Step by Step solution about How to resolve the algorithm Permutations/Rank of a permutation step by step in the Julia programming language

The provided Julia code demonstrates the generation and display of permutations and ranks of objects using built-in Julia functions. Here's a detailed explanation:

1. Generating Permutations:

   • nobjs is the number of objects we want to permute.
   • a is a vector containing integers from 1 to nobjs, representing the objects.
   • The nthperm function is used to generate the i-th permutation of the objects in a.
   • A loop iterates through all possible permutations (from 1 to factorial(nobjs)) and prints the permutation and its rank.

2. Calculating Rank:

   • The nthperm function is also used to find the rank of a given permutation.
   • prank represents the rank of the current permutation.

3. Generating Random Permutations:

   • This part of the code generates nsamp random permutations of nobjs objects without repeating any ranks.
   • The randperm function creates a random permutation of the objects.
   • The nthperm function calculates the rank of the random permutation.
   • The loop ensures that no rank is repeated by checking if prank is already in ptaken, an array that stores the ranks of previously generated permutations.
   • If prank is already in ptaken, the code generates a new random permutation until a unique rank is found.

4. Printing Output:

   • The code prints the generated permutations and their ranks.

Overall, this code provides a clear demonstration of how to use Julia's built-in functions to manipulate permutations and ranks, a common operation in combinatorial problems.

using Printf

nobjs = 4
a = collect(1:nobjs)
println("All permutations of ", nobjs, " objects:")
for i in 1:factorial(nobjs)
    p = nthperm(a, i)
    prank = nthperm(p)
    print(@sprintf("%5d => ", i))
    println(p, " (", prank, ")")
end

nobjs = 12
nsamp = 4
ptaken = Int[]
println()
println(nsamp, " random permutations of ", nobjs, " objects:")
for i in 1:nsamp
    p = randperm(nobjs)
    prank = nthperm(p)
    while prank in ptaken
        p = randperm(nobjs)
        prank = nthperm(p)
    end
    push!(ptaken, prank)
    println("         ", p, " (", prank, ")")
end