Identifier
Values
[1,0,1,0] => [1,2] => [1,2] => ([(0,1)],2) => 1
[1,1,0,1,0,0] => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3) => 2
[1,1,1,0,0,0] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => 2
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 0
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 0
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5) => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 0
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6) => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [2,6,1,3,5,4] => ([(0,5),(1,2),(1,5),(5,3),(5,4)],6) => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [2,6,1,4,3,5] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(4,3),(5,3)],6) => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [2,6,1,4,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6) => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6) => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [2,3,6,1,5,4] => ([(0,4),(0,5),(1,2),(2,3),(2,4),(2,5)],6) => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6) => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6) => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [2,4,1,3,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [2,5,1,4,3,6] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [3,1,2,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [3,1,2,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [3,1,2,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [3,1,2,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [4,1,2,3,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6) => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [5,1,2,4,3,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [4,1,3,2,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [4,1,3,2,5,6] => ([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6) => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,6,4,3,5,2] => [5,1,3,2,4,6] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6) => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => [5,1,3,4,2,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6) => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [5,1,4,3,2,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [1,6,2,3,4,5] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6) => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6) => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6) => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [1,6,2,5,4,3] => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6) => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6) => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => [1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => 1
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => [1,2,6,4,5,3] => ([(0,5),(4,3),(5,1),(5,2),(5,4)],6) => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => [1,2,6,5,4,3] => ([(0,5),(5,1),(5,2),(5,3),(5,4)],6) => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => [2,1,6,3,5,4] => ([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6) => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [2,6,3,1,4,5] => ([(0,5),(1,2),(1,3),(3,5),(5,4)],6) => 1
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [2,6,3,4,1,5] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [2,6,3,1,5,4] => ([(0,4),(0,5),(1,2),(1,3),(3,4),(3,5)],6) => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => [2,1,6,4,3,5] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6) => 0
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [2,6,4,3,1,5] => ([(0,5),(1,2),(1,3),(1,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => [2,6,5,3,1,4] => ([(0,5),(1,2),(1,3),(1,4),(4,5)],6) => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,2,3,1,5,6] => [3,1,6,2,4,5] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6) => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,2,3,1,6,5] => [3,1,6,2,5,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6) => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,2,3,5,1,6] => [3,6,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,5),(5,4)],6) => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [3,6,2,4,1,5] => ([(0,5),(1,4),(2,3),(2,4),(4,5)],6) => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,2,3,6,5,1] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,3,4,1,6] => [4,6,2,1,3,5] => ([(0,4),(1,4),(2,3),(2,5),(4,5)],6) => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [4,6,2,3,1,5] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6) => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,2,4,3,1,6] => [4,1,6,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6) => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => [4,6,3,2,1,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [3,2,1,6,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6) => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,2,6,4,1,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(4,5)],6) => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [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
>>> Load all 117 entries. <<<
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,3,2,4,1,6] => [4,2,1,6,3,5] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6) => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [4,2,6,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6) => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => [5,2,6,3,1,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,3,4,2,1,6] => [4,2,1,3,6,5] => ([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => [4,2,3,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6) => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => [5,2,3,1,6,4] => ([(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6) => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => [5,2,1,4,3,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,6,5,4,3,2,1] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 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
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$