Calculator Input
Example Data Table
| One-line permutation | Cycle notation | Cycle lengths | Order | Parity |
|---|---|---|---|---|
| 2, 3, 1, 5, 4 | (1 2 3)(4 5) | 3, 2 | 6 | Odd |
| 2, 1, 4, 3 | (1 2)(3 4) | 2, 2 | 2 | Even |
| 1, 3, 4, 2, 5 | (2 3 4) | 3 | 3 | Even |
| 3, 1, 2, 5, 4 | (1 3 2)(4 5) | 3, 2 | 6 | Odd |
| 1, 2, 3, 4 | Identity | 1 | 1 | Even |
Formula Used
Let a permutation be written as disjoint cycles:
σ = c₁c₂...cᵣ.
If the cycle lengths are
|c₁|, |c₂|, ..., |cᵣ|,
then the order of the permutation is:
ord(σ) = lcm(|c₁|, |c₂|, ..., |cᵣ|)
This works because each cycle returns to its starting position after its own length many applications. The whole permutation becomes the identity when all cycles reset together, which happens at their least common multiple.
The calculator also reports inversion count and parity:
sign(σ) = (-1)^{inv(σ)}.
How to Use This Calculator
- Choose whether you want to enter one-line notation or cycle notation.
- Type a valid permutation, such as
2, 3, 1, 5, 4or(1 2 3)(4 5). - Add degree
nwhen cycle notation omits extra fixed points. - Press the calculate button to view the order, cycles, parity, inversions, and powers.
- Use the export buttons to save the computed results as CSV or PDF.
FAQs
1) What does the order of a permutation mean?
It is the smallest positive integer k such that applying the permutation k times returns every element to its original position.
2) Why does the calculator use the LCM?
Each disjoint cycle resets after its own length. The full permutation resets when all cycles reset together, so the least common multiple gives the order.
3) Can I use one-line notation here?
Yes. Enter a rearrangement of 1 through n, separated by commas or spaces. The tool validates that every number appears exactly once.
4) Can I use cycle notation instead?
Yes. Enter disjoint cycles like (1 2 3)(4 5). If some fixed points are omitted, provide the degree n so the calculator knows the full permutation size.
5) What is a fixed point?
A fixed point is an element that maps to itself. Fixed points do not change the order, because a 1-cycle contributes a length of 1.
6) What is permutation parity?
Parity tells whether the permutation is even or odd. This tool computes it from the inversion count, where an even number of inversions means even parity.
7) Why does the powers table stop early?
The table displays powers from 0 up to 12 or the order, whichever is smaller. This keeps the output readable while still showing the repetition pattern.
8) What should I do if the input is rejected?
Check that the permutation is valid, uses positive integers, repeats no element, and follows the chosen notation format correctly.