Finding all paths down stairs?

Question

I was given the following problem in an interview:

Given a staircase with N steps, you can go up with 1 or 2 steps each time. Output all possible way

Answer 1

Your solution sounds right.

S(n):
    If n = 1 return {1}
    If n = 2 return {2, (1,1)}
    Return S(n-1)x{1} U S(n-2)x{2}

(U is Union, x is Cartesian Product)

Memoizing this is trivial, and would make it O(Fib(n)).