OMCPLP 2015 Problem 2

Determine all ten-digit numbers whose decimal $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation. That is: the first digit is equal to the amount of " $0$ " in the writing of that number, the second digit is equal to the amount of " $1$ " in the writing of that number, the third digit is equal to the amount of " $2$ " in the writing of that number, ... , the tenth digit is equal to the number of " $9$ " in the writing of that number.