Step by Step solution about How to resolve the algorithm Bell numbers step by step in the C++ programming language

The provided code is a C++ program that generates and prints Bell numbers. Bell numbers represent the number of ways to partition a set of n distinct elements into non-empty subsets. The code utilizes dynamic programming to efficiently compute Bell numbers for a given input size. A brief explanation of the code:

1. It includes the necessary headers for input/output and multiprecision arithmetic.

2. The make_bell_triangle function is defined with an integer n as an argument. It constructs a 2D vector named bell, where each row represents a Bell number.

3. The function initializes the first row of the bell triangle with a single 1.

4. It populates the remaining rows of the triangle using dynamic programming. For each row, it calculates the corresponding Bell number based on the values in the previous row. Specifically, it adds the values from the previous row to obtain the values in the current row.

5. The main function calls the make_bell_triangle function to compute Bell numbers for a specified size, which is defined in the program.

6. It prints the first 15 Bell numbers and the 50th Bell number (if the Boost library is available).

7. Finally, it prints the first 10 rows of the Bell triangle, where each row represents a set of Bell numbers.

The output of the program will display the Bell numbers up to the specified size and the first 10 rows of the Bell triangle.

#include <iostream>
#include <vector>

#ifdef HAVE_BOOST
#include <boost/multiprecision/cpp_int.hpp>
typedef boost::multiprecision::cpp_int integer;
#else
typedef unsigned int integer;
#endif

auto make_bell_triangle(int n) {
    std::vector<std::vector<integer>> bell(n);
    for (int i = 0; i < n; ++i)
        bell[i].assign(i + 1, 0);
    bell[0][0] = 1;
    for (int i = 1; i < n; ++i) {
        std::vector<integer>& row = bell[i];
        std::vector<integer>& prev_row = bell[i - 1];
        row[0] = prev_row[i - 1];
        for (int j = 1; j <= i; ++j)
            row[j] = row[j - 1] + prev_row[j - 1];
    }
    return bell;
}

int main() {
#ifdef HAVE_BOOST
    const int size = 50;
#else
    const int size = 15;
#endif
    auto bell(make_bell_triangle(size));
    
    const int limit = 15;
    std::cout << "First " << limit << " Bell numbers:\n";
    for (int i = 0; i < limit; ++i)
        std::cout << bell[i][0] << '\n';

#ifdef HAVE_BOOST
    std::cout << "\n50th Bell number is " << bell[49][0] << "\n\n";
#endif

    std::cout << "First 10 rows of the Bell triangle:\n";
    for (int i = 0; i < 10; ++i) {
        std::cout << bell[i][0];
        for (int j = 1; j <= i; ++j)
            std::cout << ' ' << bell[i][j];
        std::cout << '\n';
    }
    return 0;
}