4. If the digits $a_{i}(i=1,2, \cdots, 9)$ satisfy
$$
a_{9}a_{4}>\cdots>a_{1} \text {, }
$$

then the nine-digit positive integer $\overline{a_{9} a_{8} \cdots a_{1}}$ is called a "nine-digit peak number", for example 134698752. Then, the number of all nine-digit peak numbers is . $\qquad$
4. 11875 .

From the conditions, we know that the middle number of the nine-digit mountain number can only be 9, 8, 7, 6, 5.
When the middle number is 9, there are $\mathrm{C}_{8}^{4} \mathrm{C}_{9}^{4}$ nine-digit mountain numbers; when the middle number is 8, there are $\mathrm{C}_{7}^{4} \mathrm{C}_{8}^{4}$ nine-digit mountain numbers; when the middle number is 7, there are $\mathrm{C}_{6}^{4} \mathrm{C}_{7}^{4}$ nine-digit mountain numbers; when the middle number is 6, there are $\mathrm{C}_{5}^{4} \mathrm{C}_{6}^{4}$ nine-digit mountain numbers; when the middle number is 5, there are $\mathrm{C}_{4}^{4} \mathrm{C}_{5}^{4}$ nine-digit mountain numbers. Therefore, the total number of nine-digit mountain numbers is 11875.