Thirteen, for $\{1,2,, 3 \cdots, n\}$ and each of its non-empty subsets, we define the alternating sum as follows: arrange the numbers in the subset in descending order, then alternately add and subtract the numbers starting from the largest (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$, and the alternating sum of $\{5\}$ is 5). For $n=7$, find the total of all these alternating sums.
Solution: Obviously, the number of non-empty subsets of $\{1,2,3,4,5,6,7\}$ is $2^{7}-1=127$. Each element appears 64 times in these subsets. According to the problem's requirements, the numbers $1,2,3,4,5,6$ each appear 32 times in odd positions and 32 times in even positions, so the alternating sum of these numbers in the subsets is 0. The number 7 also appears 64 times and always takes a positive value, so the total alternating sum of all subsets is $7 \times 64=448$.