Identifier
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3) => 2
[2,3,1,4] => [2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
[2,4,1,3] => [2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 0
[3,4,1,2] => [3,4,1,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 2
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 0
[2,3,4,1,5] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[2,3,5,1,4] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[2,4,3,1,5] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[2,4,5,1,3] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[2,5,3,1,4] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[3,2,4,1,5] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[3,2,5,1,4] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[3,4,1,2,5] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,4,1,5,2] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,4,2,1,5] => [3,5,2,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[3,4,5,1,2] => [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 0
[3,5,1,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[3,5,2,1,4] => [3,5,2,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[3,5,4,1,2] => [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 0
[4,2,3,1,5] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 0
[4,2,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 0
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[4,3,5,1,2] => [4,3,5,1,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[4,5,1,2,3] => [4,5,1,3,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[4,5,1,3,2] => [4,5,1,3,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[4,5,2,1,3] => [4,5,2,1,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 0
[4,5,3,1,2] => [4,5,3,1,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 0
[5,2,3,1,4] => [5,2,4,1,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[5,2,4,1,3] => [5,2,4,1,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[5,3,2,1,4] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[5,3,4,1,2] => [5,3,4,1,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[5,4,2,1,3] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 0
[2,3,4,5,1,6] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,3,4,6,1,5] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,3,5,4,1,6] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,3,5,6,1,4] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,3,6,4,1,5] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,3,6,5,1,4] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,4,3,5,1,6] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,4,3,6,1,5] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,4,5,3,1,6] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,4,5,6,1,3] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,4,6,3,1,5] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,4,6,5,1,3] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,5,3,4,1,6] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,5,3,6,1,4] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,5,4,3,1,6] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,5,4,6,1,3] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,5,6,3,1,4] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,5,6,4,1,3] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,6,3,4,1,5] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,6,3,5,1,4] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,6,4,3,1,5] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,6,4,5,1,3] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,6,5,3,1,4] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[2,6,5,4,1,3] => [2,6,5,4,1,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[3,2,4,5,1,6] => [3,2,6,5,1,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,2,4,6,1,5] => [3,2,6,5,1,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,2,5,4,1,6] => [3,2,6,5,1,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,2,5,6,1,4] => [3,2,6,5,1,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,2,6,4,1,5] => [3,2,6,5,1,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,2,6,5,1,4] => [3,2,6,5,1,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,4,2,5,1,6] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,4,2,6,1,5] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,4,5,1,2,6] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,4,5,1,6,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,4,5,2,1,6] => [3,6,5,2,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,4,5,6,1,2] => [3,6,5,4,1,2] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[3,4,6,1,2,5] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,4,6,1,5,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,4,6,2,1,5] => [3,6,5,2,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,4,6,5,1,2] => [3,6,5,4,1,2] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,5,4,1,2,6] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,5,4,1,6,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,5,4,2,1,6] => [3,6,5,2,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,5,4,6,1,2] => [3,6,5,4,1,2] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[3,5,6,1,2,4] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,5,6,1,4,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,5,6,2,1,4] => [3,6,5,2,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,5,6,4,1,2] => [3,6,5,4,1,2] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[3,6,2,4,1,5] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,6,4,1,2,5] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,6,4,1,5,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,6,4,2,1,5] => [3,6,5,2,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,6,4,5,1,2] => [3,6,5,4,1,2] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[3,6,5,1,2,4] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,6,5,1,4,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 0
[3,6,5,2,1,4] => [3,6,5,2,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[3,6,5,4,1,2] => [3,6,5,4,1,2] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
>>> Load all 207 entries. <<<
[4,2,3,5,1,6] => [4,2,6,5,1,3] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => 0
[4,2,3,6,1,5] => [4,2,6,5,1,3] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => 0
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => 0
[4,2,5,6,1,3] => [4,2,6,5,1,3] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => 0
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,2,6,3,1,5] => [4,2,6,5,1,3] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => 0
[4,2,6,5,1,3] => [4,2,6,5,1,3] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => 0
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,3,5,1,2,6] => [4,3,6,1,5,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,3,5,1,6,2] => [4,3,6,1,5,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,3,5,2,1,6] => [4,3,6,2,1,5] => [4,3,2,6,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,3,5,6,1,2] => [4,3,6,5,1,2] => [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 0
[4,3,6,1,2,5] => [4,3,6,1,5,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,3,6,1,5,2] => [4,3,6,1,5,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,3,6,2,1,5] => [4,3,6,2,1,5] => [4,3,2,6,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,3,6,5,1,2] => [4,3,6,5,1,2] => [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 0
[4,5,2,1,3,6] => [4,6,2,1,5,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,5,2,1,6,3] => [4,6,2,1,5,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,5,2,3,1,6] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6) => 0
[4,5,2,6,1,3] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6) => 0
[4,5,3,2,1,6] => [4,6,3,2,1,5] => [4,3,6,2,1,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,5,3,6,1,2] => [4,6,3,5,1,2] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,5,6,2,1,3] => [4,6,5,2,1,3] => [2,6,4,5,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[4,5,6,3,1,2] => [4,6,5,3,1,2] => [1,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[4,6,2,1,3,5] => [4,6,2,1,5,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,6,2,1,5,3] => [4,6,2,1,5,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 0
[4,6,2,3,1,5] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6) => 0
[4,6,2,5,1,3] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6) => 0
[4,6,3,2,1,5] => [4,6,3,2,1,5] => [4,3,6,2,1,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,6,3,5,1,2] => [4,6,3,5,1,2] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6) => 0
[4,6,5,2,1,3] => [4,6,5,2,1,3] => [2,6,4,5,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[4,6,5,3,1,2] => [4,6,5,3,1,2] => [1,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[5,2,3,1,4,6] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[5,2,3,1,6,4] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[5,2,3,4,1,6] => [5,2,6,4,1,3] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[5,2,3,6,1,4] => [5,2,6,4,1,3] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[5,2,4,1,3,6] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[5,2,4,1,6,3] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[5,2,4,3,1,6] => [5,2,6,4,1,3] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[5,2,4,6,1,3] => [5,2,6,4,1,3] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[5,2,6,1,3,4] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[5,2,6,1,4,3] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[5,2,6,3,1,4] => [5,2,6,4,1,3] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[5,2,6,4,1,3] => [5,2,6,4,1,3] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[5,3,2,4,1,6] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6) => 0
[5,3,2,6,1,4] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6) => 0
[5,3,4,1,2,6] => [5,3,6,1,4,2] => [3,5,6,1,4,2] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 0
[5,3,4,1,6,2] => [5,3,6,1,4,2] => [3,5,6,1,4,2] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 0
[5,3,4,2,1,6] => [5,3,6,2,1,4] => [3,5,2,6,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 0
[5,3,4,6,1,2] => [5,3,6,4,1,2] => [1,5,6,3,4,2] => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => 0
[5,3,6,1,2,4] => [5,3,6,1,4,2] => [3,5,6,1,4,2] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 0
[5,3,6,1,4,2] => [5,3,6,1,4,2] => [3,5,6,1,4,2] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 0
[5,3,6,2,1,4] => [5,3,6,2,1,4] => [3,5,2,6,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 0
[5,3,6,4,1,2] => [5,3,6,4,1,2] => [1,5,6,3,4,2] => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => 0
[5,4,2,3,1,6] => [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6) => 0
[5,4,2,6,1,3] => [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6) => 0
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[5,4,3,6,1,2] => [5,4,3,6,1,2] => [1,5,4,3,6,2] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => 1
[5,4,6,1,2,3] => [5,4,6,1,3,2] => [5,1,4,6,3,2] => ([(0,5),(1,2),(1,3),(1,4),(4,5)],6) => 0
[5,4,6,1,3,2] => [5,4,6,1,3,2] => [5,1,4,6,3,2] => ([(0,5),(1,2),(1,3),(1,4),(4,5)],6) => 0
[5,4,6,2,1,3] => [5,4,6,2,1,3] => [2,5,4,6,1,3] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5)],6) => 0
[5,4,6,3,1,2] => [5,4,6,3,1,2] => [1,5,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 0
[5,6,2,1,3,4] => [5,6,2,1,4,3] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6) => 0
[5,6,2,1,4,3] => [5,6,2,1,4,3] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6) => 0
[5,6,2,3,1,4] => [5,6,2,4,1,3] => [2,1,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6) => 0
[5,6,2,4,1,3] => [5,6,2,4,1,3] => [2,1,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6) => 0
[5,6,3,2,1,4] => [5,6,3,2,1,4] => [3,5,6,2,1,4] => ([(0,5),(1,5),(2,3),(2,5),(3,4)],6) => 0
[5,6,3,4,1,2] => [5,6,3,4,1,2] => [1,3,5,6,4,2] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6) => 1
[5,6,4,1,2,3] => [5,6,4,1,3,2] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[5,6,4,1,3,2] => [5,6,4,1,3,2] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[5,6,4,2,1,3] => [5,6,4,2,1,3] => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[5,6,4,3,1,2] => [5,6,4,3,1,2] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0
[6,2,3,4,1,5] => [6,2,5,4,1,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,2,3,5,1,4] => [6,2,5,4,1,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,2,4,3,1,5] => [6,2,5,4,1,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,2,4,5,1,3] => [6,2,5,4,1,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,2,5,3,1,4] => [6,2,5,4,1,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,2,5,4,1,3] => [6,2,5,4,1,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,3,2,4,1,5] => [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[6,3,2,5,1,4] => [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[6,3,4,1,2,5] => [6,3,5,1,4,2] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6) => 0
[6,3,4,1,5,2] => [6,3,5,1,4,2] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6) => 0
[6,3,4,2,1,5] => [6,3,5,2,1,4] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[6,3,4,5,1,2] => [6,3,5,4,1,2] => [1,6,3,5,4,2] => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 0
[6,3,5,1,2,4] => [6,3,5,1,4,2] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6) => 0
[6,3,5,1,4,2] => [6,3,5,1,4,2] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6) => 0
[6,3,5,2,1,4] => [6,3,5,2,1,4] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[6,3,5,4,1,2] => [6,3,5,4,1,2] => [1,6,3,5,4,2] => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 0
[6,4,2,3,1,5] => [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6) => 0
[6,4,2,5,1,3] => [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6) => 0
[6,4,3,2,1,5] => [6,4,3,2,1,5] => [4,6,3,2,1,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0
[6,4,3,5,1,2] => [6,4,3,5,1,2] => [1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6) => 0
[6,4,5,1,2,3] => [6,4,5,1,3,2] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,4,5,1,3,2] => [6,4,5,1,3,2] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,4,5,2,1,3] => [6,4,5,2,1,3] => [2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6) => 0
[6,4,5,3,1,2] => [6,4,5,3,1,2] => [1,4,6,5,3,2] => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 0
[6,5,2,3,1,4] => [6,5,2,4,1,3] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,5,2,4,1,3] => [6,5,2,4,1,3] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,5,3,2,1,4] => [6,5,3,2,1,4] => [3,6,5,2,1,4] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[6,5,3,4,1,2] => [6,5,3,4,1,2] => [1,3,6,5,4,2] => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6) => 1
[6,5,4,2,1,3] => [6,5,4,2,1,3] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 0
search for individual values
searching the database for the individual values of this statistic
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].