How to use the permutation cycle notation calculator
- Enter a permutation in cycle notation, like (1 2 3)(4 5), or in one-line notation, like 2 3 1 5 4. Points are numbered from 1.
- Points left out of cycle notation stay fixed. The size of the permutation is the largest point you mention.
- Optionally enter a second permutation to compose with the first, and a power to raise the first to.
- Read the cycle and one-line forms, the cycle type, the order, the sign, the inverse, the power and the compositions.
Formula
A cycle (a b c) sends a to b, b to c and c back to a
Order = least common multiple of the cycle lengths
Sign = (-1)^(n - number of cycles), so a cycle of length L counts L - 1 swaps; even is +1 and odd is -1
Composition: the first permutation is applied first, then the second
Every permutation splits into disjoint cycles in exactly one way, apart from the order of the cycles and the starting point of each.
The permutation (1 2 3)(4 5)
In one-line form this permutation is 2 3 1 5 4. It has a cycle of length 3 and one of length 2, so its cycle type is 3 + 2, its order is lcm(3, 2) = 6 and its sign is (-1)^(5 - 2) = -1, so it is odd. It has 3 inversions and no fixed points. Its inverse is (1 3 2)(4 5), and its square is (1 3 2), because the 2-cycle squares to the identity.
Composing (1 2 3) first and then (3 4) gives (1 2 4 3). Composing them the other way, (3 4) first and then (1 2 3), gives (1 2 3 4). The one-line permutation 3 1 2 5 4 6 has the cycles (1 3 2)(4 5) with point 6 fixed, and its cube is (4 5).
Why cycles are useful
Cycle notation makes the structure of a permutation visible. The order tells how many times you must repeat it before everything is back in place, and the sign tells whether it can be built from an even or odd number of swaps. Both are central in group theory, in the solution of puzzles such as the Rubik's cube and in the definition of the determinant.
To count the permutations of a set, use the combinations and permutations calculator. When some items repeat, see the multiset permutation calculator.
Conventions to watch
Books disagree on whether the left or the right permutation is applied first when two are written side by side. The page applies the first one first and shows both orders, so you can match the convention of your text. Points start at 1, and you can write a cycle with commas as well as spaces.
The order of a permutation is the least common multiple of its cycle lengths, which you can explore with the GCF and LCM calculator.
Assumptions and limits
- Points are whole numbers from 1 to 200, and each appears at most once in cycle notation.
- One-line notation must use every number from 1 to n exactly once.
- Composition applies the first permutation first. Both orders are shown.
- A negative power uses the inverse of the first permutation.
Frequently asked questions
How do I find the order of a permutation?
Split it into cycles and take the least common multiple of their lengths. A 3-cycle and a 2-cycle give an order of 6.
What does the sign of a permutation mean?
It is +1 if the permutation is a product of an even number of swaps and -1 if odd. A cycle of length L is odd when L is even.
What is the difference between cycle and one-line notation?
One-line notation lists where each point goes. Cycle notation groups the points that rotate into each other, and leaves out the fixed points.
Does the order of composition matter?
Yes, in general. The page shows both orders because permutations usually do not commute.