Algorithm for finding numerical permutation given lexicographic index

Question

I am looking for an algorithm that given a set of numbers (for example 1 2 3) and an index (for example 2) will get me the second permutation of those numbers according to a

Answer 1

Here is a simple solution:

from math import factorial # python math library

i = 5               # i is the lexicographic index (counting starts from 0)
n = 3               # n is the length of the permutation
p = range(1, n + 1) # p is a list from 1 to n

for k in range(1, n + 1): # k goes from 1 to n
    f = factorial(n - k)  # compute factorial once per iteration
    d = i // f            # use integer division (like division + floor)
    print(p[d]),          # print permuted number with trailing space
    p.remove(p[d])        # delete p[d] from p
    i = i % f             # reduce i to its remainder

Output:

3 2 1

The time complexity is O(n^2) if p is a list, and O(n) amortized if p is a hash table and factorial is precomputed.