Cycle Decomposition Tool Calculator

Calculator Inputs

Supports one-line, mapping pairs, and cycle notation.

Example Data Table

Sample permutation on n = 8, shown in one-line and cycle forms.

Sample input

[2 3 1 5 4 6 8 7]

Disjoint cycles

(1 2 3)(4 5)(7 8)

Order = lcm(3,2,2) = 6, parity is even.

Formula Used

Cycle construction: start at the smallest unused element a, follow a → σ(a) → σ²(a) until returning to a. Each loop is one disjoint cycle.
Order: the order of σ is the least common multiple of all cycle lengths: ord(σ) = lcm(ℓ₁, ℓ₂, …).
Parity and sign: a cycle of length ℓ equals (ℓ−1) transpositions, so the sign is (-1)^{Σ(ℓ−1)}. Even gives +1, odd gives −1.
Inverse: σ^-1(σ(i)) = i, so the inverse mapping is built by swapping inputs and outputs.
Power: within a cycle (a₀ a₁ … a_ℓ−1), σ^k(a_j) = a_{(j+k) mod ℓ}.

How to Use This Calculator