Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 5
[2,1,3,4] => 1
[2,1,4,3] => 3
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 6
[1,2,5,3,4] => 6
[1,2,5,4,3] => 14
[1,3,2,4,5] => 2
[1,3,2,5,4] => 8
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[1,3,5,2,4] => 8
[1,3,5,4,2] => 11
[1,4,2,3,5] => 3
[1,4,2,5,3] => 8
[1,4,3,2,5] => 5
[1,4,3,5,2] => 7
[1,4,5,2,3] => 6
[1,4,5,3,2] => 9
[1,5,2,3,4] => 4
[1,5,2,4,3] => 11
[1,5,3,2,4] => 7
[1,5,3,4,2] => 10
[1,5,4,2,3] => 9
[1,5,4,3,2] => 14
[2,1,3,4,5] => 1
[2,1,3,5,4] => 4
[2,1,4,3,5] => 3
[2,1,4,5,3] => 6
[2,1,5,3,4] => 6
[2,1,5,4,3] => 14
[2,3,1,4,5] => 1
[2,3,1,5,4] => 4
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 3
[2,3,5,4,1] => 3
[2,4,1,3,5] => 2
[2,4,1,5,3] => 5
[2,4,3,1,5] => 2
[2,4,3,5,1] => 2
[2,4,5,1,3] => 3
[2,4,5,3,1] => 3
[2,5,1,3,4] => 3
[2,5,1,4,3] => 8
[2,5,3,1,4] => 3
[2,5,3,4,1] => 3
[2,5,4,1,3] => 5
[2,5,4,3,1] => 5
[3,1,2,4,5] => 1
[3,1,2,5,4] => 4
[3,1,4,2,5] => 2
[3,1,4,5,2] => 3
[3,1,5,2,4] => 5
[3,1,5,4,2] => 8
[3,2,1,4,5] => 1
[3,2,1,5,4] => 4
[3,2,4,1,5] => 1
[3,2,4,5,1] => 1
[3,2,5,1,4] => 3
[3,2,5,4,1] => 3
[3,4,1,2,5] => 1
[3,4,1,5,2] => 2
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 1
[3,4,5,2,1] => 1
[3,5,1,2,4] => 2
[3,5,1,4,2] => 4
[3,5,2,1,4] => 2
[3,5,2,4,1] => 2
[3,5,4,1,2] => 2
[3,5,4,2,1] => 2
[4,1,2,3,5] => 1
[4,1,2,5,3] => 3
[4,1,3,2,5] => 2
[4,1,3,5,2] => 3
[4,1,5,2,3] => 3
[4,1,5,3,2] => 5
[4,2,1,3,5] => 1
[4,2,1,5,3] => 3
[4,2,3,1,5] => 1
[4,2,3,5,1] => 1
[4,2,5,1,3] => 2
[4,2,5,3,1] => 2
[4,3,1,2,5] => 1
[4,3,1,5,2] => 2
[4,3,2,1,5] => 1
[4,3,2,5,1] => 1
[4,3,5,1,2] => 1
[4,3,5,2,1] => 1
[4,5,1,2,3] => 1
[4,5,1,3,2] => 2
[4,5,2,1,3] => 1
[4,5,2,3,1] => 1
[4,5,3,1,2] => 1
[4,5,3,2,1] => 1
[5,1,2,3,4] => 1
[5,1,2,4,3] => 3
[5,1,3,2,4] => 2
[5,1,3,4,2] => 3
[5,1,4,2,3] => 3
[5,1,4,3,2] => 5
[5,2,1,3,4] => 1
[5,2,1,4,3] => 3
[5,2,3,1,4] => 1
[5,2,3,4,1] => 1
[5,2,4,1,3] => 2
[5,2,4,3,1] => 2
[5,3,1,2,4] => 1
[5,3,1,4,2] => 2
[5,3,2,1,4] => 1
[5,3,2,4,1] => 1
[5,3,4,1,2] => 1
[5,3,4,2,1] => 1
[5,4,1,2,3] => 1
[5,4,1,3,2] => 2
[5,4,2,1,3] => 1
[5,4,2,3,1] => 1
[5,4,3,1,2] => 1
[5,4,3,2,1] => 1
[1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => 5
[1,2,3,5,4,6] => 4
[1,2,3,5,6,4] => 10
[1,2,3,6,4,5] => 10
[1,2,3,6,5,4] => 30
[1,2,4,3,5,6] => 3
[1,2,4,3,6,5] => 15
[1,2,4,5,3,6] => 6
[1,2,4,5,6,3] => 10
[1,2,4,6,3,5] => 20
[1,2,4,6,5,3] => 35
[1,2,5,3,4,6] => 6
[1,2,5,3,6,4] => 20
[1,2,5,4,3,6] => 14
[1,2,5,4,6,3] => 25
[1,2,5,6,3,4] => 20
[1,2,5,6,4,3] => 40
[1,2,6,3,4,5] => 10
[1,2,6,3,5,4] => 35
[1,2,6,4,3,5] => 25
[1,2,6,4,5,3] => 46
[1,2,6,5,3,4] => 40
[1,2,6,5,4,3] => 84
[1,3,2,4,5,6] => 2
[1,3,2,4,6,5] => 10
[1,3,2,5,4,6] => 8
[1,3,2,5,6,4] => 20
[1,3,2,6,4,5] => 20
[1,3,2,6,5,4] => 60
[1,3,4,2,5,6] => 3
[1,3,4,2,6,5] => 15
[1,3,4,5,2,6] => 4
[1,3,4,5,6,2] => 5
[1,3,4,6,2,5] => 15
[1,3,4,6,5,2] => 19
[1,3,5,2,4,6] => 8
[1,3,5,2,6,4] => 25
[1,3,5,4,2,6] => 11
[1,3,5,4,6,2] => 14
[1,3,5,6,2,4] => 20
[1,3,5,6,4,2] => 26
[1,3,6,2,4,5] => 15
[1,3,6,2,5,4] => 51
[1,3,6,4,2,5] => 21
[1,3,6,4,5,2] => 27
[1,3,6,5,2,4] => 44
[1,3,6,5,4,2] => 58
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 15
[1,4,2,5,3,6] => 8
[1,4,2,5,6,3] => 15
[1,4,2,6,3,5] => 25
[1,4,2,6,5,3] => 51
[1,4,3,2,5,6] => 5
[1,4,3,2,6,5] => 25
[1,4,3,5,2,6] => 7
[1,4,3,5,6,2] => 9
[1,4,3,6,2,5] => 26
[1,4,3,6,5,2] => 34
[1,4,5,2,3,6] => 6
[1,4,5,2,6,3] => 15
[1,4,5,3,2,6] => 9
[1,4,5,3,6,2] => 12
[1,4,5,6,2,3] => 10
[1,4,5,6,3,2] => 14
[1,4,6,2,3,5] => 15
[1,4,6,2,5,3] => 38
[1,4,6,3,2,5] => 23
[1,4,6,3,5,2] => 31
[1,4,6,5,2,3] => 26
[1,4,6,5,3,2] => 37
[1,5,2,3,4,6] => 4
[1,5,2,3,6,4] => 15
[1,5,2,4,3,6] => 11
[1,5,2,4,6,3] => 21
[1,5,2,6,3,4] => 20
[1,5,2,6,4,3] => 44
[1,5,3,2,4,6] => 7
[1,5,3,2,6,4] => 26
[1,5,3,4,2,6] => 10
[1,5,3,4,6,2] => 13
[1,5,3,6,2,4] => 24
[1,5,3,6,4,2] => 32
[1,5,4,2,3,6] => 9
[1,5,4,2,6,3] => 23
[1,5,4,3,2,6] => 14
[1,5,4,3,6,2] => 19
[1,5,4,6,2,3] => 16
[1,5,4,6,3,2] => 23
[1,5,6,2,3,4] => 10
[1,5,6,2,4,3] => 26
[1,5,6,3,2,4] => 16
[1,5,6,3,4,2] => 22
[1,5,6,4,2,3] => 19
[1,5,6,4,3,2] => 28
[1,6,2,3,4,5] => 5
[1,6,2,3,5,4] => 19
[1,6,2,4,3,5] => 14
[1,6,2,4,5,3] => 27
[1,6,2,5,3,4] => 26
[1,6,2,5,4,3] => 58
[1,6,3,2,4,5] => 9
[1,6,3,2,5,4] => 34
[1,6,3,4,2,5] => 13
[1,6,3,4,5,2] => 17
[1,6,3,5,2,4] => 32
[1,6,3,5,4,2] => 43
[1,6,4,2,3,5] => 12
[1,6,4,2,5,3] => 31
[1,6,4,3,2,5] => 19
[1,6,4,3,5,2] => 26
[1,6,4,5,2,3] => 22
[1,6,4,5,3,2] => 32
[1,6,5,2,3,4] => 14
[1,6,5,2,4,3] => 37
[1,6,5,3,2,4] => 23
[1,6,5,3,4,2] => 32
[1,6,5,4,2,3] => 28
[1,6,5,4,3,2] => 42
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 5
[2,1,3,5,4,6] => 4
[2,1,3,5,6,4] => 10
[2,1,3,6,4,5] => 10
[2,1,3,6,5,4] => 30
[2,1,4,3,5,6] => 3
[2,1,4,3,6,5] => 15
[2,1,4,5,3,6] => 6
[2,1,4,5,6,3] => 10
[2,1,4,6,3,5] => 20
[2,1,4,6,5,3] => 35
[2,1,5,3,4,6] => 6
[2,1,5,3,6,4] => 20
[2,1,5,4,3,6] => 14
[2,1,5,4,6,3] => 25
[2,1,5,6,3,4] => 20
[2,1,5,6,4,3] => 40
[2,1,6,3,4,5] => 10
[2,1,6,3,5,4] => 35
[2,1,6,4,3,5] => 25
[2,1,6,4,5,3] => 46
[2,1,6,5,3,4] => 40
[2,1,6,5,4,3] => 84
[2,3,1,4,5,6] => 1
[2,3,1,4,6,5] => 5
[2,3,1,5,4,6] => 4
[2,3,1,5,6,4] => 10
[2,3,1,6,4,5] => 10
[2,3,1,6,5,4] => 30
[2,3,4,1,5,6] => 1
[2,3,4,1,6,5] => 5
[2,3,4,5,1,6] => 1
[2,3,4,5,6,1] => 1
[2,3,4,6,1,5] => 4
[2,3,4,6,5,1] => 4
[2,3,5,1,4,6] => 3
[2,3,5,1,6,4] => 9
[2,3,5,4,1,6] => 3
[2,3,5,4,6,1] => 3
[2,3,5,6,1,4] => 6
[2,3,5,6,4,1] => 6
[2,3,6,1,4,5] => 6
[2,3,6,1,5,4] => 20
[2,3,6,4,1,5] => 6
[2,3,6,4,5,1] => 6
[2,3,6,5,1,4] => 14
[2,3,6,5,4,1] => 14
[2,4,1,3,5,6] => 2
[2,4,1,3,6,5] => 10
[2,4,1,5,3,6] => 5
[2,4,1,5,6,3] => 9
[2,4,1,6,3,5] => 16
[2,4,1,6,5,3] => 31
[2,4,3,1,5,6] => 2
[2,4,3,1,6,5] => 10
[2,4,3,5,1,6] => 2
[2,4,3,5,6,1] => 2
[2,4,3,6,1,5] => 8
[2,4,3,6,5,1] => 8
[2,4,5,1,3,6] => 3
[2,4,5,1,6,3] => 7
[2,4,5,3,1,6] => 3
[2,4,5,3,6,1] => 3
[2,4,5,6,1,3] => 4
[2,4,5,6,3,1] => 4
[2,4,6,1,3,5] => 8
[2,4,6,1,5,3] => 19
[2,4,6,3,1,5] => 8
[2,4,6,3,5,1] => 8
[2,4,6,5,1,3] => 11
[2,4,6,5,3,1] => 11
[2,5,1,3,4,6] => 3
[2,5,1,3,6,4] => 11
[2,5,1,4,3,6] => 8
[2,5,1,4,6,3] => 15
[2,5,1,6,3,4] => 14
[2,5,1,6,4,3] => 30
[2,5,3,1,4,6] => 3
[2,5,3,1,6,4] => 11
[2,5,3,4,1,6] => 3
[2,5,3,4,6,1] => 3
[2,5,3,6,1,4] => 8
[2,5,3,6,4,1] => 8
[2,5,4,1,3,6] => 5
[2,5,4,1,6,3] => 12
[2,5,4,3,1,6] => 5
[2,5,4,3,6,1] => 5
[2,5,4,6,1,3] => 7
[2,5,4,6,3,1] => 7
[2,5,6,1,3,4] => 6
[2,5,6,1,4,3] => 15
[2,5,6,3,1,4] => 6
[2,5,6,3,4,1] => 6
[2,5,6,4,1,3] => 9
[2,5,6,4,3,1] => 9
[2,6,1,3,4,5] => 4
[2,6,1,3,5,4] => 15
[2,6,1,4,3,5] => 11
[2,6,1,4,5,3] => 21
[2,6,1,5,3,4] => 20
[2,6,1,5,4,3] => 44
[2,6,3,1,4,5] => 4
[2,6,3,1,5,4] => 15
[2,6,3,4,1,5] => 4
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 11
[2,6,3,5,4,1] => 11
[2,6,4,1,3,5] => 7
[2,6,4,1,5,3] => 17
[2,6,4,3,1,5] => 7
[2,6,4,3,5,1] => 7
[2,6,4,5,1,3] => 10
[2,6,4,5,3,1] => 10
[2,6,5,1,3,4] => 9
[2,6,5,1,4,3] => 23
[2,6,5,3,1,4] => 9
[2,6,5,3,4,1] => 9
[2,6,5,4,1,3] => 14
[2,6,5,4,3,1] => 14
[3,1,2,4,5,6] => 1
[3,1,2,4,6,5] => 5
[3,1,2,5,4,6] => 4
[3,1,2,5,6,4] => 10
[3,1,2,6,4,5] => 10
[3,1,2,6,5,4] => 30
[3,1,4,2,5,6] => 2
[3,1,4,2,6,5] => 10
[3,1,4,5,2,6] => 3
[3,1,4,5,6,2] => 4
[3,1,4,6,2,5] => 11
[3,1,4,6,5,2] => 15
[3,1,5,2,4,6] => 5
[3,1,5,2,6,4] => 16
[3,1,5,4,2,6] => 8
[3,1,5,4,6,2] => 11
[3,1,5,6,2,4] => 14
[3,1,5,6,4,2] => 20
[3,1,6,2,4,5] => 9
[3,1,6,2,5,4] => 31
[3,1,6,4,2,5] => 15
[3,1,6,4,5,2] => 21
[3,1,6,5,2,4] => 30
[3,1,6,5,4,2] => 44
[3,2,1,4,5,6] => 1
[3,2,1,4,6,5] => 5
[3,2,1,5,4,6] => 4
[3,2,1,5,6,4] => 10
[3,2,1,6,4,5] => 10
[3,2,1,6,5,4] => 30
[3,2,4,1,5,6] => 1
[3,2,4,1,6,5] => 5
[3,2,4,5,1,6] => 1
[3,2,4,5,6,1] => 1
[3,2,4,6,1,5] => 4
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 3
[3,2,5,1,6,4] => 9
[3,2,5,4,1,6] => 3
[3,2,5,4,6,1] => 3
[3,2,5,6,1,4] => 6
[3,2,5,6,4,1] => 6
[3,2,6,1,4,5] => 6
[3,2,6,1,5,4] => 20
[3,2,6,4,1,5] => 6
[3,2,6,4,5,1] => 6
[3,2,6,5,1,4] => 14
[3,2,6,5,4,1] => 14
[3,4,1,2,5,6] => 1
[3,4,1,2,6,5] => 5
[3,4,1,5,2,6] => 2
[3,4,1,5,6,2] => 3
[3,4,1,6,2,5] => 7
[3,4,1,6,5,2] => 11
[3,4,2,1,5,6] => 1
[3,4,2,1,6,5] => 5
[3,4,2,5,1,6] => 1
[3,4,2,5,6,1] => 1
[3,4,2,6,1,5] => 4
[3,4,2,6,5,1] => 4
[3,4,5,1,2,6] => 1
[3,4,5,1,6,2] => 2
[3,4,5,2,1,6] => 1
[3,4,5,2,6,1] => 1
[3,4,5,6,1,2] => 1
[3,4,5,6,2,1] => 1
[3,4,6,1,2,5] => 3
[3,4,6,1,5,2] => 6
[3,4,6,2,1,5] => 3
[3,4,6,2,5,1] => 3
[3,4,6,5,1,2] => 3
[3,4,6,5,2,1] => 3
[3,5,1,2,4,6] => 2
[3,5,1,2,6,4] => 7
[3,5,1,4,2,6] => 4
[3,5,1,4,6,2] => 6
[3,5,1,6,2,4] => 8
[3,5,1,6,4,2] => 13
[3,5,2,1,4,6] => 2
[3,5,2,1,6,4] => 7
[3,5,2,4,1,6] => 2
[3,5,2,4,6,1] => 2
[3,5,2,6,1,4] => 5
[3,5,2,6,4,1] => 5
[3,5,4,1,2,6] => 2
[3,5,4,1,6,2] => 4
[3,5,4,2,1,6] => 2
[3,5,4,2,6,1] => 2
[3,5,4,6,1,2] => 2
[3,5,4,6,2,1] => 2
[3,5,6,1,2,4] => 3
[3,5,6,1,4,2] => 6
[3,5,6,2,1,4] => 3
[3,5,6,2,4,1] => 3
[3,5,6,4,1,2] => 3
[3,5,6,4,2,1] => 3
[3,6,1,2,4,5] => 3
[3,6,1,2,5,4] => 11
[3,6,1,4,2,5] => 6
[3,6,1,4,5,2] => 9
[3,6,1,5,2,4] => 13
[3,6,1,5,4,2] => 21
[3,6,2,1,4,5] => 3
[3,6,2,1,5,4] => 11
[3,6,2,4,1,5] => 3
[3,6,2,4,5,1] => 3
[3,6,2,5,1,4] => 8
[3,6,2,5,4,1] => 8
[3,6,4,1,2,5] => 3
[3,6,4,1,5,2] => 6
[3,6,4,2,1,5] => 3
[3,6,4,2,5,1] => 3
[3,6,4,5,1,2] => 3
[3,6,4,5,2,1] => 3
[3,6,5,1,2,4] => 5
[3,6,5,1,4,2] => 10
[3,6,5,2,1,4] => 5
[3,6,5,2,4,1] => 5
[3,6,5,4,1,2] => 5
[3,6,5,4,2,1] => 5
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 5
[4,1,2,5,3,6] => 3
[4,1,2,5,6,3] => 6
[4,1,2,6,3,5] => 9
[4,1,2,6,5,3] => 20
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 10
[4,1,3,5,2,6] => 3
[4,1,3,5,6,2] => 4
[4,1,3,6,2,5] => 11
[4,1,3,6,5,2] => 15
[4,1,5,2,3,6] => 3
[4,1,5,2,6,3] => 8
[4,1,5,3,2,6] => 5
[4,1,5,3,6,2] => 7
[4,1,5,6,2,3] => 6
[4,1,5,6,3,2] => 9
[4,1,6,2,3,5] => 7
[4,1,6,2,5,3] => 19
[4,1,6,3,2,5] => 12
[4,1,6,3,5,2] => 17
[4,1,6,5,2,3] => 15
[4,1,6,5,3,2] => 23
[4,2,1,3,5,6] => 1
[4,2,1,3,6,5] => 5
[4,2,1,5,3,6] => 3
[4,2,1,5,6,3] => 6
[4,2,1,6,3,5] => 9
[4,2,1,6,5,3] => 20
[4,2,3,1,5,6] => 1
[4,2,3,1,6,5] => 5
[4,2,3,5,1,6] => 1
[4,2,3,5,6,1] => 1
[4,2,3,6,1,5] => 4
[4,2,3,6,5,1] => 4
[4,2,5,1,3,6] => 2
[4,2,5,1,6,3] => 5
[4,2,5,3,1,6] => 2
[4,2,5,3,6,1] => 2
[4,2,5,6,1,3] => 3
[4,2,5,6,3,1] => 3
[4,2,6,1,3,5] => 5
[4,2,6,1,5,3] => 13
[4,2,6,3,1,5] => 5
[4,2,6,3,5,1] => 5
[4,2,6,5,1,3] => 8
[4,2,6,5,3,1] => 8
[4,3,1,2,5,6] => 1
[4,3,1,2,6,5] => 5
[4,3,1,5,2,6] => 2
[4,3,1,5,6,2] => 3
[4,3,1,6,2,5] => 7
[4,3,1,6,5,2] => 11
[4,3,2,1,5,6] => 1
[4,3,2,1,6,5] => 5
[4,3,2,5,1,6] => 1
[4,3,2,5,6,1] => 1
[4,3,2,6,1,5] => 4
[4,3,2,6,5,1] => 4
[4,3,5,1,2,6] => 1
[4,3,5,1,6,2] => 2
[4,3,5,2,1,6] => 1
[4,3,5,2,6,1] => 1
[4,3,5,6,1,2] => 1
[4,3,5,6,2,1] => 1
[4,3,6,1,2,5] => 3
[4,3,6,1,5,2] => 6
[4,3,6,2,1,5] => 3
[4,3,6,2,5,1] => 3
[4,3,6,5,1,2] => 3
[4,3,6,5,2,1] => 3
[4,5,1,2,3,6] => 1
[4,5,1,2,6,3] => 3
[4,5,1,3,2,6] => 2
[4,5,1,3,6,2] => 3
[4,5,1,6,2,3] => 3
[4,5,1,6,3,2] => 5
[4,5,2,1,3,6] => 1
[4,5,2,1,6,3] => 3
[4,5,2,3,1,6] => 1
[4,5,2,3,6,1] => 1
[4,5,2,6,1,3] => 2
[4,5,2,6,3,1] => 2
[4,5,3,1,2,6] => 1
[4,5,3,1,6,2] => 2
[4,5,3,2,1,6] => 1
[4,5,3,2,6,1] => 1
[4,5,3,6,1,2] => 1
[4,5,3,6,2,1] => 1
[4,5,6,1,2,3] => 1
[4,5,6,1,3,2] => 2
[4,5,6,2,1,3] => 1
[4,5,6,2,3,1] => 1
[4,5,6,3,1,2] => 1
[4,5,6,3,2,1] => 1
[4,6,1,2,3,5] => 2
[4,6,1,2,5,3] => 6
[4,6,1,3,2,5] => 4
[4,6,1,3,5,2] => 6
[4,6,1,5,2,3] => 6
[4,6,1,5,3,2] => 10
[4,6,2,1,3,5] => 2
[4,6,2,1,5,3] => 6
[4,6,2,3,1,5] => 2
[4,6,2,3,5,1] => 2
[4,6,2,5,1,3] => 4
[4,6,2,5,3,1] => 4
[4,6,3,1,2,5] => 2
[4,6,3,1,5,2] => 4
[4,6,3,2,1,5] => 2
[4,6,3,2,5,1] => 2
[4,6,3,5,1,2] => 2
[4,6,3,5,2,1] => 2
[4,6,5,1,2,3] => 2
[4,6,5,1,3,2] => 4
[4,6,5,2,1,3] => 2
[4,6,5,2,3,1] => 2
[4,6,5,3,1,2] => 2
[4,6,5,3,2,1] => 2
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 4
[5,1,2,4,3,6] => 3
[5,1,2,4,6,3] => 6
[5,1,2,6,3,4] => 6
[5,1,2,6,4,3] => 14
[5,1,3,2,4,6] => 2
[5,1,3,2,6,4] => 8
[5,1,3,4,2,6] => 3
[5,1,3,4,6,2] => 4
[5,1,3,6,2,4] => 8
[5,1,3,6,4,2] => 11
[5,1,4,2,3,6] => 3
[5,1,4,2,6,3] => 8
[5,1,4,3,2,6] => 5
[5,1,4,3,6,2] => 7
[5,1,4,6,2,3] => 6
[5,1,4,6,3,2] => 9
[5,1,6,2,3,4] => 4
[5,1,6,2,4,3] => 11
[5,1,6,3,2,4] => 7
[5,1,6,3,4,2] => 10
[5,1,6,4,2,3] => 9
[5,1,6,4,3,2] => 14
[5,2,1,3,4,6] => 1
[5,2,1,3,6,4] => 4
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 6
[5,2,1,6,3,4] => 6
[5,2,1,6,4,3] => 14
[5,2,3,1,4,6] => 1
[5,2,3,1,6,4] => 4
[5,2,3,4,1,6] => 1
[5,2,3,4,6,1] => 1
[5,2,3,6,1,4] => 3
[5,2,3,6,4,1] => 3
[5,2,4,1,3,6] => 2
[5,2,4,1,6,3] => 5
[5,2,4,3,1,6] => 2
[5,2,4,3,6,1] => 2
[5,2,4,6,1,3] => 3
[5,2,4,6,3,1] => 3
[5,2,6,1,3,4] => 3
[5,2,6,1,4,3] => 8
[5,2,6,3,1,4] => 3
[5,2,6,3,4,1] => 3
[5,2,6,4,1,3] => 5
[5,2,6,4,3,1] => 5
[5,3,1,2,4,6] => 1
[5,3,1,2,6,4] => 4
[5,3,1,4,2,6] => 2
[5,3,1,4,6,2] => 3
[5,3,1,6,2,4] => 5
[5,3,1,6,4,2] => 8
[5,3,2,1,4,6] => 1
[5,3,2,1,6,4] => 4
[5,3,2,4,1,6] => 1
[5,3,2,4,6,1] => 1
[5,3,2,6,1,4] => 3
[5,3,2,6,4,1] => 3
[5,3,4,1,2,6] => 1
[5,3,4,1,6,2] => 2
[5,3,4,2,1,6] => 1
[5,3,4,2,6,1] => 1
[5,3,4,6,1,2] => 1
[5,3,4,6,2,1] => 1
[5,3,6,1,2,4] => 2
[5,3,6,1,4,2] => 4
[5,3,6,2,1,4] => 2
[5,3,6,2,4,1] => 2
[5,3,6,4,1,2] => 2
[5,3,6,4,2,1] => 2
[5,4,1,2,3,6] => 1
[5,4,1,2,6,3] => 3
[5,4,1,3,2,6] => 2
[5,4,1,3,6,2] => 3
[5,4,1,6,2,3] => 3
[5,4,1,6,3,2] => 5
[5,4,2,1,3,6] => 1
[5,4,2,1,6,3] => 3
[5,4,2,3,1,6] => 1
[5,4,2,3,6,1] => 1
[5,4,2,6,1,3] => 2
[5,4,2,6,3,1] => 2
[5,4,3,1,2,6] => 1
[5,4,3,1,6,2] => 2
[5,4,3,2,1,6] => 1
[5,4,3,2,6,1] => 1
[5,4,3,6,1,2] => 1
[5,4,3,6,2,1] => 1
[5,4,6,1,2,3] => 1
[5,4,6,1,3,2] => 2
[5,4,6,2,1,3] => 1
[5,4,6,2,3,1] => 1
[5,4,6,3,1,2] => 1
[5,4,6,3,2,1] => 1
[5,6,1,2,3,4] => 1
[5,6,1,2,4,3] => 3
[5,6,1,3,2,4] => 2
[5,6,1,3,4,2] => 3
[5,6,1,4,2,3] => 3
[5,6,1,4,3,2] => 5
[5,6,2,1,3,4] => 1
[5,6,2,1,4,3] => 3
[5,6,2,3,1,4] => 1
[5,6,2,3,4,1] => 1
[5,6,2,4,1,3] => 2
[5,6,2,4,3,1] => 2
[5,6,3,1,2,4] => 1
[5,6,3,1,4,2] => 2
[5,6,3,2,1,4] => 1
[5,6,3,2,4,1] => 1
[5,6,3,4,1,2] => 1
[5,6,3,4,2,1] => 1
[5,6,4,1,2,3] => 1
[5,6,4,1,3,2] => 2
[5,6,4,2,1,3] => 1
[5,6,4,2,3,1] => 1
[5,6,4,3,1,2] => 1
[5,6,4,3,2,1] => 1
[6,1,2,3,4,5] => 1
[6,1,2,3,5,4] => 4
[6,1,2,4,3,5] => 3
[6,1,2,4,5,3] => 6
[6,1,2,5,3,4] => 6
[6,1,2,5,4,3] => 14
[6,1,3,2,4,5] => 2
[6,1,3,2,5,4] => 8
[6,1,3,4,2,5] => 3
[6,1,3,4,5,2] => 4
[6,1,3,5,2,4] => 8
[6,1,3,5,4,2] => 11
[6,1,4,2,3,5] => 3
[6,1,4,2,5,3] => 8
[6,1,4,3,2,5] => 5
[6,1,4,3,5,2] => 7
[6,1,4,5,2,3] => 6
[6,1,4,5,3,2] => 9
[6,1,5,2,3,4] => 4
[6,1,5,2,4,3] => 11
[6,1,5,3,2,4] => 7
[6,1,5,3,4,2] => 10
[6,1,5,4,2,3] => 9
[6,1,5,4,3,2] => 14
[6,2,1,3,4,5] => 1
[6,2,1,3,5,4] => 4
[6,2,1,4,3,5] => 3
[6,2,1,4,5,3] => 6
[6,2,1,5,3,4] => 6
[6,2,1,5,4,3] => 14
[6,2,3,1,4,5] => 1
[6,2,3,1,5,4] => 4
[6,2,3,4,1,5] => 1
[6,2,3,4,5,1] => 1
[6,2,3,5,1,4] => 3
[6,2,3,5,4,1] => 3
[6,2,4,1,3,5] => 2
[6,2,4,1,5,3] => 5
[6,2,4,3,1,5] => 2
[6,2,4,3,5,1] => 2
[6,2,4,5,1,3] => 3
[6,2,4,5,3,1] => 3
[6,2,5,1,3,4] => 3
[6,2,5,1,4,3] => 8
[6,2,5,3,1,4] => 3
[6,2,5,3,4,1] => 3
[6,2,5,4,1,3] => 5
[6,2,5,4,3,1] => 5
[6,3,1,2,4,5] => 1
[6,3,1,2,5,4] => 4
[6,3,1,4,2,5] => 2
[6,3,1,4,5,2] => 3
[6,3,1,5,2,4] => 5
[6,3,1,5,4,2] => 8
[6,3,2,1,4,5] => 1
[6,3,2,1,5,4] => 4
[6,3,2,4,1,5] => 1
[6,3,2,4,5,1] => 1
[6,3,2,5,1,4] => 3
[6,3,2,5,4,1] => 3
[6,3,4,1,2,5] => 1
[6,3,4,1,5,2] => 2
[6,3,4,2,1,5] => 1
[6,3,4,2,5,1] => 1
[6,3,4,5,1,2] => 1
[6,3,4,5,2,1] => 1
[6,3,5,1,2,4] => 2
[6,3,5,1,4,2] => 4
[6,3,5,2,1,4] => 2
[6,3,5,2,4,1] => 2
[6,3,5,4,1,2] => 2
[6,3,5,4,2,1] => 2
[6,4,1,2,3,5] => 1
[6,4,1,2,5,3] => 3
[6,4,1,3,2,5] => 2
[6,4,1,3,5,2] => 3
[6,4,1,5,2,3] => 3
[6,4,1,5,3,2] => 5
[6,4,2,1,3,5] => 1
[6,4,2,1,5,3] => 3
[6,4,2,3,1,5] => 1
[6,4,2,3,5,1] => 1
[6,4,2,5,1,3] => 2
[6,4,2,5,3,1] => 2
[6,4,3,1,2,5] => 1
[6,4,3,1,5,2] => 2
[6,4,3,2,1,5] => 1
[6,4,3,2,5,1] => 1
[6,4,3,5,1,2] => 1
[6,4,3,5,2,1] => 1
[6,4,5,1,2,3] => 1
[6,4,5,1,3,2] => 2
[6,4,5,2,1,3] => 1
[6,4,5,2,3,1] => 1
[6,4,5,3,1,2] => 1
[6,4,5,3,2,1] => 1
[6,5,1,2,3,4] => 1
[6,5,1,2,4,3] => 3
[6,5,1,3,2,4] => 2
[6,5,1,3,4,2] => 3
[6,5,1,4,2,3] => 3
[6,5,1,4,3,2] => 5
[6,5,2,1,3,4] => 1
[6,5,2,1,4,3] => 3
[6,5,2,3,1,4] => 1
[6,5,2,3,4,1] => 1
[6,5,2,4,1,3] => 2
[6,5,2,4,3,1] => 2
[6,5,3,1,2,4] => 1
[6,5,3,1,4,2] => 2
[6,5,3,2,1,4] => 1
[6,5,3,2,4,1] => 1
[6,5,3,4,1,2] => 1
[6,5,3,4,2,1] => 1
[6,5,4,1,2,3] => 1
[6,5,4,1,3,2] => 2
[6,5,4,2,1,3] => 1
[6,5,4,2,3,1] => 1
[6,5,4,3,1,2] => 1
[6,5,4,3,2,1] => 1
click to show generating function       
Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$.
References
[1] Kirichenko, V. A., Smirnov, E. Y., Timorin, V. A. Schubert calculus and Gelfand-Zetlin polytopes DOI:10.1070/rm2012v067n04abeh004804 arXiv:1101.0278
Code
def statistic(pi):
    n = len(pi)
    if pi == Permutations(n).one():
        return 1
    else:
        S = Word([j for i in range(n, 0, -1) for j in range(i, n+1)])
        return sum(Word(w).nb_subword_occurrences_in(S) for w in pi.reduced_words())
Created
Jun 15, 2015 at 18:00 by Per Alexandersson
Updated
Mar 31, 2019 at 22:17 by Martin Rubey