Suppose that $ n $ and $ k $ to be non-negative integers with $ k < n $.
Let $S(\{0,1 \})$ the number of integer solutions $ y = (y_1, \dotsc, y_{n-1 + k}) $ with $ y_j \in \{1,0 \}$ and $ j = 1,\dotsc, n -1 + k $ of the equation $ y_1+\dotso+y_{n+k-1}=k. $ Let $ S (\mathbb{N}) $ the number of integer solutions $ x = (x_1, \dotsc, x_{n}) $ with $ x_i \in \mathbb{N} $ and $ i = 1,\dotsc, n $ of the equation $ x_1+\dotso+x_n=k. $ Prove that $ S (\mathbb{N})=S(\{0,1 \}) $.
Edited 1. There is the "algorithm" with bars and stars (see eg Wikipedia) that "proves" this equality. But the purpose of this question is simply to give a rigorous mathematical justification for this algorithm.
Edited 2. I am interested in a demonstration of the level set theory. I would be satisfied with an explicit bijection between $ S (\mathbb{N})$ and $S(\{0,1 \}) $.