Given a permutation's lexicographic number, is it possible to get any item in it in O(1)

Question

I want to know whether the task explained below is even theoretically possible, and if so how I could do it.

You are given a space of N elements (i.e. all

Answer 1

After some research in Wikipedia, I desgined this algorithm:

def getPick(fact_num_list):
    """fact_num_list should be a list with the factorial number representation, 
    getPick will return a tuple"""
    result = [] #Desired pick
    #This will hold all the numbers pickable; not actually a set, but a list
    #instead
    inputset = range(len(fact_num_list)) 
    for fnl in fact_num_list:
        result.append(inputset[fnl])
        del inputset[fnl] #Make sure we can't pick the number again
    return tuple(result)

Obviously, this won't reach O(1) due the factor we need to "pick" every number. Due we do a for loop and thus, assuming all operations are O(1), getPick will run in O(n).

If we need to convert from base 10 to factorial base, this is an aux function:

import math

def base10_baseFactorial(number):
    """Converts a base10 number into a factorial base number. Output is a list
    for better handle of units over 36! (after using all 0-9 and A-Z)"""
    loop = 1
    #Make sure n! <= number
    while math.factorial(loop) <= number:
        loop += 1
    result = []
    if not math.factorial(loop) == number:
        loop -= 1 #Prevent dividing over a smaller number than denominator
    while loop > 0:
        denominator = math.factorial(loop)
        number, rem = divmod(number, denominator)
        result.append(rem)
        loop -= 1
    result.append(0) #Don't forget to divide to 0! as well!
    return result

Again, this will run in O(n) due to the whiles.

Summing all, the best time we can find is O(n).

PS: I'm not a native English speaker, so spelling and phrasing errors may appear. Apologies in advance, and let me know if you can't get around something.