I’m working on a Python3 code challenge where a message is encoded on or decoded from the shuffling order of a 52-card standard playing deck. I have already found how to encode the message, but I am having difficulty reversing the function to get a message from a given shuffling order. I have a function that finds a lexicographic permutation of a list of cards based on its index. I need one now that takes a permutation from a list of cards and outputs its index. I have some ideas, but I’m not as good at number theory as I should be. I have put some comments with my ideas.

deck = ["AC", "2C", "3C", "4C", "5C", "6C", "7C", "8C", "9C", "TC", "JC", "QC", "KC",
        "AD", "2D", "3D", "4D", "5D", "6D", "7D", "8D", "9D", "TD", "JD", "QD", "KD",
        "AH", "2H", "3H", "4H", "5H", "6H", "7H", "8H", "9H", "TH", "JH", "QH", "KH",
        "AS", "2S", "3S", "4S", "5S", "6S", "7S", "8S", "9S", "TS", "JS", "QS", "KS",]

def get_nth_permutation( p_index, in_list, out_list=[] ):

    if p_index >= factorial( len(in_list) ): return []
    if not in_list: return out_list

    f = factorial( len(in_list)-1 )
    index = p_index / f # p_index * f
    remainder = p_index % f # pow(p_index, -1, f) - reverse modulo?
    # reverse divmod by adding index + remainder in each step to increase p_index?
    out_list.append( in_list[int(index)] )
    del in_list[int(index)]

    if not remainder: # function should end when out_list + in_list matches deck
        return out_list + in_list # return p_index
        
    else: #keep recursion if possible
        return get_nth_permutation( remainder, in_list, out_list ) 

Any help is appreciated. It doesn’t even have to be code, even a pseudocode explanation or some next steps is better than where I am right now.