Identifier
Identifier
Values
[1] => 1
[1,1] => 0
[1,2] => 2
[2,1] => 1
[1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 0
[2,1,1] => 0
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 0
[2,2,1] => 0
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,1,1,1] => 0
[1,1,1,2] => 0
[1,1,2,1] => 0
[1,2,1,1] => 0
[2,1,1,1] => 0
[1,1,1,3] => 0
[1,1,3,1] => 0
[1,3,1,1] => 0
[3,1,1,1] => 0
[1,1,1,4] => 1
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 1
[1,1,2,2] => 0
[1,2,1,2] => 0
[1,2,2,1] => 0
[2,1,1,2] => 0
[2,1,2,1] => 0
[2,2,1,1] => 0
[1,1,2,3] => 0
[1,1,3,2] => 0
[1,2,1,3] => 0
[1,2,3,1] => 0
[1,3,1,2] => 0
[1,3,2,1] => 0
[2,1,1,3] => 0
[2,1,3,1] => 0
[2,3,1,1] => 0
[3,1,1,2] => 0
[3,1,2,1] => 0
[3,2,1,1] => 0
[1,1,2,4] => 1
[1,1,4,2] => 1
[1,2,1,4] => 1
[1,2,4,1] => 1
[1,4,1,2] => 1
[1,4,2,1] => 1
[2,1,1,4] => 1
[2,1,4,1] => 1
[2,4,1,1] => 1
[4,1,1,2] => 1
[4,1,2,1] => 1
[4,2,1,1] => 1
[1,1,3,3] => 0
[1,3,1,3] => 0
[1,3,3,1] => 0
[3,1,1,3] => 0
[3,1,3,1] => 0
[3,3,1,1] => 0
[1,1,3,4] => 2
[1,1,4,3] => 1
[1,3,1,4] => 2
[1,3,4,1] => 2
[1,4,1,3] => 1
[1,4,3,1] => 1
[3,1,1,4] => 2
[3,1,4,1] => 2
[3,4,1,1] => 2
[4,1,1,3] => 1
[4,1,3,1] => 1
[4,3,1,1] => 1
[1,2,2,2] => 1
[2,1,2,2] => 0
[2,2,1,2] => 0
[2,2,2,1] => 0
[1,2,2,3] => 1
[1,2,3,2] => 1
[1,3,2,2] => 1
[2,1,2,3] => 0
[2,1,3,2] => 0
[2,2,1,3] => 0
[2,2,3,1] => 0
[2,3,1,2] => 0
[2,3,2,1] => 0
[3,1,2,2] => 0
[3,2,1,2] => 0
[3,2,2,1] => 0
[1,2,2,4] => 2
[1,2,4,2] => 2
[1,4,2,2] => 2
[2,1,2,4] => 1
[2,1,4,2] => 1
[2,2,1,4] => 1
[2,2,4,1] => 1
[2,4,1,2] => 1
[2,4,2,1] => 1
[4,1,2,2] => 1
[4,2,1,2] => 1
[4,2,2,1] => 1
[1,2,3,3] => 2
[1,3,2,3] => 1
[1,3,3,2] => 1
[2,1,3,3] => 1
[2,3,1,3] => 1
[2,3,3,1] => 1
[3,1,2,3] => 0
[3,1,3,2] => 0
[3,2,1,3] => 0
[3,2,3,1] => 0
[3,3,1,2] => 0
[3,3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,1,1,1,1] => 0
[1,1,1,1,2] => 0
[1,1,1,2,1] => 0
[1,1,2,1,1] => 0
[1,2,1,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,3] => 0
[1,1,1,3,1] => 0
[1,1,3,1,1] => 0
[1,3,1,1,1] => 0
[3,1,1,1,1] => 0
[1,1,1,1,4] => 0
[1,1,1,4,1] => 0
[1,1,4,1,1] => 0
[1,4,1,1,1] => 0
[4,1,1,1,1] => 0
[1,1,1,1,5] => 1
[1,1,1,5,1] => 1
[1,1,5,1,1] => 1
[1,5,1,1,1] => 1
[5,1,1,1,1] => 1
[1,1,1,2,2] => 0
[1,1,2,1,2] => 0
[1,1,2,2,1] => 0
[1,2,1,1,2] => 0
[1,2,1,2,1] => 0
[1,2,2,1,1] => 0
[2,1,1,1,2] => 0
[2,1,1,2,1] => 0
[2,1,2,1,1] => 0
[2,2,1,1,1] => 0
[1,1,1,2,3] => 0
[1,1,1,3,2] => 0
[1,1,2,1,3] => 0
[1,1,2,3,1] => 0
[1,1,3,1,2] => 0
[1,1,3,2,1] => 0
[1,2,1,1,3] => 0
[1,2,1,3,1] => 0
[1,2,3,1,1] => 0
[1,3,1,1,2] => 0
[1,3,1,2,1] => 0
[1,3,2,1,1] => 0
[2,1,1,1,3] => 0
[2,1,1,3,1] => 0
[2,1,3,1,1] => 0
[2,3,1,1,1] => 0
[3,1,1,1,2] => 0
[3,1,1,2,1] => 0
[3,1,2,1,1] => 0
[3,2,1,1,1] => 0
[1,1,1,2,4] => 0
[1,1,1,4,2] => 0
[1,1,2,1,4] => 0
[1,1,2,4,1] => 0
[1,1,4,1,2] => 0
[1,1,4,2,1] => 0
[1,2,1,1,4] => 0
[1,2,1,4,1] => 0
[1,2,4,1,1] => 0
[1,4,1,1,2] => 0
[1,4,1,2,1] => 0
[1,4,2,1,1] => 0
[2,1,1,1,4] => 0
[2,1,1,4,1] => 0
[2,1,4,1,1] => 0
[2,4,1,1,1] => 0
[4,1,1,1,2] => 0
[4,1,1,2,1] => 0
[4,1,2,1,1] => 0
[4,2,1,1,1] => 0
[1,1,1,2,5] => 1
[1,1,1,5,2] => 1
[1,1,2,1,5] => 1
[1,1,2,5,1] => 1
[1,1,5,1,2] => 1
[1,1,5,2,1] => 1
[1,2,1,1,5] => 1
[1,2,1,5,1] => 1
[1,2,5,1,1] => 1
[1,5,1,1,2] => 1
[1,5,1,2,1] => 1
[1,5,2,1,1] => 1
[2,1,1,1,5] => 1
[2,1,1,5,1] => 1
[2,1,5,1,1] => 1
[2,5,1,1,1] => 1
[5,1,1,1,2] => 1
[5,1,1,2,1] => 1
[5,1,2,1,1] => 1
[5,2,1,1,1] => 1
[1,1,1,3,3] => 0
[1,1,3,1,3] => 0
[1,1,3,3,1] => 0
[1,3,1,1,3] => 0
[1,3,1,3,1] => 0
[1,3,3,1,1] => 0
[3,1,1,1,3] => 0
[3,1,1,3,1] => 0
[3,1,3,1,1] => 0
[3,3,1,1,1] => 0
[1,1,1,3,4] => 0
[1,1,1,4,3] => 0
[1,1,3,1,4] => 0
[1,1,3,4,1] => 0
[1,1,4,1,3] => 0
[1,1,4,3,1] => 0
[1,3,1,1,4] => 0
[1,3,1,4,1] => 0
[1,3,4,1,1] => 0
[1,4,1,1,3] => 0
[1,4,1,3,1] => 0
[1,4,3,1,1] => 0
[3,1,1,1,4] => 0
[3,1,1,4,1] => 0
[3,1,4,1,1] => 0
[3,4,1,1,1] => 0
[4,1,1,1,3] => 0
[4,1,1,3,1] => 0
[4,1,3,1,1] => 0
[4,3,1,1,1] => 0
[1,1,1,3,5] => 1
[1,1,1,5,3] => 1
[1,1,3,1,5] => 1
[1,1,3,5,1] => 1
[1,1,5,1,3] => 1
[1,1,5,3,1] => 1
[1,3,1,1,5] => 1
[1,3,1,5,1] => 1
[1,3,5,1,1] => 1
[1,5,1,1,3] => 1
[1,5,1,3,1] => 1
[1,5,3,1,1] => 1
[3,1,1,1,5] => 1
[3,1,1,5,1] => 1
[3,1,5,1,1] => 1
[3,5,1,1,1] => 1
[5,1,1,1,3] => 1
[5,1,1,3,1] => 1
[5,1,3,1,1] => 1
[5,3,1,1,1] => 1
[1,1,1,4,4] => 0
[1,1,4,1,4] => 0
[1,1,4,4,1] => 0
[1,4,1,1,4] => 0
[1,4,1,4,1] => 0
[1,4,4,1,1] => 0
[4,1,1,1,4] => 0
[4,1,1,4,1] => 0
[4,1,4,1,1] => 0
[4,4,1,1,1] => 0
[1,1,1,4,5] => 2
[1,1,1,5,4] => 1
[1,1,4,1,5] => 2
[1,1,4,5,1] => 2
[1,1,5,1,4] => 1
[1,1,5,4,1] => 1
[1,4,1,1,5] => 2
[1,4,1,5,1] => 2
[1,4,5,1,1] => 2
[1,5,1,1,4] => 1
[1,5,1,4,1] => 1
[1,5,4,1,1] => 1
[4,1,1,1,5] => 2
[4,1,1,5,1] => 2
[4,1,5,1,1] => 2
[4,5,1,1,1] => 2
[5,1,1,1,4] => 1
[5,1,1,4,1] => 1
[5,1,4,1,1] => 1
[5,4,1,1,1] => 1
[1,1,2,2,2] => 0
[1,2,1,2,2] => 0
[1,2,2,1,2] => 0
[1,2,2,2,1] => 0
[2,1,1,2,2] => 0
[2,1,2,1,2] => 0
[2,1,2,2,1] => 0
[2,2,1,1,2] => 0
[2,2,1,2,1] => 0
[2,2,2,1,1] => 0
[1,1,2,2,3] => 0
[1,1,2,3,2] => 0
[1,1,3,2,2] => 0
[1,2,1,2,3] => 0
[1,2,1,3,2] => 0
[1,2,2,1,3] => 0
[1,2,2,3,1] => 0
[1,2,3,1,2] => 0
[1,2,3,2,1] => 0
[1,3,1,2,2] => 0
[1,3,2,1,2] => 0
[1,3,2,2,1] => 0
[2,1,1,2,3] => 0
[2,1,1,3,2] => 0
[2,1,2,1,3] => 0
[2,1,2,3,1] => 0
[2,1,3,1,2] => 0
[2,1,3,2,1] => 0
[2,2,1,1,3] => 0
[2,2,1,3,1] => 0
[2,2,3,1,1] => 0
[2,3,1,1,2] => 0
[2,3,1,2,1] => 0
[2,3,2,1,1] => 0
[3,1,1,2,2] => 0
[3,1,2,1,2] => 0
[3,1,2,2,1] => 0
[3,2,1,1,2] => 0
[3,2,1,2,1] => 0
[3,2,2,1,1] => 0
[1,1,2,2,4] => 0
[1,1,2,4,2] => 0
[1,1,4,2,2] => 0
[1,2,1,2,4] => 0
[1,2,1,4,2] => 0
[1,2,2,1,4] => 0
[1,2,2,4,1] => 0
[1,2,4,1,2] => 0
[1,2,4,2,1] => 0
[1,4,1,2,2] => 0
[1,4,2,1,2] => 0
[1,4,2,2,1] => 0
[2,1,1,2,4] => 0
[2,1,1,4,2] => 0
[2,1,2,1,4] => 0
[2,1,2,4,1] => 0
[2,1,4,1,2] => 0
[2,1,4,2,1] => 0
[2,2,1,1,4] => 0
[2,2,1,4,1] => 0
[2,2,4,1,1] => 0
[2,4,1,1,2] => 0
[2,4,1,2,1] => 0
[2,4,2,1,1] => 0
[4,1,1,2,2] => 0
[4,1,2,1,2] => 0
[4,1,2,2,1] => 0
[4,2,1,1,2] => 0
[4,2,1,2,1] => 0
[4,2,2,1,1] => 0
[1,1,2,2,5] => 1
[1,1,2,5,2] => 1
[1,1,5,2,2] => 1
[1,2,1,2,5] => 1
[1,2,1,5,2] => 1
[1,2,2,1,5] => 1
[1,2,2,5,1] => 1
[1,2,5,1,2] => 1
[1,2,5,2,1] => 1
[1,5,1,2,2] => 1
[1,5,2,1,2] => 1
[1,5,2,2,1] => 1
[2,1,1,2,5] => 1
[2,1,1,5,2] => 1
[2,1,2,1,5] => 1
[2,1,2,5,1] => 1
[2,1,5,1,2] => 1
[2,1,5,2,1] => 1
[2,2,1,1,5] => 1
[2,2,1,5,1] => 1
[2,2,5,1,1] => 1
[2,5,1,1,2] => 1
[2,5,1,2,1] => 1
[2,5,2,1,1] => 1
[5,1,1,2,2] => 1
[5,1,2,1,2] => 1
[5,1,2,2,1] => 1
[5,2,1,1,2] => 1
[5,2,1,2,1] => 1
[5,2,2,1,1] => 1
[1,1,2,3,3] => 0
[1,1,3,2,3] => 0
[1,1,3,3,2] => 0
[1,2,1,3,3] => 0
[1,2,3,1,3] => 0
[1,2,3,3,1] => 0
[1,3,1,2,3] => 0
[1,3,1,3,2] => 0
[1,3,2,1,3] => 0
[1,3,2,3,1] => 0
[1,3,3,1,2] => 0
[1,3,3,2,1] => 0
[2,1,1,3,3] => 0
[2,1,3,1,3] => 0
[2,1,3,3,1] => 0
[2,3,1,1,3] => 0
[2,3,1,3,1] => 0
[2,3,3,1,1] => 0
[3,1,1,2,3] => 0
[3,1,1,3,2] => 0
[3,1,2,1,3] => 0
[3,1,2,3,1] => 0
[3,1,3,1,2] => 0
[3,1,3,2,1] => 0
[3,2,1,1,3] => 0
[3,2,1,3,1] => 0
[3,2,3,1,1] => 0
[3,3,1,1,2] => 0
[3,3,1,2,1] => 0
[3,3,2,1,1] => 0
[1,1,2,3,4] => 0
[1,1,2,4,3] => 0
[1,1,3,2,4] => 0
[1,1,3,4,2] => 0
[1,1,4,2,3] => 0
[1,1,4,3,2] => 0
[1,2,1,3,4] => 0
[1,2,1,4,3] => 0
[1,2,3,1,4] => 0
[1,2,3,4,1] => 0
[1,2,4,1,3] => 0
[1,2,4,3,1] => 0
[1,3,1,2,4] => 0
[1,3,1,4,2] => 0
[1,3,2,1,4] => 0
[1,3,2,4,1] => 0
[1,3,4,1,2] => 0
[1,3,4,2,1] => 0
[1,4,1,2,3] => 0
[1,4,1,3,2] => 0
[1,4,2,1,3] => 0
[1,4,2,3,1] => 0
[1,4,3,1,2] => 0
[1,4,3,2,1] => 0
[2,1,1,3,4] => 0
[2,1,1,4,3] => 0
[2,1,3,1,4] => 0
[2,1,3,4,1] => 0
[2,1,4,1,3] => 0
[2,1,4,3,1] => 0
[2,3,1,1,4] => 0
[2,3,1,4,1] => 0
[2,3,4,1,1] => 0
[2,4,1,1,3] => 0
[2,4,1,3,1] => 0
[2,4,3,1,1] => 0
[3,1,1,2,4] => 0
[3,1,1,4,2] => 0
[3,1,2,1,4] => 0
[3,1,2,4,1] => 0
[3,1,4,1,2] => 0
[3,1,4,2,1] => 0
[3,2,1,1,4] => 0
[3,2,1,4,1] => 0
[3,2,4,1,1] => 0
[3,4,1,1,2] => 0
[3,4,1,2,1] => 0
[3,4,2,1,1] => 0
[4,1,1,2,3] => 0
[4,1,1,3,2] => 0
[4,1,2,1,3] => 0
[4,1,2,3,1] => 0
[4,1,3,1,2] => 0
[4,1,3,2,1] => 0
[4,2,1,1,3] => 0
[4,2,1,3,1] => 0
[4,2,3,1,1] => 0
[4,3,1,1,2] => 0
[4,3,1,2,1] => 0
[4,3,2,1,1] => 0
[1,1,2,3,5] => 1
[1,1,2,5,3] => 1
[1,1,3,2,5] => 1
[1,1,3,5,2] => 1
[1,1,5,2,3] => 1
[1,1,5,3,2] => 1
[1,2,1,3,5] => 1
[1,2,1,5,3] => 1
[1,2,3,1,5] => 1
[1,2,3,5,1] => 1
[1,2,5,1,3] => 1
[1,2,5,3,1] => 1
[1,3,1,2,5] => 1
[1,3,1,5,2] => 1
[1,3,2,1,5] => 1
[1,3,2,5,1] => 1
[1,3,5,1,2] => 1
[1,3,5,2,1] => 1
[1,5,1,2,3] => 1
[1,5,1,3,2] => 1
[1,5,2,1,3] => 1
[1,5,2,3,1] => 1
[1,5,3,1,2] => 1
[1,5,3,2,1] => 1
[2,1,1,3,5] => 1
[2,1,1,5,3] => 1
[2,1,3,1,5] => 1
[2,1,3,5,1] => 1
[2,1,5,1,3] => 1
[2,1,5,3,1] => 1
[2,3,1,1,5] => 1
[2,3,1,5,1] => 1
[2,3,5,1,1] => 1
[2,5,1,1,3] => 1
[2,5,1,3,1] => 1
[2,5,3,1,1] => 1
[3,1,1,2,5] => 1
[3,1,1,5,2] => 1
[3,1,2,1,5] => 1
[3,1,2,5,1] => 1
[3,1,5,1,2] => 1
[3,1,5,2,1] => 1
[3,2,1,1,5] => 1
[3,2,1,5,1] => 1
[3,2,5,1,1] => 1
[3,5,1,1,2] => 1
[3,5,1,2,1] => 1
[3,5,2,1,1] => 1
[5,1,1,2,3] => 1
[5,1,1,3,2] => 1
[5,1,2,1,3] => 1
[5,1,2,3,1] => 1
[5,1,3,1,2] => 1
[5,1,3,2,1] => 1
[5,2,1,1,3] => 1
[5,2,1,3,1] => 1
[5,2,3,1,1] => 1
[5,3,1,1,2] => 1
[5,3,1,2,1] => 1
[5,3,2,1,1] => 1
[1,1,2,4,4] => 0
[1,1,4,2,4] => 0
[1,1,4,4,2] => 0
[1,2,1,4,4] => 0
[1,2,4,1,4] => 0
[1,2,4,4,1] => 0
[1,4,1,2,4] => 0
[1,4,1,4,2] => 0
[1,4,2,1,4] => 0
[1,4,2,4,1] => 0
[1,4,4,1,2] => 0
[1,4,4,2,1] => 0
[2,1,1,4,4] => 0
[2,1,4,1,4] => 0
[2,1,4,4,1] => 0
[2,4,1,1,4] => 0
[2,4,1,4,1] => 0
[2,4,4,1,1] => 0
[4,1,1,2,4] => 0
[4,1,1,4,2] => 0
[4,1,2,1,4] => 0
[4,1,2,4,1] => 0
[4,1,4,1,2] => 0
[4,1,4,2,1] => 0
[4,2,1,1,4] => 0
[4,2,1,4,1] => 0
[4,2,4,1,1] => 0
[4,4,1,1,2] => 0
[4,4,1,2,1] => 0
[4,4,2,1,1] => 0
[1,1,2,4,5] => 2
[1,1,2,5,4] => 1
[1,1,4,2,5] => 2
[1,1,4,5,2] => 2
[1,1,5,2,4] => 1
[1,1,5,4,2] => 1
[1,2,1,4,5] => 2
[1,2,1,5,4] => 1
[1,2,4,1,5] => 2
[1,2,4,5,1] => 2
[1,2,5,1,4] => 1
[1,2,5,4,1] => 1
[1,4,1,2,5] => 2
[1,4,1,5,2] => 2
[1,4,2,1,5] => 2
[1,4,2,5,1] => 2
[1,4,5,1,2] => 2
[1,4,5,2,1] => 2
[1,5,1,2,4] => 1
[1,5,1,4,2] => 1
[1,5,2,1,4] => 1
[1,5,2,4,1] => 1
[1,5,4,1,2] => 1
[1,5,4,2,1] => 1
[2,1,1,4,5] => 2
[2,1,1,5,4] => 1
[2,1,4,1,5] => 2
[2,1,4,5,1] => 2
[2,1,5,1,4] => 1
[2,1,5,4,1] => 1
[2,4,1,1,5] => 2
[2,4,1,5,1] => 2
[2,4,5,1,1] => 2
[2,5,1,1,4] => 1
[2,5,1,4,1] => 1
[2,5,4,1,1] => 1
[4,1,1,2,5] => 2
[4,1,1,5,2] => 2
[4,1,2,1,5] => 2
[4,1,2,5,1] => 2
[4,1,5,1,2] => 2
[4,1,5,2,1] => 2
[4,2,1,1,5] => 2
[4,2,1,5,1] => 2
[4,2,5,1,1] => 2
[4,5,1,1,2] => 2
[4,5,1,2,1] => 2
[4,5,2,1,1] => 2
[5,1,1,2,4] => 1
[5,1,1,4,2] => 1
[5,1,2,1,4] => 1
[5,1,2,4,1] => 1
[5,1,4,1,2] => 1
[5,1,4,2,1] => 1
[5,2,1,1,4] => 1
[5,2,1,4,1] => 1
[5,2,4,1,1] => 1
[5,4,1,1,2] => 1
[5,4,1,2,1] => 1
[5,4,2,1,1] => 1
[1,1,3,3,3] => 0
[1,3,1,3,3] => 0
[1,3,3,1,3] => 0
[1,3,3,3,1] => 0
[3,1,1,3,3] => 0
[3,1,3,1,3] => 0
[3,1,3,3,1] => 0
[3,3,1,1,3] => 0
[3,3,1,3,1] => 0
[3,3,3,1,1] => 0
[1,1,3,3,4] => 0
[1,1,3,4,3] => 0
[1,1,4,3,3] => 0
[1,3,1,3,4] => 0
[1,3,1,4,3] => 0
[1,3,3,1,4] => 0
[1,3,3,4,1] => 0
[1,3,4,1,3] => 0
[1,3,4,3,1] => 0
[1,4,1,3,3] => 0
[1,4,3,1,3] => 0
[1,4,3,3,1] => 0
[3,1,1,3,4] => 0
[3,1,1,4,3] => 0
[3,1,3,1,4] => 0
[3,1,3,4,1] => 0
[3,1,4,1,3] => 0
[3,1,4,3,1] => 0
[3,3,1,1,4] => 0
[3,3,1,4,1] => 0
[3,3,4,1,1] => 0
[3,4,1,1,3] => 0
[3,4,1,3,1] => 0
[3,4,3,1,1] => 0
[4,1,1,3,3] => 0
[4,1,3,1,3] => 0
[4,1,3,3,1] => 0
[4,3,1,1,3] => 0
[4,3,1,3,1] => 0
[4,3,3,1,1] => 0
[1,1,3,3,5] => 1
[1,1,3,5,3] => 1
[1,1,5,3,3] => 1
[1,3,1,3,5] => 1
[1,3,1,5,3] => 1
[1,3,3,1,5] => 1
[1,3,3,5,1] => 1
[1,3,5,1,3] => 1
[1,3,5,3,1] => 1
[1,5,1,3,3] => 1
[1,5,3,1,3] => 1
[1,5,3,3,1] => 1
[3,1,1,3,5] => 1
[3,1,1,5,3] => 1
[3,1,3,1,5] => 1
[3,1,3,5,1] => 1
[3,1,5,1,3] => 1
[3,1,5,3,1] => 1
[3,3,1,1,5] => 1
[3,3,1,5,1] => 1
[3,3,5,1,1] => 1
[3,5,1,1,3] => 1
[3,5,1,3,1] => 1
[3,5,3,1,1] => 1
[5,1,1,3,3] => 1
[5,1,3,1,3] => 1
[5,1,3,3,1] => 1
[5,3,1,1,3] => 1
[5,3,1,3,1] => 1
[5,3,3,1,1] => 1
[1,1,3,4,4] => 1
[1,1,4,3,4] => 0
[1,1,4,4,3] => 0
[1,3,1,4,4] => 1
[1,3,4,1,4] => 1
[1,3,4,4,1] => 1
[1,4,1,3,4] => 0
[1,4,1,4,3] => 0
[1,4,3,1,4] => 0
[1,4,3,4,1] => 0
[1,4,4,1,3] => 0
[1,4,4,3,1] => 0
[3,1,1,4,4] => 1
[3,1,4,1,4] => 1
[3,1,4,4,1] => 1
[3,4,1,1,4] => 1
[3,4,1,4,1] => 1
[3,4,4,1,1] => 1
[4,1,1,3,4] => 0
[4,1,1,4,3] => 0
[4,1,3,1,4] => 0
[4,1,3,4,1] => 0
[4,1,4,1,3] => 0
[4,1,4,3,1] => 0
[4,3,1,1,4] => 0
[4,3,1,4,1] => 0
[4,3,4,1,1] => 0
[4,4,1,1,3] => 0
[4,4,1,3,1] => 0
[4,4,3,1,1] => 0
[1,1,3,4,5] => 3
[1,1,3,5,4] => 2
[1,1,4,3,5] => 2
[1,1,4,5,3] => 2
[1,1,5,3,4] => 1
[1,1,5,4,3] => 1
[1,3,1,4,5] => 3
[1,3,1,5,4] => 2
[1,3,4,1,5] => 3
[1,3,4,5,1] => 3
[1,3,5,1,4] => 2
[1,3,5,4,1] => 2
[1,4,1,3,5] => 2
[1,4,1,5,3] => 2
[1,4,3,1,5] => 2
[1,4,3,5,1] => 2
[1,4,5,1,3] => 2
[1,4,5,3,1] => 2
[1,5,1,3,4] => 1
[1,5,1,4,3] => 1
[1,5,3,1,4] => 1
[1,5,3,4,1] => 1
[1,5,4,1,3] => 1
[1,5,4,3,1] => 1
[3,1,1,4,5] => 3
[3,1,1,5,4] => 2
[3,1,4,1,5] => 3
[3,1,4,5,1] => 3
[3,1,5,1,4] => 2
[3,1,5,4,1] => 2
[3,4,1,1,5] => 3
[3,4,1,5,1] => 3
[3,4,5,1,1] => 3
[3,5,1,1,4] => 2
[3,5,1,4,1] => 2
[3,5,4,1,1] => 2
[4,1,1,3,5] => 2
[4,1,1,5,3] => 2
[4,1,3,1,5] => 2
[4,1,3,5,1] => 2
[4,1,5,1,3] => 2
[4,1,5,3,1] => 2
[4,3,1,1,5] => 2
[4,3,1,5,1] => 2
[4,3,5,1,1] => 2
[4,5,1,1,3] => 2
[4,5,1,3,1] => 2
[4,5,3,1,1] => 2
[5,1,1,3,4] => 1
[5,1,1,4,3] => 1
[5,1,3,1,4] => 1
[5,1,3,4,1] => 1
[5,1,4,1,3] => 1
[5,1,4,3,1] => 1
[5,3,1,1,4] => 1
[5,3,1,4,1] => 1
[5,3,4,1,1] => 1
[5,4,1,1,3] => 1
[5,4,1,3,1] => 1
[5,4,3,1,1] => 1
[1,2,2,2,2] => 1
[2,1,2,2,2] => 0
[2,2,1,2,2] => 0
[2,2,2,1,2] => 0
[2,2,2,2,1] => 0
[1,2,2,2,3] => 1
[1,2,2,3,2] => 1
[1,2,3,2,2] => 1
[1,3,2,2,2] => 1
[2,1,2,2,3] => 0
[2,1,2,3,2] => 0
[2,1,3,2,2] => 0
[2,2,1,2,3] => 0
[2,2,1,3,2] => 0
[2,2,2,1,3] => 0
[2,2,2,3,1] => 0
[2,2,3,1,2] => 0
[2,2,3,2,1] => 0
[2,3,1,2,2] => 0
[2,3,2,1,2] => 0
[2,3,2,2,1] => 0
[3,1,2,2,2] => 0
[3,2,1,2,2] => 0
[3,2,2,1,2] => 0
[3,2,2,2,1] => 0
[1,2,2,2,4] => 1
[1,2,2,4,2] => 1
[1,2,4,2,2] => 1
[1,4,2,2,2] => 1
[2,1,2,2,4] => 0
[2,1,2,4,2] => 0
[2,1,4,2,2] => 0
[2,2,1,2,4] => 0
[2,2,1,4,2] => 0
[2,2,2,1,4] => 0
[2,2,2,4,1] => 0
[2,2,4,1,2] => 0
[2,2,4,2,1] => 0
[2,4,1,2,2] => 0
[2,4,2,1,2] => 0
[2,4,2,2,1] => 0
[4,1,2,2,2] => 0
[4,2,1,2,2] => 0
[4,2,2,1,2] => 0
[4,2,2,2,1] => 0
[1,2,2,2,5] => 2
[1,2,2,5,2] => 2
[1,2,5,2,2] => 2
[1,5,2,2,2] => 2
[2,1,2,2,5] => 1
[2,1,2,5,2] => 1
[2,1,5,2,2] => 1
[2,2,1,2,5] => 1
[2,2,1,5,2] => 1
[2,2,2,1,5] => 1
[2,2,2,5,1] => 1
[2,2,5,1,2] => 1
[2,2,5,2,1] => 1
[2,5,1,2,2] => 1
[2,5,2,1,2] => 1
[2,5,2,2,1] => 1
[5,1,2,2,2] => 1
[5,2,1,2,2] => 1
[5,2,2,1,2] => 1
[5,2,2,2,1] => 1
[1,2,2,3,3] => 1
[1,2,3,2,3] => 1
[1,2,3,3,2] => 1
[1,3,2,2,3] => 1
[1,3,2,3,2] => 1
[1,3,3,2,2] => 1
[2,1,2,3,3] => 0
[2,1,3,2,3] => 0
[2,1,3,3,2] => 0
[2,2,1,3,3] => 0
[2,2,3,1,3] => 0
[2,2,3,3,1] => 0
[2,3,1,2,3] => 0
[2,3,1,3,2] => 0
[2,3,2,1,3] => 0
[2,3,2,3,1] => 0
[2,3,3,1,2] => 0
[2,3,3,2,1] => 0
[3,1,2,2,3] => 0
[3,1,2,3,2] => 0
[3,1,3,2,2] => 0
[3,2,1,2,3] => 0
[3,2,1,3,2] => 0
[3,2,2,1,3] => 0
[3,2,2,3,1] => 0
[3,2,3,1,2] => 0
[3,2,3,2,1] => 0
[3,3,1,2,2] => 0
[3,3,2,1,2] => 0
[3,3,2,2,1] => 0
[1,2,2,3,4] => 1
[1,2,2,4,3] => 1
[1,2,3,2,4] => 1
[1,2,3,4,2] => 1
[1,2,4,2,3] => 1
[1,2,4,3,2] => 1
[1,3,2,2,4] => 1
[1,3,2,4,2] => 1
[1,3,4,2,2] => 1
[1,4,2,2,3] => 1
[1,4,2,3,2] => 1
[1,4,3,2,2] => 1
[2,1,2,3,4] => 0
[2,1,2,4,3] => 0
[2,1,3,2,4] => 0
[2,1,3,4,2] => 0
[2,1,4,2,3] => 0
[2,1,4,3,2] => 0
[2,2,1,3,4] => 0
[2,2,1,4,3] => 0
[2,2,3,1,4] => 0
[2,2,3,4,1] => 0
[2,2,4,1,3] => 0
[2,2,4,3,1] => 0
[2,3,1,2,4] => 0
[2,3,1,4,2] => 0
[2,3,2,1,4] => 0
[2,3,2,4,1] => 0
[2,3,4,1,2] => 0
[2,3,4,2,1] => 0
[2,4,1,2,3] => 0
[2,4,1,3,2] => 0
[2,4,2,1,3] => 0
[2,4,2,3,1] => 0
[2,4,3,1,2] => 0
[2,4,3,2,1] => 0
[3,1,2,2,4] => 0
[3,1,2,4,2] => 0
[3,1,4,2,2] => 0
[3,2,1,2,4] => 0
[3,2,1,4,2] => 0
[3,2,2,1,4] => 0
[3,2,2,4,1] => 0
[3,2,4,1,2] => 0
[3,2,4,2,1] => 0
[3,4,1,2,2] => 0
[3,4,2,1,2] => 0
[3,4,2,2,1] => 0
[4,1,2,2,3] => 0
[4,1,2,3,2] => 0
[4,1,3,2,2] => 0
[4,2,1,2,3] => 0
[4,2,1,3,2] => 0
[4,2,2,1,3] => 0
[4,2,2,3,1] => 0
[4,2,3,1,2] => 0
[4,2,3,2,1] => 0
[4,3,1,2,2] => 0
[4,3,2,1,2] => 0
[4,3,2,2,1] => 0
[1,2,2,3,5] => 2
[1,2,2,5,3] => 2
[1,2,3,2,5] => 2
[1,2,3,5,2] => 2
[1,2,5,2,3] => 2
[1,2,5,3,2] => 2
[1,3,2,2,5] => 2
[1,3,2,5,2] => 2
[1,3,5,2,2] => 2
[1,5,2,2,3] => 2
[1,5,2,3,2] => 2
[1,5,3,2,2] => 2
[2,1,2,3,5] => 1
[2,1,2,5,3] => 1
[2,1,3,2,5] => 1
[2,1,3,5,2] => 1
[2,1,5,2,3] => 1
[2,1,5,3,2] => 1
[2,2,1,3,5] => 1
[2,2,1,5,3] => 1
[2,2,3,1,5] => 1
[2,2,3,5,1] => 1
[2,2,5,1,3] => 1
[2,2,5,3,1] => 1
[2,3,1,2,5] => 1
[2,3,1,5,2] => 1
[2,3,2,1,5] => 1
[2,3,2,5,1] => 1
[2,3,5,1,2] => 1
[2,3,5,2,1] => 1
[2,5,1,2,3] => 1
[2,5,1,3,2] => 1
[2,5,2,1,3] => 1
[2,5,2,3,1] => 1
[2,5,3,1,2] => 1
[2,5,3,2,1] => 1
[3,1,2,2,5] => 1
[3,1,2,5,2] => 1
[3,1,5,2,2] => 1
[3,2,1,2,5] => 1
[3,2,1,5,2] => 1
[3,2,2,1,5] => 1
[3,2,2,5,1] => 1
[3,2,5,1,2] => 1
[3,2,5,2,1] => 1
[3,5,1,2,2] => 1
[3,5,2,1,2] => 1
[3,5,2,2,1] => 1
[5,1,2,2,3] => 1
[5,1,2,3,2] => 1
[5,1,3,2,2] => 1
[5,2,1,2,3] => 1
[5,2,1,3,2] => 1
[5,2,2,1,3] => 1
[5,2,2,3,1] => 1
[5,2,3,1,2] => 1
[5,2,3,2,1] => 1
[5,3,1,2,2] => 1
[5,3,2,1,2] => 1
[5,3,2,2,1] => 1
[1,2,2,4,4] => 1
[1,2,4,2,4] => 1
[1,2,4,4,2] => 1
[1,4,2,2,4] => 1
[1,4,2,4,2] => 1
[1,4,4,2,2] => 1
[2,1,2,4,4] => 0
[2,1,4,2,4] => 0
[2,1,4,4,2] => 0
[2,2,1,4,4] => 0
[2,2,4,1,4] => 0
[2,2,4,4,1] => 0
[2,4,1,2,4] => 0
[2,4,1,4,2] => 0
[2,4,2,1,4] => 0
[2,4,2,4,1] => 0
[2,4,4,1,2] => 0
[2,4,4,2,1] => 0
[4,1,2,2,4] => 0
[4,1,2,4,2] => 0
[4,1,4,2,2] => 0
[4,2,1,2,4] => 0
[4,2,1,4,2] => 0
[4,2,2,1,4] => 0
[4,2,2,4,1] => 0
[4,2,4,1,2] => 0
[4,2,4,2,1] => 0
[4,4,1,2,2] => 0
[4,4,2,1,2] => 0
[4,4,2,2,1] => 0
[1,2,2,4,5] => 3
[1,2,2,5,4] => 2
[1,2,4,2,5] => 3
[1,2,4,5,2] => 3
[1,2,5,2,4] => 2
[1,2,5,4,2] => 2
[1,4,2,2,5] => 3
[1,4,2,5,2] => 3
[1,4,5,2,2] => 3
[1,5,2,2,4] => 2
[1,5,2,4,2] => 2
[1,5,4,2,2] => 2
[2,1,2,4,5] => 2
[2,1,2,5,4] => 1
[2,1,4,2,5] => 2
[2,1,4,5,2] => 2
[2,1,5,2,4] => 1
[2,1,5,4,2] => 1
[2,2,1,4,5] => 2
[2,2,1,5,4] => 1
[2,2,4,1,5] => 2
[2,2,4,5,1] => 2
[2,2,5,1,4] => 1
[2,2,5,4,1] => 1
[2,4,1,2,5] => 2
[2,4,1,5,2] => 2
[2,4,2,1,5] => 2
[2,4,2,5,1] => 2
[2,4,5,1,2] => 2
[2,4,5,2,1] => 2
[2,5,1,2,4] => 1
[2,5,1,4,2] => 1
[2,5,2,1,4] => 1
[2,5,2,4,1] => 1
[2,5,4,1,2] => 1
[2,5,4,2,1] => 1
[4,1,2,2,5] => 2
[4,1,2,5,2] => 2
[4,1,5,2,2] => 2
[4,2,1,2,5] => 2
[4,2,1,5,2] => 2
[4,2,2,1,5] => 2
[4,2,2,5,1] => 2
[4,2,5,1,2] => 2
[4,2,5,2,1] => 2
[4,5,1,2,2] => 2
[4,5,2,1,2] => 2
[4,5,2,2,1] => 2
[5,1,2,2,4] => 1
[5,1,2,4,2] => 1
[5,1,4,2,2] => 1
[5,2,1,2,4] => 1
[5,2,1,4,2] => 1
[5,2,2,1,4] => 1
[5,2,2,4,1] => 1
[5,2,4,1,2] => 1
[5,2,4,2,1] => 1
[5,4,1,2,2] => 1
[5,4,2,1,2] => 1
[5,4,2,2,1] => 1
[1,2,3,3,3] => 2
[1,3,2,3,3] => 1
[1,3,3,2,3] => 1
[1,3,3,3,2] => 1
[2,1,3,3,3] => 1
[2,3,1,3,3] => 1
[2,3,3,1,3] => 1
[2,3,3,3,1] => 1
[3,1,2,3,3] => 0
[3,1,3,2,3] => 0
[3,1,3,3,2] => 0
[3,2,1,3,3] => 0
[3,2,3,1,3] => 0
[3,2,3,3,1] => 0
[3,3,1,2,3] => 0
[3,3,1,3,2] => 0
[3,3,2,1,3] => 0
[3,3,2,3,1] => 0
[3,3,3,1,2] => 0
[3,3,3,2,1] => 0
[1,2,3,3,4] => 2
[1,2,3,4,3] => 2
[1,2,4,3,3] => 2
[1,3,2,3,4] => 1
[1,3,2,4,3] => 1
[1,3,3,2,4] => 1
[1,3,3,4,2] => 1
[1,3,4,2,3] => 1
[1,3,4,3,2] => 1
[1,4,2,3,3] => 1
[1,4,3,2,3] => 1
[1,4,3,3,2] => 1
[2,1,3,3,4] => 1
[2,1,3,4,3] => 1
[2,1,4,3,3] => 1
[2,3,1,3,4] => 1
[2,3,1,4,3] => 1
[2,3,3,1,4] => 1
[2,3,3,4,1] => 1
[2,3,4,1,3] => 1
[2,3,4,3,1] => 1
[2,4,1,3,3] => 1
[2,4,3,1,3] => 1
[2,4,3,3,1] => 1
[3,1,2,3,4] => 0
[3,1,2,4,3] => 0
[3,1,3,2,4] => 0
[3,1,3,4,2] => 0
[3,1,4,2,3] => 0
[3,1,4,3,2] => 0
[3,2,1,3,4] => 0
[3,2,1,4,3] => 0
[3,2,3,1,4] => 0
[3,2,3,4,1] => 0
[3,2,4,1,3] => 0
[3,2,4,3,1] => 0
[3,3,1,2,4] => 0
[3,3,1,4,2] => 0
[3,3,2,1,4] => 0
[3,3,2,4,1] => 0
[3,3,4,1,2] => 0
[3,3,4,2,1] => 0
[3,4,1,2,3] => 0
[3,4,1,3,2] => 0
[3,4,2,1,3] => 0
[3,4,2,3,1] => 0
[3,4,3,1,2] => 0
[3,4,3,2,1] => 0
[4,1,2,3,3] => 0
[4,1,3,2,3] => 0
[4,1,3,3,2] => 0
[4,2,1,3,3] => 0
[4,2,3,1,3] => 0
[4,2,3,3,1] => 0
[4,3,1,2,3] => 0
[4,3,1,3,2] => 0
[4,3,2,1,3] => 0
[4,3,2,3,1] => 0
[4,3,3,1,2] => 0
click to show generating function       
Description
The number of critical left to right maxima of the parking functions.
An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it.
This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via
$$ \sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P}, $$
where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].
References
[1] Handbook of enumerative combinatorics MathSciNet:3408702
Code
def statistic(P):
    """
    sage: R. = PolynomialRing(ZZ)
    sage: n=3; sum(x^statistic(P)*y^(binomial(len(P)+1,2)-sum(P)) for P in ParkingFunctions(n))
    x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y

    sage: G = graphs.CompleteGraph(n+1)
    sage: G.tutte_polynomial()
    x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y
    
    """    
    return len([1 for i, j in enumerate(P)
                if ((i == 0 or max(P[:i]) < j)
                    and len([1 for k in P if k < j]) == j-1
                    and len([1 for k in P if k > j]) == len(P)-j)])

Created
Aug 12, 2017 at 23:31 by Martin Rubey
Updated
Aug 12, 2017 at 23:31 by Martin Rubey