You have a cartesian product of a countable number of countable sets and you want to index the elements of that cartesian product. The goal is to find explicit formulas to go either direction in this one-to-one mapping.

space.py

"""
Problem -- you have a bounded space defined by the cardinality of each dimension.
You want to:
    a) Go from coordinates inside that space to an index of inside that space (map_point)
    b) Reverse that (go from an index to a set of coordinates) (unmap_point)

"""
import operator

def get_multipliers(cardinalities):
    multiplier, multipliers = 1, []
    for dimension, cardinality in enumerate(cardinalities):
        multipliers.append(multiplier)
        multiplier *= cardinality
    return multipliers

def map_point(X, cardinalities):
    """X is an n-dimensional input vector. Find its index in space."""
    assert len(X) == len(cardinalities)

    multipliers = get_multipliers(cardinalities)

    i = 0
    for dimension, value in enumerate(X):
        i += multipliers[dimension] * value

    return i

def unmap_point(i, cardinalities):
    """i is an index in a space defined by cardinalities. Return its coordinates"""
    X = []
    N = reduce(operator.mul, cardinalities, 1)  # Max Valid I
    assert i < N

    for cardinality in cardinalities:
        i, value = divmod(i, cardinality)
        X.append(value)

    return X


def test():
    cardinalities = 6, 5
    X = (3, 2)
    i = map_point(X, cardinalities)
    print(X, cardinalities, i, unmap_point(i, cardinalities))


if __name__ == '__main__':
    test()

foldl (\(i, acc) c -> let (i', x) = divMod i c in (i', (x:acc))) (2,[]) [3 :: Int, 4]