Rook Polynomial Calculator Form
Example Data Table
This example uses the default 4 × 4 board shown in the calculator.
| k | Meaning | Example Output Type |
|---|---|---|
| 0 | No rooks placed | Always contributes 1 |
| 1 | One rook on an available square | Counts all allowed single placements |
| 2 | Two nonattacking rooks | Counts valid pairs on distinct rows and columns |
| 3 | Three nonattacking rooks | Counts valid triples respecting blocked cells |
| 4 | Four nonattacking rooks | Only exists if four placements are feasible |
Formula Used
Rook polynomial: R(x) = Σ rkxk
Coefficient meaning: rk is the number of ways to place exactly k nonattacking rooks on available cells.
Nonattacking rule: No two rooks may share the same row or column.
Degree: The degree equals the largest k for which rk is positive.
Evaluation: For any chosen x-value, compute R(x) = r0 + r1x + r2x2 + ...
How to Use This Calculator
- Enter the number of rows and columns for your board.
- Type the board matrix using 1 for allowed cells.
- Use 0 for blocked cells that cannot hold rooks.
- Choose an x-value if you want polynomial evaluation.
- Click the calculate button to generate coefficients and graph.
- Review the polynomial, degree, and placement counts.
- Export the coefficient table as CSV or PDF.
Frequently Asked Questions
1. What does a rook polynomial represent?
A rook polynomial summarizes how many valid ways exist to place nonattacking rooks on a board with allowed and blocked cells. Each coefficient counts placements of a specific size.
2. What does the coefficient rk mean?
The coefficient rk counts the number of distinct placements of exactly k rooks so that no two share a row or a column.
3. Why is the constant term always 1?
The constant term counts the empty placement, where no rook is placed. There is exactly one way to choose nothing, so r0 is always 1.
4. Why do blocked cells matter?
Blocked cells reduce the number of legal placements. They can lower coefficients and may also reduce the maximum number of rooks that fit without attacks.
5. What is the degree of the rook polynomial?
The degree is the highest number of nonattacking rooks that can be placed on the allowed cells. It equals the largest k with a positive coefficient.
6. Can I use rectangular boards?
Yes. The calculator supports rectangular boards, not only square boards. The practical limit here is 8 rows by 8 columns for stable performance.
7. What does evaluating R(x) help with?
Evaluating the polynomial at a chosen x-value can help compare boards, inspect growth behavior, or build related combinatorial generating-function analyses.
8. How should I format the matrix input?
Enter one board row per line. Use spaces or commas between values. Every row must match the column count, and each cell must be 0 or 1.