Level 5

solution.py

def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n-1)

def counter(lst):
    count_dict = {}
    for item in lst:
        if item in count_dict:
            count_dict[item] += 1
        else:
            count_dict[item] = 1
    return count_dict

def gcd(a, b):
    while b:
        a, b = b, a % b
    return a

def cycle_count(c, n):
    cc = factorial(n)
    for a, b in counter(c).items():
        cc //= (a**b) * factorial(b)
    return cc

def cycle_partitions(n, i=1):
    yield [n]
    for i in range(i, n//2 + 1):
        for p in cycle_partitions(n-i, i):
            yield [i] + p

def solution(w, h, s):
    grid = 0
    for cpw in cycle_partitions(w):
        for cph in cycle_partitions(h):
            m = cycle_count(cpw, w) * cycle_count(cph, h)
            grid += m * (s**sum([sum([gcd(i, j) for i in cpw]) for j in cph]))
    return str(grid // (factorial(w) * factorial(h)))