Calculator Input
Example Data Table
This example shows how each position maps through the permutation and back through its inverse.
| Index i | Original p(i) | Inverse p-1(i) |
|---|---|---|
| 1 | 3 | 2 |
| 2 | 1 | 4 |
| 3 | 4 | 1 |
| 4 | 2 | 3 |
Example: For p = [3, 1, 4, 2], the inverse is p-1 = [2, 4, 1, 3].
Formula Used
An inverse permutation reverses the action of the original permutation.
If p maps index i to value p(i), then the inverse q = p-1 satisfies:
q(p(i)) = i for every index i.
In one-line notation, if p = [p1, p2, ..., pn], then:
p-1(pi) = i
This calculator builds the inverse by placing each index i into position p(i).
How to Use This Calculator
- Enter the permutation in one-line notation.
- Separate values with commas, spaces, or semicolons.
- Make sure the values run from 1 to n.
- Click the calculate button.
- Read the inverse permutation above the form.
- Review cycle form, parity, and verification tables.
- Use the graph for a visual comparison.
- Export the result as CSV or PDF.
Frequently Asked Questions
1. What is an inverse permutation?
It is the permutation that undoes another permutation. When you apply a permutation and then its inverse, every element returns to its original position.
2. What input format should I use?
Use one-line notation, such as 3, 1, 4, 2. The list must contain every integer from 1 through n exactly once.
3. Why does the calculator reject repeated values?
Repeated values break the one-to-one structure required for permutations. Without unique images, the inverse cannot be defined correctly.
4. What does the identity check mean?
It verifies that p-1(p(i)) = i for every position. Passing this check confirms that the computed inverse is valid.
5. Why are cycle forms useful?
Cycle notation shows how values move in loops. It often makes permutation structure easier to understand than one-line notation.
6. What are fixed points?
Fixed points are positions where p(i) = i. These elements stay in place under the permutation and also remain fixed in the inverse.
7. What does parity tell me?
Parity tells whether the permutation has an even or odd number of inversions. This is useful in algebra, combinatorics, and determinant theory.
8. Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet-friendly output or the PDF button for a printable report.