We can compute the inverse of a permutation.
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The order of a permutation $p$ is its order in the symmetric group $\mathfrak{S}_n$, i.e., the smallest positive integer $k$ such that $p^k$ is the identity permutation.
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Every permutation can be written as a product of transpositions. One definition for the sign of a permutation $p$ is $1$ if it can be written as a product of an even number of transpositions and is $-1$ if it can be written as an odd number of transpositions. If $\text{sign}(p) = 1$, the permutation is called even and if $\text{sign}(p) = -1 $, it is called pdd.
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A permutation $p$ is a derangement if $p(i) \neq i$ for all $i$. We can determine if $p$ is a derangement as well its fixed points.
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A permutation $p$ has an inversion $(i,j)$ if $i < j$ and $p(i) > p(j)$. We can compute all the inversions of a permutation.
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The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Permutations/Documentation/packageDocs.m2:386:0.