Identifier
Identifier
Values
[1] => 0
[1,1] => 0
[1,2] => 1
[2,1] => 1
[1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[2,1,1] => 1
[1,1,3] => 2
[1,3,1] => 2
[3,1,1] => 2
[1,2,2] => 2
[2,1,2] => 2
[2,2,1] => 2
[1,2,3] => 3
[1,3,2] => 3
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 3
[1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 1
[1,1,1,3] => 2
[1,1,3,1] => 2
[1,3,1,1] => 2
[3,1,1,1] => 2
[1,1,1,4] => 3
[1,1,4,1] => 3
[1,4,1,1] => 3
[4,1,1,1] => 3
[1,1,2,2] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[2,1,1,2] => 2
[2,1,2,1] => 2
[2,2,1,1] => 2
[1,1,2,3] => 3
[1,1,3,2] => 3
[1,2,1,3] => 3
[1,2,3,1] => 3
[1,3,1,2] => 3
[1,3,2,1] => 3
[2,1,1,3] => 3
[2,1,3,1] => 3
[2,3,1,1] => 3
[3,1,1,2] => 3
[3,1,2,1] => 3
[3,2,1,1] => 3
[1,1,2,4] => 4
[1,1,4,2] => 4
[1,2,1,4] => 4
[1,2,4,1] => 4
[1,4,1,2] => 4
[1,4,2,1] => 4
[2,1,1,4] => 4
[2,1,4,1] => 4
[2,4,1,1] => 4
[4,1,1,2] => 4
[4,1,2,1] => 4
[4,2,1,1] => 4
[1,1,3,3] => 4
[1,3,1,3] => 4
[1,3,3,1] => 4
[3,1,1,3] => 4
[3,1,3,1] => 4
[3,3,1,1] => 4
[1,1,3,4] => 5
[1,1,4,3] => 5
[1,3,1,4] => 5
[1,3,4,1] => 5
[1,4,1,3] => 5
[1,4,3,1] => 5
[3,1,1,4] => 5
[3,1,4,1] => 5
[3,4,1,1] => 5
[4,1,1,3] => 5
[4,1,3,1] => 5
[4,3,1,1] => 5
[1,2,2,2] => 3
[2,1,2,2] => 3
[2,2,1,2] => 3
[2,2,2,1] => 3
[1,2,2,3] => 4
[1,2,3,2] => 4
[1,3,2,2] => 4
[2,1,2,3] => 4
[2,1,3,2] => 4
[2,2,1,3] => 4
[2,2,3,1] => 4
[2,3,1,2] => 4
[2,3,2,1] => 4
[3,1,2,2] => 4
[3,2,1,2] => 4
[3,2,2,1] => 4
[1,2,2,4] => 5
[1,2,4,2] => 5
[1,4,2,2] => 5
[2,1,2,4] => 5
[2,1,4,2] => 5
[2,2,1,4] => 5
[2,2,4,1] => 5
[2,4,1,2] => 5
[2,4,2,1] => 5
[4,1,2,2] => 5
[4,2,1,2] => 5
[4,2,2,1] => 5
[1,2,3,3] => 5
[1,3,2,3] => 5
[1,3,3,2] => 5
[2,1,3,3] => 5
[2,3,1,3] => 5
[2,3,3,1] => 5
[3,1,2,3] => 5
[3,1,3,2] => 5
[3,2,1,3] => 5
[3,2,3,1] => 5
[3,3,1,2] => 5
[3,3,2,1] => 5
[1,2,3,4] => 6
[1,2,4,3] => 6
[1,3,2,4] => 6
[1,3,4,2] => 6
[1,4,2,3] => 6
[1,4,3,2] => 6
[2,1,3,4] => 6
[2,1,4,3] => 6
[2,3,1,4] => 6
[2,3,4,1] => 6
[2,4,1,3] => 6
[2,4,3,1] => 6
[3,1,2,4] => 6
[3,1,4,2] => 6
[3,2,1,4] => 6
[3,2,4,1] => 6
[3,4,1,2] => 6
[3,4,2,1] => 6
[4,1,2,3] => 6
[4,1,3,2] => 6
[4,2,1,3] => 6
[4,2,3,1] => 6
[4,3,1,2] => 6
[4,3,2,1] => 6
[1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,2,1,1] => 1
[1,2,1,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,3] => 2
[1,1,1,3,1] => 2
[1,1,3,1,1] => 2
[1,3,1,1,1] => 2
[3,1,1,1,1] => 2
[1,1,1,1,4] => 3
[1,1,1,4,1] => 3
[1,1,4,1,1] => 3
[1,4,1,1,1] => 3
[4,1,1,1,1] => 3
[1,1,1,1,5] => 4
[1,1,1,5,1] => 4
[1,1,5,1,1] => 4
[1,5,1,1,1] => 4
[5,1,1,1,1] => 4
[1,1,1,2,2] => 2
[1,1,2,1,2] => 2
[1,1,2,2,1] => 2
[1,2,1,1,2] => 2
[1,2,1,2,1] => 2
[1,2,2,1,1] => 2
[2,1,1,1,2] => 2
[2,1,1,2,1] => 2
[2,1,2,1,1] => 2
[2,2,1,1,1] => 2
[1,1,1,2,3] => 3
[1,1,1,3,2] => 3
[1,1,2,1,3] => 3
[1,1,2,3,1] => 3
[1,1,3,1,2] => 3
[1,1,3,2,1] => 3
[1,2,1,1,3] => 3
[1,2,1,3,1] => 3
[1,2,3,1,1] => 3
[1,3,1,1,2] => 3
[1,3,1,2,1] => 3
[1,3,2,1,1] => 3
[2,1,1,1,3] => 3
[2,1,1,3,1] => 3
[2,1,3,1,1] => 3
[2,3,1,1,1] => 3
[3,1,1,1,2] => 3
[3,1,1,2,1] => 3
[3,1,2,1,1] => 3
[3,2,1,1,1] => 3
[1,1,1,2,4] => 4
[1,1,1,4,2] => 4
[1,1,2,1,4] => 4
[1,1,2,4,1] => 4
[1,1,4,1,2] => 4
[1,1,4,2,1] => 4
[1,2,1,1,4] => 4
[1,2,1,4,1] => 4
[1,2,4,1,1] => 4
[1,4,1,1,2] => 4
[1,4,1,2,1] => 4
[1,4,2,1,1] => 4
[2,1,1,1,4] => 4
[2,1,1,4,1] => 4
[2,1,4,1,1] => 4
[2,4,1,1,1] => 4
[4,1,1,1,2] => 4
[4,1,1,2,1] => 4
[4,1,2,1,1] => 4
[4,2,1,1,1] => 4
[1,1,1,2,5] => 5
[1,1,1,5,2] => 5
[1,1,2,1,5] => 5
[1,1,2,5,1] => 5
[1,1,5,1,2] => 5
[1,1,5,2,1] => 5
[1,2,1,1,5] => 5
[1,2,1,5,1] => 5
[1,2,5,1,1] => 5
[1,5,1,1,2] => 5
[1,5,1,2,1] => 5
[1,5,2,1,1] => 5
[2,1,1,1,5] => 5
[2,1,1,5,1] => 5
[2,1,5,1,1] => 5
[2,5,1,1,1] => 5
[5,1,1,1,2] => 5
[5,1,1,2,1] => 5
[5,1,2,1,1] => 5
[5,2,1,1,1] => 5
[1,1,1,3,3] => 4
[1,1,3,1,3] => 4
[1,1,3,3,1] => 4
[1,3,1,1,3] => 4
[1,3,1,3,1] => 4
[1,3,3,1,1] => 4
[3,1,1,1,3] => 4
[3,1,1,3,1] => 4
[3,1,3,1,1] => 4
[3,3,1,1,1] => 4
[1,1,1,3,4] => 5
[1,1,1,4,3] => 5
[1,1,3,1,4] => 5
[1,1,3,4,1] => 5
[1,1,4,1,3] => 5
[1,1,4,3,1] => 5
[1,3,1,1,4] => 5
[1,3,1,4,1] => 5
[1,3,4,1,1] => 5
[1,4,1,1,3] => 5
[1,4,1,3,1] => 5
[1,4,3,1,1] => 5
[3,1,1,1,4] => 5
[3,1,1,4,1] => 5
[3,1,4,1,1] => 5
[3,4,1,1,1] => 5
[4,1,1,1,3] => 5
[4,1,1,3,1] => 5
[4,1,3,1,1] => 5
[4,3,1,1,1] => 5
[1,1,1,3,5] => 6
[1,1,1,5,3] => 6
[1,1,3,1,5] => 6
[1,1,3,5,1] => 6
[1,1,5,1,3] => 6
[1,1,5,3,1] => 6
[1,3,1,1,5] => 6
[1,3,1,5,1] => 6
[1,3,5,1,1] => 6
[1,5,1,1,3] => 6
[1,5,1,3,1] => 6
[1,5,3,1,1] => 6
[3,1,1,1,5] => 6
[3,1,1,5,1] => 6
[3,1,5,1,1] => 6
[3,5,1,1,1] => 6
[5,1,1,1,3] => 6
[5,1,1,3,1] => 6
[5,1,3,1,1] => 6
[5,3,1,1,1] => 6
[1,1,1,4,4] => 6
[1,1,4,1,4] => 6
[1,1,4,4,1] => 6
[1,4,1,1,4] => 6
[1,4,1,4,1] => 6
[1,4,4,1,1] => 6
[4,1,1,1,4] => 6
[4,1,1,4,1] => 6
[4,1,4,1,1] => 6
[4,4,1,1,1] => 6
[1,1,1,4,5] => 7
[1,1,1,5,4] => 7
[1,1,4,1,5] => 7
[1,1,4,5,1] => 7
[1,1,5,1,4] => 7
[1,1,5,4,1] => 7
[1,4,1,1,5] => 7
[1,4,1,5,1] => 7
[1,4,5,1,1] => 7
[1,5,1,1,4] => 7
[1,5,1,4,1] => 7
[1,5,4,1,1] => 7
[4,1,1,1,5] => 7
[4,1,1,5,1] => 7
[4,1,5,1,1] => 7
[4,5,1,1,1] => 7
[5,1,1,1,4] => 7
[5,1,1,4,1] => 7
[5,1,4,1,1] => 7
[5,4,1,1,1] => 7
[1,1,2,2,2] => 3
[1,2,1,2,2] => 3
[1,2,2,1,2] => 3
[1,2,2,2,1] => 3
[2,1,1,2,2] => 3
[2,1,2,1,2] => 3
[2,1,2,2,1] => 3
[2,2,1,1,2] => 3
[2,2,1,2,1] => 3
[2,2,2,1,1] => 3
[1,1,2,2,3] => 4
[1,1,2,3,2] => 4
[1,1,3,2,2] => 4
[1,2,1,2,3] => 4
[1,2,1,3,2] => 4
[1,2,2,1,3] => 4
[1,2,2,3,1] => 4
[1,2,3,1,2] => 4
[1,2,3,2,1] => 4
[1,3,1,2,2] => 4
[1,3,2,1,2] => 4
[1,3,2,2,1] => 4
[2,1,1,2,3] => 4
[2,1,1,3,2] => 4
[2,1,2,1,3] => 4
[2,1,2,3,1] => 4
[2,1,3,1,2] => 4
[2,1,3,2,1] => 4
[2,2,1,1,3] => 4
[2,2,1,3,1] => 4
[2,2,3,1,1] => 4
[2,3,1,1,2] => 4
[2,3,1,2,1] => 4
[2,3,2,1,1] => 4
[3,1,1,2,2] => 4
[3,1,2,1,2] => 4
[3,1,2,2,1] => 4
[3,2,1,1,2] => 4
[3,2,1,2,1] => 4
[3,2,2,1,1] => 4
[1,1,2,2,4] => 5
[1,1,2,4,2] => 5
[1,1,4,2,2] => 5
[1,2,1,2,4] => 5
[1,2,1,4,2] => 5
[1,2,2,1,4] => 5
[1,2,2,4,1] => 5
[1,2,4,1,2] => 5
[1,2,4,2,1] => 5
[1,4,1,2,2] => 5
[1,4,2,1,2] => 5
[1,4,2,2,1] => 5
[2,1,1,2,4] => 5
[2,1,1,4,2] => 5
[2,1,2,1,4] => 5
[2,1,2,4,1] => 5
[2,1,4,1,2] => 5
[2,1,4,2,1] => 5
[2,2,1,1,4] => 5
[2,2,1,4,1] => 5
[2,2,4,1,1] => 5
[2,4,1,1,2] => 5
[2,4,1,2,1] => 5
[2,4,2,1,1] => 5
[4,1,1,2,2] => 5
[4,1,2,1,2] => 5
[4,1,2,2,1] => 5
[4,2,1,1,2] => 5
[4,2,1,2,1] => 5
[4,2,2,1,1] => 5
[1,1,2,2,5] => 6
[1,1,2,5,2] => 6
[1,1,5,2,2] => 6
[1,2,1,2,5] => 6
[1,2,1,5,2] => 6
[1,2,2,1,5] => 6
[1,2,2,5,1] => 6
[1,2,5,1,2] => 6
[1,2,5,2,1] => 6
[1,5,1,2,2] => 6
[1,5,2,1,2] => 6
[1,5,2,2,1] => 6
[2,1,1,2,5] => 6
[2,1,1,5,2] => 6
[2,1,2,1,5] => 6
[2,1,2,5,1] => 6
[2,1,5,1,2] => 6
[2,1,5,2,1] => 6
[2,2,1,1,5] => 6
[2,2,1,5,1] => 6
[2,2,5,1,1] => 6
[2,5,1,1,2] => 6
[2,5,1,2,1] => 6
[2,5,2,1,1] => 6
[5,1,1,2,2] => 6
[5,1,2,1,2] => 6
[5,1,2,2,1] => 6
[5,2,1,1,2] => 6
[5,2,1,2,1] => 6
[5,2,2,1,1] => 6
[1,1,2,3,3] => 5
[1,1,3,2,3] => 5
[1,1,3,3,2] => 5
[1,2,1,3,3] => 5
[1,2,3,1,3] => 5
[1,2,3,3,1] => 5
[1,3,1,2,3] => 5
[1,3,1,3,2] => 5
[1,3,2,1,3] => 5
[1,3,2,3,1] => 5
[1,3,3,1,2] => 5
[1,3,3,2,1] => 5
[2,1,1,3,3] => 5
[2,1,3,1,3] => 5
[2,1,3,3,1] => 5
[2,3,1,1,3] => 5
[2,3,1,3,1] => 5
[2,3,3,1,1] => 5
[3,1,1,2,3] => 5
[3,1,1,3,2] => 5
[3,1,2,1,3] => 5
[3,1,2,3,1] => 5
[3,1,3,1,2] => 5
[3,1,3,2,1] => 5
[3,2,1,1,3] => 5
[3,2,1,3,1] => 5
[3,2,3,1,1] => 5
[3,3,1,1,2] => 5
[3,3,1,2,1] => 5
[3,3,2,1,1] => 5
[1,1,2,3,4] => 6
[1,1,2,4,3] => 6
[1,1,3,2,4] => 6
[1,1,3,4,2] => 6
[1,1,4,2,3] => 6
[1,1,4,3,2] => 6
[1,2,1,3,4] => 6
[1,2,1,4,3] => 6
[1,2,3,1,4] => 6
[1,2,3,4,1] => 6
[1,2,4,1,3] => 6
[1,2,4,3,1] => 6
[1,3,1,2,4] => 6
[1,3,1,4,2] => 6
[1,3,2,1,4] => 6
[1,3,2,4,1] => 6
[1,3,4,1,2] => 6
[1,3,4,2,1] => 6
[1,4,1,2,3] => 6
[1,4,1,3,2] => 6
[1,4,2,1,3] => 6
[1,4,2,3,1] => 6
[1,4,3,1,2] => 6
[1,4,3,2,1] => 6
[2,1,1,3,4] => 6
[2,1,1,4,3] => 6
[2,1,3,1,4] => 6
[2,1,3,4,1] => 6
[2,1,4,1,3] => 6
[2,1,4,3,1] => 6
[2,3,1,1,4] => 6
[2,3,1,4,1] => 6
[2,3,4,1,1] => 6
[2,4,1,1,3] => 6
[2,4,1,3,1] => 6
[2,4,3,1,1] => 6
[3,1,1,2,4] => 6
[3,1,1,4,2] => 6
[3,1,2,1,4] => 6
[3,1,2,4,1] => 6
[3,1,4,1,2] => 6
[3,1,4,2,1] => 6
[3,2,1,1,4] => 6
[3,2,1,4,1] => 6
[3,2,4,1,1] => 6
[3,4,1,1,2] => 6
[3,4,1,2,1] => 6
[3,4,2,1,1] => 6
[4,1,1,2,3] => 6
[4,1,1,3,2] => 6
[4,1,2,1,3] => 6
[4,1,2,3,1] => 6
[4,1,3,1,2] => 6
[4,1,3,2,1] => 6
[4,2,1,1,3] => 6
[4,2,1,3,1] => 6
[4,2,3,1,1] => 6
[4,3,1,1,2] => 6
[4,3,1,2,1] => 6
[4,3,2,1,1] => 6
[1,1,2,3,5] => 7
[1,1,2,5,3] => 7
[1,1,3,2,5] => 7
[1,1,3,5,2] => 7
[1,1,5,2,3] => 7
[1,1,5,3,2] => 7
[1,2,1,3,5] => 7
[1,2,1,5,3] => 7
[1,2,3,1,5] => 7
[1,2,3,5,1] => 7
[1,2,5,1,3] => 7
[1,2,5,3,1] => 7
[1,3,1,2,5] => 7
[1,3,1,5,2] => 7
[1,3,2,1,5] => 7
[1,3,2,5,1] => 7
[1,3,5,1,2] => 7
[1,3,5,2,1] => 7
[1,5,1,2,3] => 7
[1,5,1,3,2] => 7
[1,5,2,1,3] => 7
[1,5,2,3,1] => 7
[1,5,3,1,2] => 7
[1,5,3,2,1] => 7
[2,1,1,3,5] => 7
[2,1,1,5,3] => 7
[2,1,3,1,5] => 7
[2,1,3,5,1] => 7
[2,1,5,1,3] => 7
[2,1,5,3,1] => 7
[2,3,1,1,5] => 7
[2,3,1,5,1] => 7
[2,3,5,1,1] => 7
[2,5,1,1,3] => 7
[2,5,1,3,1] => 7
[2,5,3,1,1] => 7
[3,1,1,2,5] => 7
[3,1,1,5,2] => 7
[3,1,2,1,5] => 7
[3,1,2,5,1] => 7
[3,1,5,1,2] => 7
[3,1,5,2,1] => 7
[3,2,1,1,5] => 7
[3,2,1,5,1] => 7
[3,2,5,1,1] => 7
[3,5,1,1,2] => 7
[3,5,1,2,1] => 7
[3,5,2,1,1] => 7
[5,1,1,2,3] => 7
[5,1,1,3,2] => 7
[5,1,2,1,3] => 7
[5,1,2,3,1] => 7
[5,1,3,1,2] => 7
[5,1,3,2,1] => 7
[5,2,1,1,3] => 7
[5,2,1,3,1] => 7
[5,2,3,1,1] => 7
[5,3,1,1,2] => 7
[5,3,1,2,1] => 7
[5,3,2,1,1] => 7
[1,1,2,4,4] => 7
[1,1,4,2,4] => 7
[1,1,4,4,2] => 7
[1,2,1,4,4] => 7
[1,2,4,1,4] => 7
[1,2,4,4,1] => 7
[1,4,1,2,4] => 7
[1,4,1,4,2] => 7
[1,4,2,1,4] => 7
[1,4,2,4,1] => 7
[1,4,4,1,2] => 7
[1,4,4,2,1] => 7
[2,1,1,4,4] => 7
[2,1,4,1,4] => 7
[2,1,4,4,1] => 7
[2,4,1,1,4] => 7
[2,4,1,4,1] => 7
[2,4,4,1,1] => 7
[4,1,1,2,4] => 7
[4,1,1,4,2] => 7
[4,1,2,1,4] => 7
[4,1,2,4,1] => 7
[4,1,4,1,2] => 7
[4,1,4,2,1] => 7
[4,2,1,1,4] => 7
[4,2,1,4,1] => 7
[4,2,4,1,1] => 7
[4,4,1,1,2] => 7
[4,4,1,2,1] => 7
[4,4,2,1,1] => 7
[1,1,2,4,5] => 8
[1,1,2,5,4] => 8
[1,1,4,2,5] => 8
[1,1,4,5,2] => 8
[1,1,5,2,4] => 8
[1,1,5,4,2] => 8
[1,2,1,4,5] => 8
[1,2,1,5,4] => 8
[1,2,4,1,5] => 8
[1,2,4,5,1] => 8
[1,2,5,1,4] => 8
[1,2,5,4,1] => 8
[1,4,1,2,5] => 8
[1,4,1,5,2] => 8
[1,4,2,1,5] => 8
[1,4,2,5,1] => 8
[1,4,5,1,2] => 8
[1,4,5,2,1] => 8
[1,5,1,2,4] => 8
[1,5,1,4,2] => 8
[1,5,2,1,4] => 8
[1,5,2,4,1] => 8
[1,5,4,1,2] => 8
[1,5,4,2,1] => 8
[2,1,1,4,5] => 8
[2,1,1,5,4] => 8
[2,1,4,1,5] => 8
[2,1,4,5,1] => 8
[2,1,5,1,4] => 8
[2,1,5,4,1] => 8
[2,4,1,1,5] => 8
[2,4,1,5,1] => 8
[2,4,5,1,1] => 8
[2,5,1,1,4] => 8
[2,5,1,4,1] => 8
[2,5,4,1,1] => 8
[4,1,1,2,5] => 8
[4,1,1,5,2] => 8
[4,1,2,1,5] => 8
[4,1,2,5,1] => 8
[4,1,5,1,2] => 8
[4,1,5,2,1] => 8
[4,2,1,1,5] => 8
[4,2,1,5,1] => 8
[4,2,5,1,1] => 8
[4,5,1,1,2] => 8
[4,5,1,2,1] => 8
[4,5,2,1,1] => 8
[5,1,1,2,4] => 8
[5,1,1,4,2] => 8
[5,1,2,1,4] => 8
[5,1,2,4,1] => 8
[5,1,4,1,2] => 8
[5,1,4,2,1] => 8
[5,2,1,1,4] => 8
[5,2,1,4,1] => 8
[5,2,4,1,1] => 8
[5,4,1,1,2] => 8
[5,4,1,2,1] => 8
[5,4,2,1,1] => 8
[1,1,3,3,3] => 6
[1,3,1,3,3] => 6
[1,3,3,1,3] => 6
[1,3,3,3,1] => 6
[3,1,1,3,3] => 6
[3,1,3,1,3] => 6
[3,1,3,3,1] => 6
[3,3,1,1,3] => 6
[3,3,1,3,1] => 6
[3,3,3,1,1] => 6
[1,1,3,3,4] => 7
[1,1,3,4,3] => 7
[1,1,4,3,3] => 7
[1,3,1,3,4] => 7
[1,3,1,4,3] => 7
[1,3,3,1,4] => 7
[1,3,3,4,1] => 7
[1,3,4,1,3] => 7
[1,3,4,3,1] => 7
[1,4,1,3,3] => 7
[1,4,3,1,3] => 7
[1,4,3,3,1] => 7
[3,1,1,3,4] => 7
[3,1,1,4,3] => 7
[3,1,3,1,4] => 7
[3,1,3,4,1] => 7
[3,1,4,1,3] => 7
[3,1,4,3,1] => 7
[3,3,1,1,4] => 7
[3,3,1,4,1] => 7
[3,3,4,1,1] => 7
[3,4,1,1,3] => 7
[3,4,1,3,1] => 7
[3,4,3,1,1] => 7
[4,1,1,3,3] => 7
[4,1,3,1,3] => 7
[4,1,3,3,1] => 7
[4,3,1,1,3] => 7
[4,3,1,3,1] => 7
[4,3,3,1,1] => 7
[1,1,3,3,5] => 8
[1,1,3,5,3] => 8
[1,1,5,3,3] => 8
[1,3,1,3,5] => 8
[1,3,1,5,3] => 8
[1,3,3,1,5] => 8
[1,3,3,5,1] => 8
[1,3,5,1,3] => 8
[1,3,5,3,1] => 8
[1,5,1,3,3] => 8
[1,5,3,1,3] => 8
[1,5,3,3,1] => 8
[3,1,1,3,5] => 8
[3,1,1,5,3] => 8
[3,1,3,1,5] => 8
[3,1,3,5,1] => 8
[3,1,5,1,3] => 8
[3,1,5,3,1] => 8
[3,3,1,1,5] => 8
[3,3,1,5,1] => 8
[3,3,5,1,1] => 8
[3,5,1,1,3] => 8
[3,5,1,3,1] => 8
[3,5,3,1,1] => 8
[5,1,1,3,3] => 8
[5,1,3,1,3] => 8
[5,1,3,3,1] => 8
[5,3,1,1,3] => 8
[5,3,1,3,1] => 8
[5,3,3,1,1] => 8
[1,1,3,4,4] => 8
[1,1,4,3,4] => 8
[1,1,4,4,3] => 8
[1,3,1,4,4] => 8
[1,3,4,1,4] => 8
[1,3,4,4,1] => 8
[1,4,1,3,4] => 8
[1,4,1,4,3] => 8
[1,4,3,1,4] => 8
[1,4,3,4,1] => 8
[1,4,4,1,3] => 8
[1,4,4,3,1] => 8
[3,1,1,4,4] => 8
[3,1,4,1,4] => 8
[3,1,4,4,1] => 8
[3,4,1,1,4] => 8
[3,4,1,4,1] => 8
[3,4,4,1,1] => 8
[4,1,1,3,4] => 8
[4,1,1,4,3] => 8
[4,1,3,1,4] => 8
[4,1,3,4,1] => 8
[4,1,4,1,3] => 8
[4,1,4,3,1] => 8
[4,3,1,1,4] => 8
[4,3,1,4,1] => 8
[4,3,4,1,1] => 8
[4,4,1,1,3] => 8
[4,4,1,3,1] => 8
[4,4,3,1,1] => 8
[1,1,3,4,5] => 9
[1,1,3,5,4] => 9
[1,1,4,3,5] => 9
[1,1,4,5,3] => 9
[1,1,5,3,4] => 9
[1,1,5,4,3] => 9
[1,3,1,4,5] => 9
[1,3,1,5,4] => 9
[1,3,4,1,5] => 9
[1,3,4,5,1] => 9
[1,3,5,1,4] => 9
[1,3,5,4,1] => 9
[1,4,1,3,5] => 9
[1,4,1,5,3] => 9
[1,4,3,1,5] => 9
[1,4,3,5,1] => 9
[1,4,5,1,3] => 9
[1,4,5,3,1] => 9
[1,5,1,3,4] => 9
[1,5,1,4,3] => 9
[1,5,3,1,4] => 9
[1,5,3,4,1] => 9
[1,5,4,1,3] => 9
[1,5,4,3,1] => 9
[3,1,1,4,5] => 9
[3,1,1,5,4] => 9
[3,1,4,1,5] => 9
[3,1,4,5,1] => 9
[3,1,5,1,4] => 9
[3,1,5,4,1] => 9
[3,4,1,1,5] => 9
[3,4,1,5,1] => 9
[3,4,5,1,1] => 9
[3,5,1,1,4] => 9
[3,5,1,4,1] => 9
[3,5,4,1,1] => 9
[4,1,1,3,5] => 9
[4,1,1,5,3] => 9
[4,1,3,1,5] => 9
[4,1,3,5,1] => 9
[4,1,5,1,3] => 9
[4,1,5,3,1] => 9
[4,3,1,1,5] => 9
[4,3,1,5,1] => 9
[4,3,5,1,1] => 9
[4,5,1,1,3] => 9
[4,5,1,3,1] => 9
[4,5,3,1,1] => 9
[5,1,1,3,4] => 9
[5,1,1,4,3] => 9
[5,1,3,1,4] => 9
[5,1,3,4,1] => 9
[5,1,4,1,3] => 9
[5,1,4,3,1] => 9
[5,3,1,1,4] => 9
[5,3,1,4,1] => 9
[5,3,4,1,1] => 9
[5,4,1,1,3] => 9
[5,4,1,3,1] => 9
[5,4,3,1,1] => 9
[1,2,2,2,2] => 4
[2,1,2,2,2] => 4
[2,2,1,2,2] => 4
[2,2,2,1,2] => 4
[2,2,2,2,1] => 4
[1,2,2,2,3] => 5
[1,2,2,3,2] => 5
[1,2,3,2,2] => 5
[1,3,2,2,2] => 5
[2,1,2,2,3] => 5
[2,1,2,3,2] => 5
[2,1,3,2,2] => 5
[2,2,1,2,3] => 5
[2,2,1,3,2] => 5
[2,2,2,1,3] => 5
[2,2,2,3,1] => 5
[2,2,3,1,2] => 5
[2,2,3,2,1] => 5
[2,3,1,2,2] => 5
[2,3,2,1,2] => 5
[2,3,2,2,1] => 5
[3,1,2,2,2] => 5
[3,2,1,2,2] => 5
[3,2,2,1,2] => 5
[3,2,2,2,1] => 5
[1,2,2,2,4] => 6
[1,2,2,4,2] => 6
[1,2,4,2,2] => 6
[1,4,2,2,2] => 6
[2,1,2,2,4] => 6
[2,1,2,4,2] => 6
[2,1,4,2,2] => 6
[2,2,1,2,4] => 6
[2,2,1,4,2] => 6
[2,2,2,1,4] => 6
[2,2,2,4,1] => 6
[2,2,4,1,2] => 6
[2,2,4,2,1] => 6
[2,4,1,2,2] => 6
[2,4,2,1,2] => 6
[2,4,2,2,1] => 6
[4,1,2,2,2] => 6
[4,2,1,2,2] => 6
[4,2,2,1,2] => 6
[4,2,2,2,1] => 6
[1,2,2,2,5] => 7
[1,2,2,5,2] => 7
[1,2,5,2,2] => 7
[1,5,2,2,2] => 7
[2,1,2,2,5] => 7
[2,1,2,5,2] => 7
[2,1,5,2,2] => 7
[2,2,1,2,5] => 7
[2,2,1,5,2] => 7
[2,2,2,1,5] => 7
[2,2,2,5,1] => 7
[2,2,5,1,2] => 7
[2,2,5,2,1] => 7
[2,5,1,2,2] => 7
[2,5,2,1,2] => 7
[2,5,2,2,1] => 7
[5,1,2,2,2] => 7
[5,2,1,2,2] => 7
[5,2,2,1,2] => 7
[5,2,2,2,1] => 7
[1,2,2,3,3] => 6
[1,2,3,2,3] => 6
[1,2,3,3,2] => 6
[1,3,2,2,3] => 6
[1,3,2,3,2] => 6
[1,3,3,2,2] => 6
[2,1,2,3,3] => 6
[2,1,3,2,3] => 6
[2,1,3,3,2] => 6
[2,2,1,3,3] => 6
[2,2,3,1,3] => 6
[2,2,3,3,1] => 6
[2,3,1,2,3] => 6
[2,3,1,3,2] => 6
[2,3,2,1,3] => 6
[2,3,2,3,1] => 6
[2,3,3,1,2] => 6
[2,3,3,2,1] => 6
[3,1,2,2,3] => 6
[3,1,2,3,2] => 6
[3,1,3,2,2] => 6
[3,2,1,2,3] => 6
[3,2,1,3,2] => 6
[3,2,2,1,3] => 6
[3,2,2,3,1] => 6
[3,2,3,1,2] => 6
[3,2,3,2,1] => 6
[3,3,1,2,2] => 6
[3,3,2,1,2] => 6
[3,3,2,2,1] => 6
[1,2,2,3,4] => 7
[1,2,2,4,3] => 7
[1,2,3,2,4] => 7
[1,2,3,4,2] => 7
[1,2,4,2,3] => 7
[1,2,4,3,2] => 7
[1,3,2,2,4] => 7
[1,3,2,4,2] => 7
[1,3,4,2,2] => 7
[1,4,2,2,3] => 7
[1,4,2,3,2] => 7
[1,4,3,2,2] => 7
[2,1,2,3,4] => 7
[2,1,2,4,3] => 7
[2,1,3,2,4] => 7
[2,1,3,4,2] => 7
[2,1,4,2,3] => 7
[2,1,4,3,2] => 7
[2,2,1,3,4] => 7
[2,2,1,4,3] => 7
[2,2,3,1,4] => 7
[2,2,3,4,1] => 7
[2,2,4,1,3] => 7
[2,2,4,3,1] => 7
[2,3,1,2,4] => 7
[2,3,1,4,2] => 7
[2,3,2,1,4] => 7
[2,3,2,4,1] => 7
[2,3,4,1,2] => 7
[2,3,4,2,1] => 7
[2,4,1,2,3] => 7
[2,4,1,3,2] => 7
[2,4,2,1,3] => 7
[2,4,2,3,1] => 7
[2,4,3,1,2] => 7
[2,4,3,2,1] => 7
[3,1,2,2,4] => 7
[3,1,2,4,2] => 7
[3,1,4,2,2] => 7
[3,2,1,2,4] => 7
[3,2,1,4,2] => 7
[3,2,2,1,4] => 7
[3,2,2,4,1] => 7
[3,2,4,1,2] => 7
[3,2,4,2,1] => 7
[3,4,1,2,2] => 7
[3,4,2,1,2] => 7
[3,4,2,2,1] => 7
[4,1,2,2,3] => 7
[4,1,2,3,2] => 7
[4,1,3,2,2] => 7
[4,2,1,2,3] => 7
[4,2,1,3,2] => 7
[4,2,2,1,3] => 7
[4,2,2,3,1] => 7
[4,2,3,1,2] => 7
[4,2,3,2,1] => 7
[4,3,1,2,2] => 7
[4,3,2,1,2] => 7
[4,3,2,2,1] => 7
[1,2,2,3,5] => 8
[1,2,2,5,3] => 8
[1,2,3,2,5] => 8
[1,2,3,5,2] => 8
[1,2,5,2,3] => 8
[1,2,5,3,2] => 8
[1,3,2,2,5] => 8
[1,3,2,5,2] => 8
[1,3,5,2,2] => 8
[1,5,2,2,3] => 8
[1,5,2,3,2] => 8
[1,5,3,2,2] => 8
[2,1,2,3,5] => 8
[2,1,2,5,3] => 8
[2,1,3,2,5] => 8
[2,1,3,5,2] => 8
[2,1,5,2,3] => 8
[2,1,5,3,2] => 8
[2,2,1,3,5] => 8
[2,2,1,5,3] => 8
[2,2,3,1,5] => 8
[2,2,3,5,1] => 8
[2,2,5,1,3] => 8
[2,2,5,3,1] => 8
[2,3,1,2,5] => 8
[2,3,1,5,2] => 8
[2,3,2,1,5] => 8
[2,3,2,5,1] => 8
[2,3,5,1,2] => 8
[2,3,5,2,1] => 8
[2,5,1,2,3] => 8
[2,5,1,3,2] => 8
[2,5,2,1,3] => 8
[2,5,2,3,1] => 8
[2,5,3,1,2] => 8
[2,5,3,2,1] => 8
[3,1,2,2,5] => 8
[3,1,2,5,2] => 8
[3,1,5,2,2] => 8
[3,2,1,2,5] => 8
[3,2,1,5,2] => 8
[3,2,2,1,5] => 8
[3,2,2,5,1] => 8
[3,2,5,1,2] => 8
[3,2,5,2,1] => 8
[3,5,1,2,2] => 8
[3,5,2,1,2] => 8
[3,5,2,2,1] => 8
[5,1,2,2,3] => 8
[5,1,2,3,2] => 8
[5,1,3,2,2] => 8
[5,2,1,2,3] => 8
[5,2,1,3,2] => 8
[5,2,2,1,3] => 8
[5,2,2,3,1] => 8
[5,2,3,1,2] => 8
[5,2,3,2,1] => 8
[5,3,1,2,2] => 8
[5,3,2,1,2] => 8
[5,3,2,2,1] => 8
[1,2,2,4,4] => 8
[1,2,4,2,4] => 8
[1,2,4,4,2] => 8
[1,4,2,2,4] => 8
[1,4,2,4,2] => 8
[1,4,4,2,2] => 8
[2,1,2,4,4] => 8
[2,1,4,2,4] => 8
[2,1,4,4,2] => 8
[2,2,1,4,4] => 8
[2,2,4,1,4] => 8
[2,2,4,4,1] => 8
[2,4,1,2,4] => 8
[2,4,1,4,2] => 8
[2,4,2,1,4] => 8
[2,4,2,4,1] => 8
[2,4,4,1,2] => 8
[2,4,4,2,1] => 8
[4,1,2,2,4] => 8
[4,1,2,4,2] => 8
[4,1,4,2,2] => 8
[4,2,1,2,4] => 8
[4,2,1,4,2] => 8
[4,2,2,1,4] => 8
[4,2,2,4,1] => 8
[4,2,4,1,2] => 8
[4,2,4,2,1] => 8
[4,4,1,2,2] => 8
[4,4,2,1,2] => 8
[4,4,2,2,1] => 8
[1,2,2,4,5] => 9
[1,2,2,5,4] => 9
[1,2,4,2,5] => 9
[1,2,4,5,2] => 9
[1,2,5,2,4] => 9
[1,2,5,4,2] => 9
[1,4,2,2,5] => 9
[1,4,2,5,2] => 9
[1,4,5,2,2] => 9
[1,5,2,2,4] => 9
[1,5,2,4,2] => 9
[1,5,4,2,2] => 9
[2,1,2,4,5] => 9
[2,1,2,5,4] => 9
[2,1,4,2,5] => 9
[2,1,4,5,2] => 9
[2,1,5,2,4] => 9
[2,1,5,4,2] => 9
[2,2,1,4,5] => 9
[2,2,1,5,4] => 9
[2,2,4,1,5] => 9
[2,2,4,5,1] => 9
[2,2,5,1,4] => 9
[2,2,5,4,1] => 9
[2,4,1,2,5] => 9
[2,4,1,5,2] => 9
[2,4,2,1,5] => 9
[2,4,2,5,1] => 9
[2,4,5,1,2] => 9
[2,4,5,2,1] => 9
[2,5,1,2,4] => 9
[2,5,1,4,2] => 9
[2,5,2,1,4] => 9
[2,5,2,4,1] => 9
[2,5,4,1,2] => 9
[2,5,4,2,1] => 9
[4,1,2,2,5] => 9
[4,1,2,5,2] => 9
[4,1,5,2,2] => 9
[4,2,1,2,5] => 9
[4,2,1,5,2] => 9
[4,2,2,1,5] => 9
[4,2,2,5,1] => 9
[4,2,5,1,2] => 9
[4,2,5,2,1] => 9
[4,5,1,2,2] => 9
[4,5,2,1,2] => 9
[4,5,2,2,1] => 9
[5,1,2,2,4] => 9
[5,1,2,4,2] => 9
[5,1,4,2,2] => 9
[5,2,1,2,4] => 9
[5,2,1,4,2] => 9
[5,2,2,1,4] => 9
[5,2,2,4,1] => 9
[5,2,4,1,2] => 9
[5,2,4,2,1] => 9
[5,4,1,2,2] => 9
[5,4,2,1,2] => 9
[5,4,2,2,1] => 9
[1,2,3,3,3] => 7
[1,3,2,3,3] => 7
[1,3,3,2,3] => 7
[1,3,3,3,2] => 7
[2,1,3,3,3] => 7
[2,3,1,3,3] => 7
[2,3,3,1,3] => 7
[2,3,3,3,1] => 7
[3,1,2,3,3] => 7
[3,1,3,2,3] => 7
[3,1,3,3,2] => 7
[3,2,1,3,3] => 7
[3,2,3,1,3] => 7
[3,2,3,3,1] => 7
[3,3,1,2,3] => 7
[3,3,1,3,2] => 7
[3,3,2,1,3] => 7
[3,3,2,3,1] => 7
[3,3,3,1,2] => 7
[3,3,3,2,1] => 7
[1,2,3,3,4] => 8
[1,2,3,4,3] => 8
[1,2,4,3,3] => 8
[1,3,2,3,4] => 8
[1,3,2,4,3] => 8
[1,3,3,2,4] => 8
[1,3,3,4,2] => 8
[1,3,4,2,3] => 8
[1,3,4,3,2] => 8
[1,4,2,3,3] => 8
[1,4,3,2,3] => 8
[1,4,3,3,2] => 8
[2,1,3,3,4] => 8
[2,1,3,4,3] => 8
[2,1,4,3,3] => 8
[2,3,1,3,4] => 8
[2,3,1,4,3] => 8
[2,3,3,1,4] => 8
[2,3,3,4,1] => 8
[2,3,4,1,3] => 8
[2,3,4,3,1] => 8
[2,4,1,3,3] => 8
[2,4,3,1,3] => 8
[2,4,3,3,1] => 8
[3,1,2,3,4] => 8
[3,1,2,4,3] => 8
[3,1,3,2,4] => 8
[3,1,3,4,2] => 8
[3,1,4,2,3] => 8
[3,1,4,3,2] => 8
[3,2,1,3,4] => 8
[3,2,1,4,3] => 8
[3,2,3,1,4] => 8
[3,2,3,4,1] => 8
[3,2,4,1,3] => 8
[3,2,4,3,1] => 8
[3,3,1,2,4] => 8
[3,3,1,4,2] => 8
[3,3,2,1,4] => 8
[3,3,2,4,1] => 8
[3,3,4,1,2] => 8
[3,3,4,2,1] => 8
[3,4,1,2,3] => 8
[3,4,1,3,2] => 8
[3,4,2,1,3] => 8
[3,4,2,3,1] => 8
[3,4,3,1,2] => 8
[3,4,3,2,1] => 8
[4,1,2,3,3] => 8
[4,1,3,2,3] => 8
[4,1,3,3,2] => 8
[4,2,1,3,3] => 8
[4,2,3,1,3] => 8
[4,2,3,3,1] => 8
[4,3,1,2,3] => 8
[4,3,1,3,2] => 8
[4,3,2,1,3] => 8
[4,3,2,3,1] => 8
[4,3,3,1,2] => 8
click to show generating function       
Description
The sum of the entries of a parking function minus its length.
Under the standard map to labelled rooted forests, this statistic is sent to the number of inversions.
References
[1] Stanley, R. P. Hyperplane arrangements, parking functions and tree inversions MathSciNet:1627378
Code
def statistic(PF):
    return sum(PF) - len(PF)

Created
Jun 26, 2016 at 21:01 by Christian Stump
Updated
Jun 26, 2016 at 21:01 by Christian Stump