Identifier
Identifier
Values
[1] => 0
[1,1] => 1
[1,2] => 0
[2,1] => 0
[1,1,1] => 3
[1,1,2] => 2
[1,2,1] => 2
[2,1,1] => 2
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 1
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,1,1,1] => 6
[1,1,1,2] => 5
[1,1,2,1] => 5
[1,2,1,1] => 5
[2,1,1,1] => 5
[1,1,1,3] => 4
[1,1,3,1] => 4
[1,3,1,1] => 4
[3,1,1,1] => 4
[1,1,1,4] => 3
[1,1,4,1] => 3
[1,4,1,1] => 3
[4,1,1,1] => 3
[1,1,2,2] => 4
[1,2,1,2] => 4
[1,2,2,1] => 4
[2,1,1,2] => 4
[2,1,2,1] => 4
[2,2,1,1] => 4
[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] => 2
[1,1,4,2] => 2
[1,2,1,4] => 2
[1,2,4,1] => 2
[1,4,1,2] => 2
[1,4,2,1] => 2
[2,1,1,4] => 2
[2,1,4,1] => 2
[2,4,1,1] => 2
[4,1,1,2] => 2
[4,1,2,1] => 2
[4,2,1,1] => 2
[1,1,3,3] => 2
[1,3,1,3] => 2
[1,3,3,1] => 2
[3,1,1,3] => 2
[3,1,3,1] => 2
[3,3,1,1] => 2
[1,1,3,4] => 1
[1,1,4,3] => 1
[1,3,1,4] => 1
[1,3,4,1] => 1
[1,4,1,3] => 1
[1,4,3,1] => 1
[3,1,1,4] => 1
[3,1,4,1] => 1
[3,4,1,1] => 1
[4,1,1,3] => 1
[4,1,3,1] => 1
[4,3,1,1] => 1
[1,2,2,2] => 3
[2,1,2,2] => 3
[2,2,1,2] => 3
[2,2,2,1] => 3
[1,2,2,3] => 2
[1,2,3,2] => 2
[1,3,2,2] => 2
[2,1,2,3] => 2
[2,1,3,2] => 2
[2,2,1,3] => 2
[2,2,3,1] => 2
[2,3,1,2] => 2
[2,3,2,1] => 2
[3,1,2,2] => 2
[3,2,1,2] => 2
[3,2,2,1] => 2
[1,2,2,4] => 1
[1,2,4,2] => 1
[1,4,2,2] => 1
[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] => 1
[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] => 1
[3,1,3,2] => 1
[3,2,1,3] => 1
[3,2,3,1] => 1
[3,3,1,2] => 1
[3,3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,1,1,1,1] => 10
[1,1,1,1,2] => 9
[1,1,1,2,1] => 9
[1,1,2,1,1] => 9
[1,2,1,1,1] => 9
[2,1,1,1,1] => 9
[1,1,1,1,3] => 8
[1,1,1,3,1] => 8
[1,1,3,1,1] => 8
[1,3,1,1,1] => 8
[3,1,1,1,1] => 8
[1,1,1,1,4] => 7
[1,1,1,4,1] => 7
[1,1,4,1,1] => 7
[1,4,1,1,1] => 7
[4,1,1,1,1] => 7
[1,1,1,1,5] => 6
[1,1,1,5,1] => 6
[1,1,5,1,1] => 6
[1,5,1,1,1] => 6
[5,1,1,1,1] => 6
[1,1,1,2,2] => 8
[1,1,2,1,2] => 8
[1,1,2,2,1] => 8
[1,2,1,1,2] => 8
[1,2,1,2,1] => 8
[1,2,2,1,1] => 8
[2,1,1,1,2] => 8
[2,1,1,2,1] => 8
[2,1,2,1,1] => 8
[2,2,1,1,1] => 8
[1,1,1,2,3] => 7
[1,1,1,3,2] => 7
[1,1,2,1,3] => 7
[1,1,2,3,1] => 7
[1,1,3,1,2] => 7
[1,1,3,2,1] => 7
[1,2,1,1,3] => 7
[1,2,1,3,1] => 7
[1,2,3,1,1] => 7
[1,3,1,1,2] => 7
[1,3,1,2,1] => 7
[1,3,2,1,1] => 7
[2,1,1,1,3] => 7
[2,1,1,3,1] => 7
[2,1,3,1,1] => 7
[2,3,1,1,1] => 7
[3,1,1,1,2] => 7
[3,1,1,2,1] => 7
[3,1,2,1,1] => 7
[3,2,1,1,1] => 7
[1,1,1,2,4] => 6
[1,1,1,4,2] => 6
[1,1,2,1,4] => 6
[1,1,2,4,1] => 6
[1,1,4,1,2] => 6
[1,1,4,2,1] => 6
[1,2,1,1,4] => 6
[1,2,1,4,1] => 6
[1,2,4,1,1] => 6
[1,4,1,1,2] => 6
[1,4,1,2,1] => 6
[1,4,2,1,1] => 6
[2,1,1,1,4] => 6
[2,1,1,4,1] => 6
[2,1,4,1,1] => 6
[2,4,1,1,1] => 6
[4,1,1,1,2] => 6
[4,1,1,2,1] => 6
[4,1,2,1,1] => 6
[4,2,1,1,1] => 6
[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] => 6
[1,1,3,1,3] => 6
[1,1,3,3,1] => 6
[1,3,1,1,3] => 6
[1,3,1,3,1] => 6
[1,3,3,1,1] => 6
[3,1,1,1,3] => 6
[3,1,1,3,1] => 6
[3,1,3,1,1] => 6
[3,3,1,1,1] => 6
[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] => 4
[1,1,1,5,3] => 4
[1,1,3,1,5] => 4
[1,1,3,5,1] => 4
[1,1,5,1,3] => 4
[1,1,5,3,1] => 4
[1,3,1,1,5] => 4
[1,3,1,5,1] => 4
[1,3,5,1,1] => 4
[1,5,1,1,3] => 4
[1,5,1,3,1] => 4
[1,5,3,1,1] => 4
[3,1,1,1,5] => 4
[3,1,1,5,1] => 4
[3,1,5,1,1] => 4
[3,5,1,1,1] => 4
[5,1,1,1,3] => 4
[5,1,1,3,1] => 4
[5,1,3,1,1] => 4
[5,3,1,1,1] => 4
[1,1,1,4,4] => 4
[1,1,4,1,4] => 4
[1,1,4,4,1] => 4
[1,4,1,1,4] => 4
[1,4,1,4,1] => 4
[1,4,4,1,1] => 4
[4,1,1,1,4] => 4
[4,1,1,4,1] => 4
[4,1,4,1,1] => 4
[4,4,1,1,1] => 4
[1,1,1,4,5] => 3
[1,1,1,5,4] => 3
[1,1,4,1,5] => 3
[1,1,4,5,1] => 3
[1,1,5,1,4] => 3
[1,1,5,4,1] => 3
[1,4,1,1,5] => 3
[1,4,1,5,1] => 3
[1,4,5,1,1] => 3
[1,5,1,1,4] => 3
[1,5,1,4,1] => 3
[1,5,4,1,1] => 3
[4,1,1,1,5] => 3
[4,1,1,5,1] => 3
[4,1,5,1,1] => 3
[4,5,1,1,1] => 3
[5,1,1,1,4] => 3
[5,1,1,4,1] => 3
[5,1,4,1,1] => 3
[5,4,1,1,1] => 3
[1,1,2,2,2] => 7
[1,2,1,2,2] => 7
[1,2,2,1,2] => 7
[1,2,2,2,1] => 7
[2,1,1,2,2] => 7
[2,1,2,1,2] => 7
[2,1,2,2,1] => 7
[2,2,1,1,2] => 7
[2,2,1,2,1] => 7
[2,2,2,1,1] => 7
[1,1,2,2,3] => 6
[1,1,2,3,2] => 6
[1,1,3,2,2] => 6
[1,2,1,2,3] => 6
[1,2,1,3,2] => 6
[1,2,2,1,3] => 6
[1,2,2,3,1] => 6
[1,2,3,1,2] => 6
[1,2,3,2,1] => 6
[1,3,1,2,2] => 6
[1,3,2,1,2] => 6
[1,3,2,2,1] => 6
[2,1,1,2,3] => 6
[2,1,1,3,2] => 6
[2,1,2,1,3] => 6
[2,1,2,3,1] => 6
[2,1,3,1,2] => 6
[2,1,3,2,1] => 6
[2,2,1,1,3] => 6
[2,2,1,3,1] => 6
[2,2,3,1,1] => 6
[2,3,1,1,2] => 6
[2,3,1,2,1] => 6
[2,3,2,1,1] => 6
[3,1,1,2,2] => 6
[3,1,2,1,2] => 6
[3,1,2,2,1] => 6
[3,2,1,1,2] => 6
[3,2,1,2,1] => 6
[3,2,2,1,1] => 6
[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] => 4
[1,1,2,5,2] => 4
[1,1,5,2,2] => 4
[1,2,1,2,5] => 4
[1,2,1,5,2] => 4
[1,2,2,1,5] => 4
[1,2,2,5,1] => 4
[1,2,5,1,2] => 4
[1,2,5,2,1] => 4
[1,5,1,2,2] => 4
[1,5,2,1,2] => 4
[1,5,2,2,1] => 4
[2,1,1,2,5] => 4
[2,1,1,5,2] => 4
[2,1,2,1,5] => 4
[2,1,2,5,1] => 4
[2,1,5,1,2] => 4
[2,1,5,2,1] => 4
[2,2,1,1,5] => 4
[2,2,1,5,1] => 4
[2,2,5,1,1] => 4
[2,5,1,1,2] => 4
[2,5,1,2,1] => 4
[2,5,2,1,1] => 4
[5,1,1,2,2] => 4
[5,1,2,1,2] => 4
[5,1,2,2,1] => 4
[5,2,1,1,2] => 4
[5,2,1,2,1] => 4
[5,2,2,1,1] => 4
[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] => 4
[1,1,2,4,3] => 4
[1,1,3,2,4] => 4
[1,1,3,4,2] => 4
[1,1,4,2,3] => 4
[1,1,4,3,2] => 4
[1,2,1,3,4] => 4
[1,2,1,4,3] => 4
[1,2,3,1,4] => 4
[1,2,3,4,1] => 4
[1,2,4,1,3] => 4
[1,2,4,3,1] => 4
[1,3,1,2,4] => 4
[1,3,1,4,2] => 4
[1,3,2,1,4] => 4
[1,3,2,4,1] => 4
[1,3,4,1,2] => 4
[1,3,4,2,1] => 4
[1,4,1,2,3] => 4
[1,4,1,3,2] => 4
[1,4,2,1,3] => 4
[1,4,2,3,1] => 4
[1,4,3,1,2] => 4
[1,4,3,2,1] => 4
[2,1,1,3,4] => 4
[2,1,1,4,3] => 4
[2,1,3,1,4] => 4
[2,1,3,4,1] => 4
[2,1,4,1,3] => 4
[2,1,4,3,1] => 4
[2,3,1,1,4] => 4
[2,3,1,4,1] => 4
[2,3,4,1,1] => 4
[2,4,1,1,3] => 4
[2,4,1,3,1] => 4
[2,4,3,1,1] => 4
[3,1,1,2,4] => 4
[3,1,1,4,2] => 4
[3,1,2,1,4] => 4
[3,1,2,4,1] => 4
[3,1,4,1,2] => 4
[3,1,4,2,1] => 4
[3,2,1,1,4] => 4
[3,2,1,4,1] => 4
[3,2,4,1,1] => 4
[3,4,1,1,2] => 4
[3,4,1,2,1] => 4
[3,4,2,1,1] => 4
[4,1,1,2,3] => 4
[4,1,1,3,2] => 4
[4,1,2,1,3] => 4
[4,1,2,3,1] => 4
[4,1,3,1,2] => 4
[4,1,3,2,1] => 4
[4,2,1,1,3] => 4
[4,2,1,3,1] => 4
[4,2,3,1,1] => 4
[4,3,1,1,2] => 4
[4,3,1,2,1] => 4
[4,3,2,1,1] => 4
[1,1,2,3,5] => 3
[1,1,2,5,3] => 3
[1,1,3,2,5] => 3
[1,1,3,5,2] => 3
[1,1,5,2,3] => 3
[1,1,5,3,2] => 3
[1,2,1,3,5] => 3
[1,2,1,5,3] => 3
[1,2,3,1,5] => 3
[1,2,3,5,1] => 3
[1,2,5,1,3] => 3
[1,2,5,3,1] => 3
[1,3,1,2,5] => 3
[1,3,1,5,2] => 3
[1,3,2,1,5] => 3
[1,3,2,5,1] => 3
[1,3,5,1,2] => 3
[1,3,5,2,1] => 3
[1,5,1,2,3] => 3
[1,5,1,3,2] => 3
[1,5,2,1,3] => 3
[1,5,2,3,1] => 3
[1,5,3,1,2] => 3
[1,5,3,2,1] => 3
[2,1,1,3,5] => 3
[2,1,1,5,3] => 3
[2,1,3,1,5] => 3
[2,1,3,5,1] => 3
[2,1,5,1,3] => 3
[2,1,5,3,1] => 3
[2,3,1,1,5] => 3
[2,3,1,5,1] => 3
[2,3,5,1,1] => 3
[2,5,1,1,3] => 3
[2,5,1,3,1] => 3
[2,5,3,1,1] => 3
[3,1,1,2,5] => 3
[3,1,1,5,2] => 3
[3,1,2,1,5] => 3
[3,1,2,5,1] => 3
[3,1,5,1,2] => 3
[3,1,5,2,1] => 3
[3,2,1,1,5] => 3
[3,2,1,5,1] => 3
[3,2,5,1,1] => 3
[3,5,1,1,2] => 3
[3,5,1,2,1] => 3
[3,5,2,1,1] => 3
[5,1,1,2,3] => 3
[5,1,1,3,2] => 3
[5,1,2,1,3] => 3
[5,1,2,3,1] => 3
[5,1,3,1,2] => 3
[5,1,3,2,1] => 3
[5,2,1,1,3] => 3
[5,2,1,3,1] => 3
[5,2,3,1,1] => 3
[5,3,1,1,2] => 3
[5,3,1,2,1] => 3
[5,3,2,1,1] => 3
[1,1,2,4,4] => 3
[1,1,4,2,4] => 3
[1,1,4,4,2] => 3
[1,2,1,4,4] => 3
[1,2,4,1,4] => 3
[1,2,4,4,1] => 3
[1,4,1,2,4] => 3
[1,4,1,4,2] => 3
[1,4,2,1,4] => 3
[1,4,2,4,1] => 3
[1,4,4,1,2] => 3
[1,4,4,2,1] => 3
[2,1,1,4,4] => 3
[2,1,4,1,4] => 3
[2,1,4,4,1] => 3
[2,4,1,1,4] => 3
[2,4,1,4,1] => 3
[2,4,4,1,1] => 3
[4,1,1,2,4] => 3
[4,1,1,4,2] => 3
[4,1,2,1,4] => 3
[4,1,2,4,1] => 3
[4,1,4,1,2] => 3
[4,1,4,2,1] => 3
[4,2,1,1,4] => 3
[4,2,1,4,1] => 3
[4,2,4,1,1] => 3
[4,4,1,1,2] => 3
[4,4,1,2,1] => 3
[4,4,2,1,1] => 3
[1,1,2,4,5] => 2
[1,1,2,5,4] => 2
[1,1,4,2,5] => 2
[1,1,4,5,2] => 2
[1,1,5,2,4] => 2
[1,1,5,4,2] => 2
[1,2,1,4,5] => 2
[1,2,1,5,4] => 2
[1,2,4,1,5] => 2
[1,2,4,5,1] => 2
[1,2,5,1,4] => 2
[1,2,5,4,1] => 2
[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] => 2
[1,5,1,4,2] => 2
[1,5,2,1,4] => 2
[1,5,2,4,1] => 2
[1,5,4,1,2] => 2
[1,5,4,2,1] => 2
[2,1,1,4,5] => 2
[2,1,1,5,4] => 2
[2,1,4,1,5] => 2
[2,1,4,5,1] => 2
[2,1,5,1,4] => 2
[2,1,5,4,1] => 2
[2,4,1,1,5] => 2
[2,4,1,5,1] => 2
[2,4,5,1,1] => 2
[2,5,1,1,4] => 2
[2,5,1,4,1] => 2
[2,5,4,1,1] => 2
[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] => 2
[5,1,1,4,2] => 2
[5,1,2,1,4] => 2
[5,1,2,4,1] => 2
[5,1,4,1,2] => 2
[5,1,4,2,1] => 2
[5,2,1,1,4] => 2
[5,2,1,4,1] => 2
[5,2,4,1,1] => 2
[5,4,1,1,2] => 2
[5,4,1,2,1] => 2
[5,4,2,1,1] => 2
[1,1,3,3,3] => 4
[1,3,1,3,3] => 4
[1,3,3,1,3] => 4
[1,3,3,3,1] => 4
[3,1,1,3,3] => 4
[3,1,3,1,3] => 4
[3,1,3,3,1] => 4
[3,3,1,1,3] => 4
[3,3,1,3,1] => 4
[3,3,3,1,1] => 4
[1,1,3,3,4] => 3
[1,1,3,4,3] => 3
[1,1,4,3,3] => 3
[1,3,1,3,4] => 3
[1,3,1,4,3] => 3
[1,3,3,1,4] => 3
[1,3,3,4,1] => 3
[1,3,4,1,3] => 3
[1,3,4,3,1] => 3
[1,4,1,3,3] => 3
[1,4,3,1,3] => 3
[1,4,3,3,1] => 3
[3,1,1,3,4] => 3
[3,1,1,4,3] => 3
[3,1,3,1,4] => 3
[3,1,3,4,1] => 3
[3,1,4,1,3] => 3
[3,1,4,3,1] => 3
[3,3,1,1,4] => 3
[3,3,1,4,1] => 3
[3,3,4,1,1] => 3
[3,4,1,1,3] => 3
[3,4,1,3,1] => 3
[3,4,3,1,1] => 3
[4,1,1,3,3] => 3
[4,1,3,1,3] => 3
[4,1,3,3,1] => 3
[4,3,1,1,3] => 3
[4,3,1,3,1] => 3
[4,3,3,1,1] => 3
[1,1,3,3,5] => 2
[1,1,3,5,3] => 2
[1,1,5,3,3] => 2
[1,3,1,3,5] => 2
[1,3,1,5,3] => 2
[1,3,3,1,5] => 2
[1,3,3,5,1] => 2
[1,3,5,1,3] => 2
[1,3,5,3,1] => 2
[1,5,1,3,3] => 2
[1,5,3,1,3] => 2
[1,5,3,3,1] => 2
[3,1,1,3,5] => 2
[3,1,1,5,3] => 2
[3,1,3,1,5] => 2
[3,1,3,5,1] => 2
[3,1,5,1,3] => 2
[3,1,5,3,1] => 2
[3,3,1,1,5] => 2
[3,3,1,5,1] => 2
[3,3,5,1,1] => 2
[3,5,1,1,3] => 2
[3,5,1,3,1] => 2
[3,5,3,1,1] => 2
[5,1,1,3,3] => 2
[5,1,3,1,3] => 2
[5,1,3,3,1] => 2
[5,3,1,1,3] => 2
[5,3,1,3,1] => 2
[5,3,3,1,1] => 2
[1,1,3,4,4] => 2
[1,1,4,3,4] => 2
[1,1,4,4,3] => 2
[1,3,1,4,4] => 2
[1,3,4,1,4] => 2
[1,3,4,4,1] => 2
[1,4,1,3,4] => 2
[1,4,1,4,3] => 2
[1,4,3,1,4] => 2
[1,4,3,4,1] => 2
[1,4,4,1,3] => 2
[1,4,4,3,1] => 2
[3,1,1,4,4] => 2
[3,1,4,1,4] => 2
[3,1,4,4,1] => 2
[3,4,1,1,4] => 2
[3,4,1,4,1] => 2
[3,4,4,1,1] => 2
[4,1,1,3,4] => 2
[4,1,1,4,3] => 2
[4,1,3,1,4] => 2
[4,1,3,4,1] => 2
[4,1,4,1,3] => 2
[4,1,4,3,1] => 2
[4,3,1,1,4] => 2
[4,3,1,4,1] => 2
[4,3,4,1,1] => 2
[4,4,1,1,3] => 2
[4,4,1,3,1] => 2
[4,4,3,1,1] => 2
[1,1,3,4,5] => 1
[1,1,3,5,4] => 1
[1,1,4,3,5] => 1
[1,1,4,5,3] => 1
[1,1,5,3,4] => 1
[1,1,5,4,3] => 1
[1,3,1,4,5] => 1
[1,3,1,5,4] => 1
[1,3,4,1,5] => 1
[1,3,4,5,1] => 1
[1,3,5,1,4] => 1
[1,3,5,4,1] => 1
[1,4,1,3,5] => 1
[1,4,1,5,3] => 1
[1,4,3,1,5] => 1
[1,4,3,5,1] => 1
[1,4,5,1,3] => 1
[1,4,5,3,1] => 1
[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] => 1
[3,1,1,5,4] => 1
[3,1,4,1,5] => 1
[3,1,4,5,1] => 1
[3,1,5,1,4] => 1
[3,1,5,4,1] => 1
[3,4,1,1,5] => 1
[3,4,1,5,1] => 1
[3,4,5,1,1] => 1
[3,5,1,1,4] => 1
[3,5,1,4,1] => 1
[3,5,4,1,1] => 1
[4,1,1,3,5] => 1
[4,1,1,5,3] => 1
[4,1,3,1,5] => 1
[4,1,3,5,1] => 1
[4,1,5,1,3] => 1
[4,1,5,3,1] => 1
[4,3,1,1,5] => 1
[4,3,1,5,1] => 1
[4,3,5,1,1] => 1
[4,5,1,1,3] => 1
[4,5,1,3,1] => 1
[4,5,3,1,1] => 1
[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] => 6
[2,1,2,2,2] => 6
[2,2,1,2,2] => 6
[2,2,2,1,2] => 6
[2,2,2,2,1] => 6
[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] => 4
[1,2,2,4,2] => 4
[1,2,4,2,2] => 4
[1,4,2,2,2] => 4
[2,1,2,2,4] => 4
[2,1,2,4,2] => 4
[2,1,4,2,2] => 4
[2,2,1,2,4] => 4
[2,2,1,4,2] => 4
[2,2,2,1,4] => 4
[2,2,2,4,1] => 4
[2,2,4,1,2] => 4
[2,2,4,2,1] => 4
[2,4,1,2,2] => 4
[2,4,2,1,2] => 4
[2,4,2,2,1] => 4
[4,1,2,2,2] => 4
[4,2,1,2,2] => 4
[4,2,2,1,2] => 4
[4,2,2,2,1] => 4
[1,2,2,2,5] => 3
[1,2,2,5,2] => 3
[1,2,5,2,2] => 3
[1,5,2,2,2] => 3
[2,1,2,2,5] => 3
[2,1,2,5,2] => 3
[2,1,5,2,2] => 3
[2,2,1,2,5] => 3
[2,2,1,5,2] => 3
[2,2,2,1,5] => 3
[2,2,2,5,1] => 3
[2,2,5,1,2] => 3
[2,2,5,2,1] => 3
[2,5,1,2,2] => 3
[2,5,2,1,2] => 3
[2,5,2,2,1] => 3
[5,1,2,2,2] => 3
[5,2,1,2,2] => 3
[5,2,2,1,2] => 3
[5,2,2,2,1] => 3
[1,2,2,3,3] => 4
[1,2,3,2,3] => 4
[1,2,3,3,2] => 4
[1,3,2,2,3] => 4
[1,3,2,3,2] => 4
[1,3,3,2,2] => 4
[2,1,2,3,3] => 4
[2,1,3,2,3] => 4
[2,1,3,3,2] => 4
[2,2,1,3,3] => 4
[2,2,3,1,3] => 4
[2,2,3,3,1] => 4
[2,3,1,2,3] => 4
[2,3,1,3,2] => 4
[2,3,2,1,3] => 4
[2,3,2,3,1] => 4
[2,3,3,1,2] => 4
[2,3,3,2,1] => 4
[3,1,2,2,3] => 4
[3,1,2,3,2] => 4
[3,1,3,2,2] => 4
[3,2,1,2,3] => 4
[3,2,1,3,2] => 4
[3,2,2,1,3] => 4
[3,2,2,3,1] => 4
[3,2,3,1,2] => 4
[3,2,3,2,1] => 4
[3,3,1,2,2] => 4
[3,3,2,1,2] => 4
[3,3,2,2,1] => 4
[1,2,2,3,4] => 3
[1,2,2,4,3] => 3
[1,2,3,2,4] => 3
[1,2,3,4,2] => 3
[1,2,4,2,3] => 3
[1,2,4,3,2] => 3
[1,3,2,2,4] => 3
[1,3,2,4,2] => 3
[1,3,4,2,2] => 3
[1,4,2,2,3] => 3
[1,4,2,3,2] => 3
[1,4,3,2,2] => 3
[2,1,2,3,4] => 3
[2,1,2,4,3] => 3
[2,1,3,2,4] => 3
[2,1,3,4,2] => 3
[2,1,4,2,3] => 3
[2,1,4,3,2] => 3
[2,2,1,3,4] => 3
[2,2,1,4,3] => 3
[2,2,3,1,4] => 3
[2,2,3,4,1] => 3
[2,2,4,1,3] => 3
[2,2,4,3,1] => 3
[2,3,1,2,4] => 3
[2,3,1,4,2] => 3
[2,3,2,1,4] => 3
[2,3,2,4,1] => 3
[2,3,4,1,2] => 3
[2,3,4,2,1] => 3
[2,4,1,2,3] => 3
[2,4,1,3,2] => 3
[2,4,2,1,3] => 3
[2,4,2,3,1] => 3
[2,4,3,1,2] => 3
[2,4,3,2,1] => 3
[3,1,2,2,4] => 3
[3,1,2,4,2] => 3
[3,1,4,2,2] => 3
[3,2,1,2,4] => 3
[3,2,1,4,2] => 3
[3,2,2,1,4] => 3
[3,2,2,4,1] => 3
[3,2,4,1,2] => 3
[3,2,4,2,1] => 3
[3,4,1,2,2] => 3
[3,4,2,1,2] => 3
[3,4,2,2,1] => 3
[4,1,2,2,3] => 3
[4,1,2,3,2] => 3
[4,1,3,2,2] => 3
[4,2,1,2,3] => 3
[4,2,1,3,2] => 3
[4,2,2,1,3] => 3
[4,2,2,3,1] => 3
[4,2,3,1,2] => 3
[4,2,3,2,1] => 3
[4,3,1,2,2] => 3
[4,3,2,1,2] => 3
[4,3,2,2,1] => 3
[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] => 2
[2,1,2,5,3] => 2
[2,1,3,2,5] => 2
[2,1,3,5,2] => 2
[2,1,5,2,3] => 2
[2,1,5,3,2] => 2
[2,2,1,3,5] => 2
[2,2,1,5,3] => 2
[2,2,3,1,5] => 2
[2,2,3,5,1] => 2
[2,2,5,1,3] => 2
[2,2,5,3,1] => 2
[2,3,1,2,5] => 2
[2,3,1,5,2] => 2
[2,3,2,1,5] => 2
[2,3,2,5,1] => 2
[2,3,5,1,2] => 2
[2,3,5,2,1] => 2
[2,5,1,2,3] => 2
[2,5,1,3,2] => 2
[2,5,2,1,3] => 2
[2,5,2,3,1] => 2
[2,5,3,1,2] => 2
[2,5,3,2,1] => 2
[3,1,2,2,5] => 2
[3,1,2,5,2] => 2
[3,1,5,2,2] => 2
[3,2,1,2,5] => 2
[3,2,1,5,2] => 2
[3,2,2,1,5] => 2
[3,2,2,5,1] => 2
[3,2,5,1,2] => 2
[3,2,5,2,1] => 2
[3,5,1,2,2] => 2
[3,5,2,1,2] => 2
[3,5,2,2,1] => 2
[5,1,2,2,3] => 2
[5,1,2,3,2] => 2
[5,1,3,2,2] => 2
[5,2,1,2,3] => 2
[5,2,1,3,2] => 2
[5,2,2,1,3] => 2
[5,2,2,3,1] => 2
[5,2,3,1,2] => 2
[5,2,3,2,1] => 2
[5,3,1,2,2] => 2
[5,3,2,1,2] => 2
[5,3,2,2,1] => 2
[1,2,2,4,4] => 2
[1,2,4,2,4] => 2
[1,2,4,4,2] => 2
[1,4,2,2,4] => 2
[1,4,2,4,2] => 2
[1,4,4,2,2] => 2
[2,1,2,4,4] => 2
[2,1,4,2,4] => 2
[2,1,4,4,2] => 2
[2,2,1,4,4] => 2
[2,2,4,1,4] => 2
[2,2,4,4,1] => 2
[2,4,1,2,4] => 2
[2,4,1,4,2] => 2
[2,4,2,1,4] => 2
[2,4,2,4,1] => 2
[2,4,4,1,2] => 2
[2,4,4,2,1] => 2
[4,1,2,2,4] => 2
[4,1,2,4,2] => 2
[4,1,4,2,2] => 2
[4,2,1,2,4] => 2
[4,2,1,4,2] => 2
[4,2,2,1,4] => 2
[4,2,2,4,1] => 2
[4,2,4,1,2] => 2
[4,2,4,2,1] => 2
[4,4,1,2,2] => 2
[4,4,2,1,2] => 2
[4,4,2,2,1] => 2
[1,2,2,4,5] => 1
[1,2,2,5,4] => 1
[1,2,4,2,5] => 1
[1,2,4,5,2] => 1
[1,2,5,2,4] => 1
[1,2,5,4,2] => 1
[1,4,2,2,5] => 1
[1,4,2,5,2] => 1
[1,4,5,2,2] => 1
[1,5,2,2,4] => 1
[1,5,2,4,2] => 1
[1,5,4,2,2] => 1
[2,1,2,4,5] => 1
[2,1,2,5,4] => 1
[2,1,4,2,5] => 1
[2,1,4,5,2] => 1
[2,1,5,2,4] => 1
[2,1,5,4,2] => 1
[2,2,1,4,5] => 1
[2,2,1,5,4] => 1
[2,2,4,1,5] => 1
[2,2,4,5,1] => 1
[2,2,5,1,4] => 1
[2,2,5,4,1] => 1
[2,4,1,2,5] => 1
[2,4,1,5,2] => 1
[2,4,2,1,5] => 1
[2,4,2,5,1] => 1
[2,4,5,1,2] => 1
[2,4,5,2,1] => 1
[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] => 1
[4,1,2,5,2] => 1
[4,1,5,2,2] => 1
[4,2,1,2,5] => 1
[4,2,1,5,2] => 1
[4,2,2,1,5] => 1
[4,2,2,5,1] => 1
[4,2,5,1,2] => 1
[4,2,5,2,1] => 1
[4,5,1,2,2] => 1
[4,5,2,1,2] => 1
[4,5,2,2,1] => 1
[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] => 3
[1,3,2,3,3] => 3
[1,3,3,2,3] => 3
[1,3,3,3,2] => 3
[2,1,3,3,3] => 3
[2,3,1,3,3] => 3
[2,3,3,1,3] => 3
[2,3,3,3,1] => 3
[3,1,2,3,3] => 3
[3,1,3,2,3] => 3
[3,1,3,3,2] => 3
[3,2,1,3,3] => 3
[3,2,3,1,3] => 3
[3,2,3,3,1] => 3
[3,3,1,2,3] => 3
[3,3,1,3,2] => 3
[3,3,2,1,3] => 3
[3,3,2,3,1] => 3
[3,3,3,1,2] => 3
[3,3,3,2,1] => 3
[1,2,3,3,4] => 2
[1,2,3,4,3] => 2
[1,2,4,3,3] => 2
[1,3,2,3,4] => 2
[1,3,2,4,3] => 2
[1,3,3,2,4] => 2
[1,3,3,4,2] => 2
[1,3,4,2,3] => 2
[1,3,4,3,2] => 2
[1,4,2,3,3] => 2
[1,4,3,2,3] => 2
[1,4,3,3,2] => 2
[2,1,3,3,4] => 2
[2,1,3,4,3] => 2
[2,1,4,3,3] => 2
[2,3,1,3,4] => 2
[2,3,1,4,3] => 2
[2,3,3,1,4] => 2
[2,3,3,4,1] => 2
[2,3,4,1,3] => 2
[2,3,4,3,1] => 2
[2,4,1,3,3] => 2
[2,4,3,1,3] => 2
[2,4,3,3,1] => 2
click to show generating function       
Description
The area of the Dyck path corresponding to a parking function.
For a parking function $p_1,\ldots,p_n$ of length $n$, this is given by $\binom{n+1}{2} - \sum_i p_i$.
References
[1] Triangle of coefficients of polynomials enumerating trees with n labeled nodes by inversions. OEIS:A052121
Code
def statistic(p):
    return p.area()
Created
May 08, 2014 at 23:40 by Joanna Heppner
Updated
Jun 14, 2018 at 11:16 by Christian Stump