Exercise · ZW-SIGN-OF-A-PERMUTATION-005 · ★★☆ Intermediate

For \(\sigma\in S_4\) with list \([2,4,1,3]\) (that is, \(1\mapsto2\), \(2\mapsto4\), \(3\mapsto1\), \(4\mapsto3\)), the sign is \((-1)^m\) with \(m\) the number of inversions. What is the sign?

Choose one option.

Show answer

B. \(-1\)

Explanation

The inversions are the value pairs \((2,1)\), \((4,1)\) and \((4,3)\): \(m=3\), odd. The sign is \(-1\).

Review the concept: Sign of a permutation

The sign \(\operatorname{sgn}(\sigma)\) of a permutation \(\sigma\) is \(+1\) or \(-1\) according to whether \(\sigma\) is a product of an even or odd number \(k\) of transpositions.

\[\operatorname{sgn}(\sigma)=(-1)^k\]

For \(\sigma\in S_4\) with list \([2,4,1,3]\) (that is, \(1\mapsto2\), \(2\mapsto4\), \(3\mapsto1\), \(4\mapsto3\)), the sign is \((-1)^m\) with \(m\) the number of inversions. What is the sign?

Explanation

Review the concept: Sign of a permutation

More practice problems: sign of a permutation