Elementary properties

Determine graphs, cycles, a factorzation into transpositions, the number of inversions, the sign, and the inverse permutations for the following permutations: $$p$$, $$q$$ and their compositions $$q\circ p$$ and $$p\circ q$$. (Permutations are composed as mappings, i.e. $$(q\circ p)(i)=q(p(i))$$.)

• Variant 1

$$p=(6{,}4,1{,}5,3{,}2)$$, $$q=(6{,}4,3{,}2,5{,}1)$$.

• Variant 2

$$p=(1{,}2,7{,}6,5{,}4,3{,}8,9)$$, $$q=(1{,}3,5{,}7,9{,}8,6{,}4,2)$$.

• Variant 3

$$p=(5{,}4,3{,}2,1{,}9,8{,}7,6)$$, $$q=(8{,}6,4{,}2,1{,}3,5{,}7,9)$$.

• Variant 4

$$p=(3{,}6,9{,}2,5{,}8,1{,}4,7)$$, $$q=(9{,}8,7{,}6,5{,}4,3{,}2,1)$$.