Lobb Number

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++

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// 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;
}

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Java

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// 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.

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Python 3

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# 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

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C#

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// 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.

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PHP

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<?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.
?>

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Output :

35


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