Step by Step solution about How to resolve the algorithm Stirling numbers of the second kind step by step in the Julia programming language

The provided code in Julia programming language computes and displays the Stirling numbers of the second kind, denoted as stirlings2(n, k), for a given range of n and k values. Stirling numbers of the second kind represent the number of ways to partition a set of n elements into k non-empty subsets.

Here's a breakdown of the code:

1. Importing Combinatorics: The Combinatorics module provides mathematical functions and data structures, including functions for combinatorics, such as binomial coefficients.

2. Cache for stirlings2: A dictionary (s2cache) is defined to store previously calculated values of stirlings2 for a given pair of (n, k) values. This caching mechanism optimizes the computation by avoiding recalculations for the same input.

3. Function stirlings2:

   • It calculates the Stirling number of the second kind, stirlings2(n, k), using a recursive formula.
   • If the pair (n, k) is already in the cache, it returns the cached value.
   • It handles special cases where n or k is 0 or k is close to n.
   • The result is stored in the cache and returned.

4. Function printstirling2table: This function prints a table of Stirling numbers of the second kind for specified values of kmax.

   • It first prints the header row with column labels for k values and the subsequent rows display values for corresponding n values.
   • sstring(n, k) is a helper function that formats the cell content based on whether stirlings2(n, k) is non-zero.

5. Printing the Stirling2 Table: The printstirling2table function is called to display a table of Stirling numbers of the second kind for kmax up to 12.

6. Finding the Maximum Value: Finally, the code calculates and prints the maximum value of stirlings2(100, k) for k in the range of 1 to 100. This gives an idea of the largest Stirling number of the second kind for n = 100.

By using caching and efficient recursive calculations, this code efficiently computes and displays Stirling numbers of the second kind for specified ranges of n and k values.

using Combinatorics

const s2cache = Dict()

function stirlings2(n, k)
    if haskey(s2cache, Pair(n, k))
        return s2cache[Pair(n, k)]
    elseif n < 0
        throw(DomainError(n, "n must be nonnegative"))
    elseif n == k == 0
        return one(n)
    elseif n == 0 || k == 0
        return zero(n)
    elseif k == n - 1
        return binomial(n, 2)
    elseif k == 2
        return 2^(n-1) - 1
    end

    ret = k * stirlings2(n - 1, k) + stirlings2(n - 1, k - 1)
    s2cache[Pair(n, k)] = ret
    return ret

end

function printstirling2table(kmax)
    println("  ", mapreduce(i -> lpad(i, 10), *, 0:kmax))

    sstring(n, k) = begin i = stirlings2(n, k); lpad(k > n && i == 0 ? "" : i, 10) end

    for n in 0:kmax
        println(rpad(n, 2) * mapreduce(k -> sstring(n, k), *, 0:kmax))
    end
end

printstirling2table(12)
println("\nThe maximum for stirling2(100, _) is: ", maximum(k-> stirlings2(BigInt(100), BigInt(k)), 1:100))