Collection of Maths Problems

Elementary properties

Task number: 2456

• Variant 1

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

• Resolution

  \(q\circ p=(1{,}2,6{,}5,3{,}4)\), \(p\circ q=(2{,}5,1{,}4,3{,}6)\)

  Permutation \(p\) has 11 inversions: \((1{,}2),(1{,}3),(1{,}4),(1{,}5),(1{,}6),\) \((2{,}3),(2{,}5),(2{,}6),\) \((4{,}5),(4{,}6),(5{,}6)\), see e.g. the the crossings in the bipartite graph.

  \(q\) has 12 inverions, both compositions \(q\circ p\) and \(p\circ q\) have 5 inverzí.

  Cycles \(p=(1{,}6,2{,}4,5{,}3)\), \(q=(1{,}6),(2{,}4),(3),(5)\),
  \(q\circ p=(1),(2),(3{,}6,4{,}5)\), \(q\circ p=(1{,}2,5{,}3),(4),(6)\)

  Factorization into transpositions: \(p=(1{,}6)\circ(6{,}2)\circ(2{,}4)\circ(4{,}5)\circ(5{,}3)\), \(q=(1{,}6)\circ(2{,}4)\), \(q\circ p=(3{,}6)\circ(6{,}4)\circ(4{,}5)\), \(p\circ q=(1{,}2)\circ(2{,}5)\circ(5{,}3)\).

  Signs are \(-1,+1,-1\) a \(-1\).

  Inverse permutations are \(p^{-1}=(3{,}6,5{,}2,4{,}1)\), \(q^{-1}=(6{,}4,3{,}2,5{,}1)\), \((q\circ p)^{-1}=(1{,}2,5{,}6,4{,}3)\), \((p\circ q)^{-1}=(3{,}1,5{,}4,2{,}6)\).

• Variant 2

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

• Result

  \(p\): \(p^{-1}=(1{,}2,7{,}6,5{,}4,3{,}8,9)\)
  10 inverses, cycles \((1),(2),(3{,}7),(4{,}6),(5),(8),(9)\), 2 transpositions \((3{,}7)\circ(4{,}6)\), sign \(+1\),

  \(q\): \(q^{-1}=(1{,}9,2{,}8,3{,}7,4{,}6,5)\)
  16 inverses, cycles \((1),(2{,}3,5{,}9),(4{,}7,6{,}8)\), 6 transpositions, sign \(+1\),

  \(q\circ p=(1{,}3,6{,}8,9{,}7,5{,}4,2)\): \((q\circ p)^{-1}=(1{,}9,2{,}8,7{,}3,6{,}4,5)\)
  18 inverses, cycles \((1),(2{,}3,6{,}7,5{,}9),(4{,}8)\), 6 transpositions, sign \(+1\),

  \(p\circ q=(1{,}7,5{,}3,9{,}8,4{,}6,2)\): \((p\circ q)^{-1}=(1{,}9,4{,}7,3{,}8,2{,}6,5)\)
  18 inverses, cycles \((1),(2{,}7,4{,}3,5{,}9),(6{,}8)\), 6 transpositions, sign \(+1\).

• Variant 3

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

• Result

  \(p\): \(p^{-1}=(5{,}4,3{,}2,1{,}9,8{,}7,6)\),
  16 inversions, cycles \((1{,}5),(2{,}4),(3),(6{,}9),(7{,}8)\), 4 transpositions, sign \(+1\),

  \(q\): \(q^{-1}=(5{,}4,6{,}3,7{,}2,8{,}1,9)\),
  16 inversions, cycles \((1{,}8,7{,}5),(2{,}6,3{,}4),(9)\), 6 transpositions, sign \(+1\),

  \(q\circ p=(1{,}2,4{,}6,8{,}9,7{,}5,3)\), \((q\circ p)^{-1}=(1{,}2,9{,}3,8{,}4,7{,}5,6)\)
  12 inversions, cycles \((1),(2),(3{,}4,6{,}9),(5{,}8),(7)\), 4 transpositions, sign \(+1\),

  \(p\circ q=(7{,}9,2{,}4,5{,}3,1{,}8,6)\), \((p\circ q)^{-1}=(7{,}3,6{,}4,5{,}9,1{,}8,2)\)
  20 inversions, cycles \((1{,}7),(2{,}9,6{,}3),(4),(5),(8)\), 4 transpositions, sign \(+1\).

• Variant 4

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

• Result

  \(p\): \(p^{-1}=(7{,}4,1{,}8,5{,}2,9{,}6,3)\),
  18 inverses, cycles \((1{,}3,9{,}7),(2{,}6,8{,}4),(5)\), 6 transpositions, sign \(+1\),

  \(q\): \(q^{-1}=(9{,}8,7{,}6,5{,}4,3{,}2,1)\),
  36 inverses, cycles \((1{,}9),(2{,}8),(3{,}7),(4{,}6),(5)\), 4 transpositions, sign \(+1\),

  \(q\circ p=(7{,}4,1{,}8,5{,}2,9{,}6,3)\), \((q\circ p)^{-1}=(3{,}6,9{,}2,5{,}8,1{,}4,6)\)
  18 inverses, cycles \((1{,}7,9{,}3),(2{,}4,8{,}6),(5)\), 6 transpositions, sign \(+1\),

  \(p\circ q=(7{,}4,1{,}8,5{,}2,9{,}6,3)\), \((p\circ q)^{-1}=(3{,}6,9{,}2,5{,}8,1{,}4,6)\)
  18 inverses, cycles \((1{,}7,9{,}3),(2{,}4,8{,}6),(5)\), 6 transpositions, sign \(+1\),

  Observe that \(q\circ p=p\circ q\).