# Generating Permutations From Factoradic Numbers

This is the title of an interesting tutorial by Prof. Joel Guilherme S. Filho. It describes how to generate permutations of a set of n elements from a given decimal index based on its representation as a “factoradic number”, i.e. a number represented in the mixed base [(n-1)!, (n-2)!, … , 2!, 1!, 0!].

Thanks to the list manipulation facilities and the support for arbitrary long integers provided by the Python programming language, it’s easy to write a short script (shown below) that generates permutations even for large indexes. Please read his tutorial for additional details.

#!/usr/bin/python3
#
# Author: Joao H de A Franco (jhafranco@acm.org)
#
# Description: Generating Permutations From Factoradic Numbers
#
# Date: 2012-02-06
#
#          (CC BY-NC-SA 3.0)
#=============================================================
from sys import argv
from math import factorial

N = int(argv[1])

n = 0
while N > factorial(n):
n += 1

S = [i for i in range(n)]
Factoradic, Permutation = [None] * n, [None] * n

v, s = N, S[:]
for i in range(n):
r, v = divmod(v, factorial(n - i - 1))

print("For N = {:d}\n".format(N))
print("n = {:d}\n".format(n))
print("Permutation[0] = {:s}\n".format(S))
print("Permutation[N] = {:s}\n".format(Permutation))

Executing this script with 51 as the command-line argument, we get the output below:

For N = 51

n = 5

Perm[0] = [0, 1, 2, 3, 4]

Factoradic[N] = [2, 0, 1, 1, 0]

Permutation[N] = [2, 0, 3, 4, 1]

Trying now a big number like the Mersenne number $\small 2^{607} - 1$ (which happens to be the Mersenne prime # 14), we get the following results:

For N = 53113799281676709868958820655246862732959311772703192319944413 ... 29486246501015346579337652707239409519978766587351943831270835393219031728127

n = 113

Permutation[0] = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112]

Factoradic[N] = [2, 77, 30, 88, 71, 102, 97, 70, 4, 12, 75, 71, 27, 9, 91, 43, 75, 58, 30, 73, 10, 11, 68, 33, 77, 79, 50, 80, 41, 77, 48, 5, 32, 78, 25, 74, 0, 9, 49, 3, 43, 14, 6, 20, 2, 6, 61, 14, 29, 7, 37, 41, 5, 15, 30, 54, 26, 0, 41, 19, 29, 50, 6, 6, 2, 25, 6, 8, 44, 10, 26, 13, 16, 0, 14, 4, 5, 24, 24, 30, 22, 6, 6, 29, 10, 24, 12, 23, 3, 20, 15, 20, 16, 11, 12, 7, 2, 15, 10, 5, 5, 8, 6, 7, 2, 0, 0, 1, 0, 1, 0, 1, 0]

Permutation[N] = [2, 78, 31, 91, 73, 107, 102, 72, 5, 14, 82, 77, 30, 11, 104, 49, 87, 65, 36, 88, 13, 16, 84, 42, 98, 101, 61, 106, 52, 103, 60, 7, 43, 111, 34, 108, 0, 17, 69, 6, 63, 24, 12, 35, 4, 18, 99, 28, 53, 20, 67, 75, 15, 37, 58, 105, 54, 1, 85, 45, 64, 110, 22, 23, 9, 62, 26, 32, 112, 39, 74, 46, 51, 3, 50, 25, 29, 83, 86, 96, 80, 38, 40, 109, 55, 94, 59, 95, 21, 92, 76, 97, 81, 66, 70, 47, 19, 100, 71, 44, 48, 79, 57, 89, 27, 8, 10, 41, 33, 68, 56, 93, 90]