$1 \cdot 3$ If a decimal positive integer with at least two digits has each of its digits smaller than the digit to its right, it is called an "ascending number". Find the total number of ascending numbers.
[Solution] According to the given information, the digits of an ascending number are all different. Since the first digit cannot be 0, an ascending number can have at most 9 digits. Therefore, the digits of an ascending number must be the numbers in a subset of the set
$$
S=\{1,2,3,4,5,6,7,8,9\}
$$
arranged in ascending order, and the subset must have at least two elements.
Since $S$ has 9 elements, the total number of all its subsets with at least two elements is $2^{9}-C_{9}^{0}-C_{9}^{1}=502$. Therefore, there are 502 ascending numbers.