I would also vote for Mosty Mostacho's solution with minor modification to his query code:
SELECT a.i, a.j, (
SELECT count(*) from test b where a.j >= b.j AND a.i = b.i
) AS row_number FROM test a
Which will give the same result:
+------+------+------------+
| i | j | row_number |
+------+------+------------+
| 1 | 11 | 1 |
| 1 | 12 | 2 |
| 1 | 13 | 3 |
| 2 | 21 | 1 |
| 2 | 22 | 2 |
| 2 | 23 | 3 |
| 3 | 31 | 1 |
| 3 | 32 | 2 |
| 3 | 33 | 3 |
| 4 | 14 | 1 |
+------+------+------------+
for the table:
+------+------+
| i | j |
+------+------+
| 1 | 11 |
| 1 | 12 |
| 1 | 13 |
| 2 | 21 |
| 2 | 22 |
| 2 | 23 |
| 3 | 31 |
| 3 | 32 |
| 3 | 33 |
| 4 | 14 |
+------+------+
With the only difference that the query doesn't use JOIN and GROUP BY, relying on nested select instead.