Given an $n \times n$ table, where $n \geq 1$, the task is to determine the maximum number of cells $k$ that can be colored such that there is a unique way to place $n$ tokens on the colored cells. Importantly, no two tokens should be in the same row or column.

To approach this problem, consider the constraints:

1. Each row and column must contain exactly one token.
2. The tokens must be placed on the colored cells only.
3. The placement must be unique, meaning there should be exactly one valid arrangement of tokens under the given constraints.

### Step-by-Step Process

1. Understand Permutation Constraint: Placing $n$ tokens such that each token is in a different row and column corresponds to finding a permutation of rows into columns.

2. Coloring Constraint: The colored cells in a row or column define possible positions for placing a token. The uniqueness constraint implies that starting from a row or column results in a deterministic path through the entire grid.

3. **Maximize $k$ for Uniqueness: To ensure a unique arrangement:
- There should be a clear path from one token to another such that any deviation from a set path results in multiple possible token arrangements.
- Conceptually, this means placing tokens in a manner that they form a unique permutation path across the colored cells.

4. Arrange the Coloring**: Consider the possible token placements when maximizing $k$:
* Each row must have precisely one colored cell that a token can occupy.
* Each column must similarly have precisely one colored cell.
* If you were to color more cells, permutations would become possible, violating the uniqueness condition.

5. **Estimate $k$**:
* Available positions correspond to a grid of $n \times n$, with permutations allowed of up to the grid dimension.
* Theoretically, the diagonal from the top left to the bottom right ensures a unique solution, but that uses only $n$ cells.
* Coloring additional cells such that they don’t offer new valid permutations while adhering to the above, results in a balanced use of color.

The maximum number of cells that can be colored to maintain uniqueness without permutation possibilities of the tokens is:
$$k = \frac{n(n+1)}{2}$$

Thus, the maximum value of $k$ for which a unique placement of tokens is achievable is:
$$\boxed{\frac{n(n+1)}{2}}$$