Question 27
Write down a row of arbitrary integers (repetitions allowed). Now construct a second row as follows. Suppose the integer $n$ is in column $k$ in the first row. In column $k$ in the second row write down the number of occurrences of $n$ in row 1 in columns 1 to $k$ inclusive. Similarly, construct a third row under the second row (using the values in the second row), and a fourth row. An example follows:
| 7 | 1 | 2 | 1 | 7 | 1 | 1 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 2 | 2 | 3 | 4 |
| 1 | 2 | 3 | 1 | 2 | 1 | 1 |
| 1 | 1 | 1 | 2 | 2 | 3 | 4 |
Show that the fourth row is always the same as the second row.
| 7 | 1 | 2 | 1 | 7 | 1 | 1 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 2 | 2 | 3 | 4 |
| 1 | 2 | 3 | 1 | 2 | 1 | 1 |
| 1 | 1 | 1 | 2 | 2 | 3 | 4 |
Show that the fourth row is always the same as the second row.