Sign of a permutation

Determine the sign of the following permutation

Variant 1
\(p=(1{,}3,5,…,2n-1{,}2,4{,}6,…,2n)\)
Hint
Count the inversions.
Result
\(sgn(p)=(-1)^{\frac{n(n-1)}{2}}\).
Variant 2
\(p=(1{,}4,7,…,3n-2{,}2,5{,}8,…,3n-1{,}3,6{,}9,…,3n)\)
Result
\(sgn(p)=(-1)^{\frac{3n(n-1)}{2}}=(-1)^{\frac{n(n-1)}{2}}\) because \(3n(n-1)\equiv -n(n-1) \pmod 4\)
Variant 3
\(p=(2{,}5,8,…,3n-1{,}3,6{,}9,…,3n,1{,}4,7,…,3n-2)\)
Result
\(sgn(p)=(-1)^{\frac{n(3n+1)}{2}}=(-1)^{\frac{n(n-1)}{2}}\) because \(3n^2+n \equiv -n^2+n=-n(n-1) \pmod 4\)
Variant 4
\(p=(3{,}6,9,…,3n,2{,}5,8,…,3n-1{,}1,4{,}7,…,3n-2)\)
Result
\(sgn(p)=(-1)^{\frac{3n(n+1)}{2}}=(-1)^{\frac{n(n+1)}{2}}\) because \(3n(n+1)\equiv -n(n+1) \pmod 4\)