What is summation notation for functions of decreasing integers?

Question

Wondering how may write following expression in sigma notation for summation? \begin{eqnarray} S &=& f(x_1,x_2,\cdots,x_n-1)+f(x_1,x_2,\cdots,x_n-2)+\cdots+f(x_1,x_2,\cdots,0)\\ &&+f(x_1,x_2-1,\cdots,0)+f(x_1,x_2-2,\cdots,0)+\cdots+f(x_1,0,\cdots,0)\\ &&+f(x_1-1,0,\cdots,0)+f(x_1-2,0,\cdots,0)+\cdots+f(0,0,\cdots,0) \end{eqnarray} $\forall x_i \in \mathbb{N}$

Example: \begin{eqnarray} S &=& f(3,3,2)+f(3,3,1)+\cdots+f(3,3,0)\\ &&+f(3,2,0)+f(3,1,0)+\cdots+f(3,0,0)\\ &&+f(2,0,0)+f(1,0,0)+\cdots+f(0,0,0) \end{eqnarray}

Answer 1

I would suggest:

$$S = \sum_{i=1}^n \sum_{t=1}^{x_i} f(x_1,..., x_{i-1}, x_i - t,0,...,0)$$