InversePermutation

The inverse $\sigma^{-1}$ of a permutation, in whichever notation it was given: a one-line word back to a one-line word, Cycles back to Cycles.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageInversePermutation!↗

InversePermutation(perm)$\sigma^{-1}$, in the same notation as perm

Typed signatures

Type	Package	Applies
(expression<Cycles> | list<integer>) -> expression<Cycles> | list<integer>	groupalgebra	always

Domain: Permutations

Details

• For a one-line word, $\sigma^{-1}(\sigma(i)) = i$ for every $i$
• For Cycles, each cycle $(i_1\,i_2\,\dots\,i_k)$ inverts to $(i_1\,i_k\,\dots\,i_2)$ — reversed but still starting at $i_1$

Examples

Basic #

live ↗

In$\mathrm{InversePermutation}(\bigl\lbrack2, 5, 3, 6, 1, 8, 7, 9, 4, 10\bigr\rbrack)$Out$\bigl\lbrack5, 1, 3, 9, 2, 4, 7, 6, 8, 10\bigr\rbrack$

Disjoint transpositions are their own inverse

live ↗

In$\mathrm{InversePermutation}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)$

live ↗

In$\mathrm{InversePermutation}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack3, 2, 5, 1\bigr\rbrack, \bigl\lbrack4, 7\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack3, 1, 5, 2\bigr\rbrack, \bigl\lbrack4, 7\bigr\rbrack\bigr\rbrack)$

Scope #

live ↗

In$\mathrm{InversePermutation}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 5\bigr\rbrack, \bigl\lbrack4, 6, 8, 9\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 5, 2\bigr\rbrack, \bigl\lbrack4, 9, 8, 6\bigr\rbrack\bigr\rbrack)$

Each cycle reverses; large points need no dense list

live ↗

In$\mathrm{InversePermutation}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 100, 10\,000\bigr\rbrack, \bigl\lbrack1\,000\,000, 100\,000\,000\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 10\,000, 100\bigr\rbrack, \bigl\lbrack1\,000\,000, 100\,000\,000\bigr\rbrack\bigr\rbrack)$

About

Examples	5
Checked against	Wolfram Language 4 agree1 differ
Crosswalk	Wolfram Language

See also: Cycles, PermutationCycles, Permute