sign w
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.
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The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Permutations/Documentation/mainDocs.m2:679:0.