import math
import numpy as np

def solution(w, h, s):
    return str(sum([s**cycle for cycle in generate_cycles(w, h)])/(math.factorial(w)*math.factorial(h)))

def generate_cycles(w, h):
    identity = np.arange(w*h).reshape((w, h))
    for i in generate_permutation(list(range(w))):
        for j in generate_permutation(list(range(h))):
            yield count_cycle(identity[i][:, j])

def generate_permutation(numbers):
    if len(numbers) == 1:
        yield numbers
    else:
        for i, n in enumerate(numbers):
            for j in generate_permutation(numbers[:i] + numbers[i+1:]):
                yield [n] + j

def count_cycle(arr):
    cycle = np.zeros(arr.shape, dtype=int)
    w, h = arr.shape
    cycle_id = 0
    for i in range(w):
        for j in range(h):
            x, y = i, j
            if cycle[x, y] == 0:
                cycle_id += 1
            while cycle[x, y] == 0:
                cycle[x, y] = cycle_id
                x, y = arr[x, y]/h, arr[x, y]%h
    return cycle_id

tests = [
        ([2, 3, 4], '430'),
        ([2, 2, 2], '7'),
        ([1, 1, 2], '2'),
        ([1, 1, 3], '3'),
        ([2, 1, 2], '3'),
        ([1, 2, 2], '3'),
        ([3, 1, 2], '4'),
        ([1, 3, 2], '4'),
        ([2, 3, 2], '13'),
        ]

for i, o in tests:
    result = solution(*i)
    print (i, result == o, result, o)