16.2.3 Sign of a Permutation

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: N/A
Constructions: 16.5 Permutation Definition of Determinant
Generalizations: 16.1 Permutation, 16.2 Symmetric Groups

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Sign of a Permutation

Using 16.2.2 Set of All Inversions for a Given pi in Sn, we can create a function $n : S_{n} \to [n]$ such that $n (π)$ gives the number of distinct inversions with respect to $π$ . Using this output, we can define the sign of a permutation $π$ to be $sgn (π) = (- 1)^{n (π)}$ . Another way to think of the sign is:

Positive if there are an even number of inversions with respect to $π$ .
Negative if there are an odd number of inversions with respect to $π$ .