public class LobbNumber {
 
    // Returns value of Binomial Coefficient C(n, k)
    static int binomialCoeff(int n, int k) {
        int[] C = new int[k + 1];
        Arrays.fill(C, 0);
        C[0] = 1; // nC0 is 1
 
        // Calculate value of Binomial Coefficient
        for (int i = 1; i <= n; i++) {
            for (int j = Math.min(i, k); j > 0; j--) {
                C[j] = C[j] + C[j - 1];
            }
        }
 
        //return final answer
        return C[k];
    }
 
    // Return the Lm, n Lobb Number.
    static int lobb(int n, int m) {
        return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
    }
 
    // Driven Program
    public static void main(String[] args) {
        int n = 5, m = 3;
 
        // function call
        System.out.println(lobb(n, m));
    }
}

                    

Python3

# Returns value of Binomial Coefficient C(n, k)
def binomialCoeff(n, k):
    C = [0] * (k+1)
    C[0] = 1  # nC0 is 1
 
    # Calculate value of Binomial Coefficient
    for i in range(1, n+1):
        j = min(i, k)
        while j > 0:
            C[j] = C[j] + C[j-1]
            j -= 1
 
    # return final answer
    return C[k]
 
# Return the Lm, n Lobb Number.
 
 
def lobb(n, m):
    return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) // (m + n + 1)
 
 
# Driven Program
if __name__ == "__main__":
    n = 5
    m = 3
 
    # function call
    print(lobb(n, m))

                    

C#

using System;
 
public class Program
{
    // Returns value of Binomial Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
        int[] C = new int[k + 1];
        Array.Fill(C, 0);
        C[0] = 1; // nC0 is 1
 
        // Calculate value of Binomial Coefficient
        for (int i = 1; i <= n; i++)
        {
            for (int j = Math.Min(i, k); j > 0; j--)
                C[j] = C[j] + C[j - 1];
        }
 
        //return final answer
        return C[k];
    }
 
    // Return the Lm, n Lobb Number.
    static int lobb(int n, int m)
    {
        return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
    }
 
    // Driven Program
    public static void Main()
    {
        int n = 5, m = 3;
 
        // function call
        Console.WriteLine(lobb(n, m));
    }
}

                    

Javascript

function binomialCoeff(n, k) {
let C = new Array(k + 1).fill(0);
C[0] = 1; // nC0 is 1
 
// Calculate value of Binomial Coefficient
for (let i = 1; i <= n; i++) {
for (let j = Math.min(i, k); j > 0; j--)
C[j] = C[j] + C[j - 1];
}
 
//return final answer
return C[k];
}
 
function lobb(n, m) {
return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
}
 
// Driven Program
let n = 5, m = 3;
console.log(lobb(n, m));

                    

Output
35

Time Complexity: O(n^2)
Auxiliary Space: O(k)



Last Updated : 15 May, 2023
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