Question 72
Determine all $n \times n$ tables of nonnegative integers with rows and columns labelled by $0, 1, 2, \dots, n-1$ such that for each $0 \le i, j \le n-1$, the number in the cell $(i, j)$ is the number of $i$'s of row $j$. One such table can be seen below.
| | column 0 | column 1 | column 2 | column 3 | column 4 |
|--------|----------|----------|----------|----------|----------|
| row 0 | 1 | 0 | 3 | 3 | 4 |
| row 1 | 1 | 3 | 2 | 1 | 1 |
| row 2 | 0 | 1 | 0 | 1 | 0 |
| row 3 | 2 | 1 | 0 | 0 | 0 |
| row 4 | 1 | 0 | 0 | 0 | 0 |
(The value in the cell (2,3) is 1, for there is one “2” in row 3; the value in the cell (4,2) is 0, for there is no “4” in row 2, etc.)
| | column 0 | column 1 | column 2 | column 3 | column 4 |
|--------|----------|----------|----------|----------|----------|
| row 0 | 1 | 0 | 3 | 3 | 4 |
| row 1 | 1 | 3 | 2 | 1 | 1 |
| row 2 | 0 | 1 | 0 | 1 | 0 |
| row 3 | 2 | 1 | 0 | 0 | 0 |
| row 4 | 1 | 0 | 0 | 0 | 0 |
(The value in the cell (2,3) is 1, for there is one “2” in row 3; the value in the cell (4,2) is 0, for there is no “4” in row 2, etc.)