Identifier
Identifier
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 1
[1,2] => 0
[2,1] => 0
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 0
[1,2,1] => 1
[1,3] => 0
[2,1,1] => 0
[2,2] => 1
[3,1] => 0
[4] => 1
[1,1,1,1,1] => 2
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 0
[1,3,1] => 2
[1,4] => 0
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 0
[2,3] => 0
[3,1,1] => 1
[3,2] => 0
[4,1] => 0
[5] => 2
[1,1,1,1,1,1] => 2
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 2
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 0
[1,2,1,1,1] => 1
[1,2,1,2] => 0
[1,2,2,1] => 2
[1,2,3] => 0
[1,3,1,1] => 1
[1,3,2] => 0
[1,4,1] => 2
[1,5] => 0
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 0
[2,1,3] => 0
[2,2,1,1] => 1
[2,2,2] => 2
[2,3,1] => 0
[2,4] => 1
[3,1,1,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[3,3] => 2
[4,1,1] => 0
[4,2] => 1
[5,1] => 0
[6] => 2
[1,1,1,1,1,1,1] => 3
[1,1,1,1,1,2] => 2
[1,1,1,1,2,1] => 2
[1,1,1,1,3] => 2
[1,1,1,2,1,1] => 2
[1,1,1,2,2] => 1
[1,1,1,3,1] => 2
[1,1,1,4] => 1
[1,1,2,1,1,1] => 2
[1,1,2,1,2] => 1
[1,1,2,2,1] => 1
[1,1,2,3] => 1
[1,1,3,1,1] => 3
[1,1,3,2] => 2
[1,1,4,1] => 1
[1,1,5] => 1
[1,2,1,1,1,1] => 2
[1,2,1,1,2] => 1
[1,2,1,2,1] => 3
[1,2,1,3] => 1
[1,2,2,1,1] => 1
[1,2,2,2] => 0
[1,2,3,1] => 1
[1,2,4] => 2
[1,3,1,1,1] => 2
[1,3,1,2] => 1
[1,3,2,1] => 1
[1,3,3] => 1
[1,4,1,1] => 1
[1,4,2] => 0
[1,5,1] => 3
[1,6] => 0
[2,1,1,1,1,1] => 2
[2,1,1,1,2] => 3
[2,1,1,2,1] => 1
[2,1,1,3] => 1
[2,1,2,1,1] => 1
[2,1,2,2] => 2
[2,1,3,1] => 1
[2,1,4] => 0
[2,2,1,1,1] => 1
[2,2,1,2] => 2
[2,2,2,1] => 0
[2,2,3] => 0
[2,3,1,1] => 2
[2,3,2] => 3
[2,4,1] => 0
[2,5] => 0
[3,1,1,1,1] => 2
[3,1,1,2] => 1
[3,1,2,1] => 1
[3,1,3] => 3
[3,2,1,1] => 1
[3,2,2] => 0
[3,3,1] => 1
[3,4] => 0
[4,1,1,1] => 1
[4,1,2] => 0
[4,2,1] => 2
[4,3] => 0
[5,1,1] => 1
[5,2] => 0
[6,1] => 0
[7] => 3
[1,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,2] => 2
[1,1,1,1,1,2,1] => 2
[1,1,1,1,1,3] => 2
[1,1,1,1,2,1,1] => 2
[1,1,1,1,2,2] => 1
[1,1,1,1,3,1] => 2
[1,1,1,1,4] => 1
[1,1,1,2,1,1,1] => 3
[1,1,1,2,1,2] => 2
[1,1,1,2,2,1] => 2
[1,1,1,2,3] => 2
[1,1,1,3,1,1] => 2
[1,1,1,3,2] => 1
[1,1,1,4,1] => 1
[1,1,1,5] => 1
[1,1,2,1,1,1,1] => 2
[1,1,2,1,1,2] => 1
[1,1,2,1,2,1] => 1
[1,1,2,1,3] => 1
[1,1,2,2,1,1] => 3
[1,1,2,2,2] => 2
[1,1,2,3,1] => 1
[1,1,2,4] => 0
[1,1,3,1,1,1] => 2
[1,1,3,1,2] => 1
[1,1,3,2,1] => 1
[1,1,3,3] => 1
[1,1,4,1,1] => 3
[1,1,4,2] => 2
[1,1,5,1] => 1
[1,1,6] => 0
[1,2,1,1,1,1,1] => 2
[1,2,1,1,1,2] => 1
[1,2,1,1,2,1] => 3
[1,2,1,1,3] => 1
[1,2,1,2,1,1] => 1
[1,2,1,2,2] => 0
[1,2,1,3,1] => 1
[1,2,1,4] => 2
[1,2,2,1,1,1] => 2
[1,2,2,1,2] => 1
[1,2,2,2,1] => 3
[1,2,2,3] => 1
[1,2,3,1,1] => 1
[1,2,3,2] => 0
[1,2,4,1] => 2
[1,2,5] => 0
[1,3,1,1,1,1] => 2
[1,3,1,1,2] => 1
[1,3,1,2,1] => 1
[1,3,1,3] => 1
[1,3,2,1,1] => 1
[1,3,2,2] => 0
[1,3,3,1] => 3
[1,3,4] => 0
[1,4,1,1,1] => 1
[1,4,1,2] => 0
[1,4,2,1] => 2
[1,4,3] => 0
[1,5,1,1] => 1
[1,5,2] => 0
[1,6,1] => 3
[1,7] => 0
[2,1,1,1,1,1,1] => 2
[2,1,1,1,1,2] => 3
[2,1,1,1,2,1] => 1
[2,1,1,1,3] => 1
[2,1,1,2,1,1] => 1
[2,1,1,2,2] => 2
[2,1,1,3,1] => 1
[2,1,1,4] => 0
[2,1,2,1,1,1] => 2
[2,1,2,1,2] => 3
[2,1,2,2,1] => 1
[2,1,2,3] => 1
[2,1,3,1,1] => 1
[2,1,3,2] => 2
[2,1,4,1] => 0
[2,1,5] => 2
[2,2,1,1,1,1] => 1
[2,2,1,1,2] => 2
[2,2,1,2,1] => 0
[2,2,1,3] => 0
[2,2,2,1,1] => 2
[2,2,2,2] => 3
[2,2,3,1] => 0
[2,2,4] => 1
[2,3,1,1,1] => 1
[2,3,1,2] => 2
[2,3,2,1] => 0
[2,3,3] => 0
[2,4,1,1] => 2
[2,4,2] => 3
[2,5,1] => 0
[2,6] => 1
[3,1,1,1,1,1] => 2
[3,1,1,1,2] => 1
[3,1,1,2,1] => 1
[3,1,1,3] => 3
[3,1,2,1,1] => 1
[3,1,2,2] => 0
[3,1,3,1] => 1
[3,1,4] => 0
[3,2,1,1,1] => 2
[3,2,1,2] => 1
[3,2,2,1] => 1
[3,2,3] => 3
[3,3,1,1] => 1
[3,3,2] => 0
[3,4,1] => 0
[3,5] => 0
[4,1,1,1,1] => 1
[4,1,1,2] => 0
[4,1,2,1] => 2
[4,1,3] => 0
[4,2,1,1] => 0
[4,2,2] => 1
[4,3,1] => 0
[4,4] => 3
[5,1,1,1] => 1
[5,1,2] => 2
[5,2,1] => 0
[5,3] => 0
[6,1,1] => 0
[6,2] => 1
[7,1] => 0
[8] => 3
[1,1,1,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,2] => 3
[1,1,1,1,1,1,2,1] => 3
[1,1,1,1,1,1,3] => 3
[1,1,1,1,1,2,1,1] => 3
[1,1,1,1,1,2,2] => 2
[1,1,1,1,1,3,1] => 3
[1,1,1,1,1,4] => 2
[1,1,1,1,2,1,1,1] => 3
[1,1,1,1,2,1,2] => 2
[1,1,1,1,2,2,1] => 2
[1,1,1,1,2,3] => 2
[1,1,1,1,3,1,1] => 3
[1,1,1,1,3,2] => 2
[1,1,1,1,4,1] => 2
[1,1,1,1,5] => 2
[1,1,1,2,1,1,1,1] => 3
[1,1,1,2,1,1,2] => 2
[1,1,1,2,1,2,1] => 2
[1,1,1,2,1,3] => 2
[1,1,1,2,2,1,1] => 2
[1,1,1,2,2,2] => 1
[1,1,1,2,3,1] => 2
[1,1,1,2,4] => 1
[1,1,1,3,1,1,1] => 4
[1,1,1,3,1,2] => 3
[1,1,1,3,2,1] => 3
[1,1,1,3,3] => 3
[1,1,1,4,1,1] => 2
[1,1,1,4,2] => 1
[1,1,1,5,1] => 2
[1,1,1,6] => 1
[1,1,2,1,1,1,1,1] => 3
[1,1,2,1,1,1,2] => 2
[1,1,2,1,1,2,1] => 2
[1,1,2,1,1,3] => 2
[1,1,2,1,2,1,1] => 4
[1,1,2,1,2,2] => 3
[1,1,2,1,3,1] => 2
[1,1,2,1,4] => 1
[1,1,2,2,1,1,1] => 2
[1,1,2,2,1,2] => 1
[1,1,2,2,2,1] => 1
[1,1,2,2,3] => 1
[1,1,2,3,1,1] => 2
[1,1,2,3,2] => 1
[1,1,2,4,1] => 3
[1,1,2,5] => 1
[1,1,3,1,1,1,1] => 3
[1,1,3,1,1,2] => 2
[1,1,3,1,2,1] => 2
[1,1,3,1,3] => 2
[1,1,3,2,1,1] => 2
[1,1,3,2,2] => 1
[1,1,3,3,1] => 2
[1,1,3,4] => 1
[1,1,4,1,1,1] => 2
[1,1,4,1,2] => 1
[1,1,4,2,1] => 1
[1,1,4,3] => 1
[1,1,5,1,1] => 4
[1,1,5,2] => 3
[1,1,6,1] => 1
[1,1,7] => 1
[1,2,1,1,1,1,1,1] => 3
[1,2,1,1,1,1,2] => 2
[1,2,1,1,1,2,1] => 4
[1,2,1,1,1,3] => 2
[1,2,1,1,2,1,1] => 2
[1,2,1,1,2,2] => 1
[1,2,1,1,3,1] => 2
[1,2,1,1,4] => 3
[1,2,1,2,1,1,1] => 2
[1,2,1,2,1,2] => 1
[1,2,1,2,2,1] => 3
[1,2,1,2,3] => 1
[1,2,1,3,1,1] => 2
[1,2,1,3,2] => 1
[1,2,1,4,1] => 1
[1,2,1,5] => 1
[1,2,2,1,1,1,1] => 2
[1,2,2,1,1,2] => 1
[1,2,2,1,2,1] => 3
[1,2,2,1,3] => 1
[1,2,2,2,1,1] => 1
[1,2,2,2,2] => 0
[1,2,2,3,1] => 1
[1,2,2,4] => 2
[1,2,3,1,1,1] => 3
[1,2,3,1,2] => 2
[1,2,3,2,1] => 4
[1,2,3,3] => 2
[1,2,4,1,1] => 1
[1,2,4,2] => 0
[1,2,5,1] => 1
[1,2,6] => 0
[1,3,1,1,1,1,1] => 3
[1,3,1,1,1,2] => 2
[1,3,1,1,2,1] => 2
[1,3,1,1,3] => 2
[1,3,1,2,1,1] => 2
[1,3,1,2,2] => 1
[1,3,1,3,1] => 4
[1,3,1,4] => 1
[1,3,2,1,1,1] => 2
[1,3,2,1,2] => 1
[1,3,2,2,1] => 1
[1,3,2,3] => 1
[1,3,3,1,1] => 2
[1,3,3,2] => 1
[1,3,4,1] => 1
[1,3,5] => 3
[1,4,1,1,1,1] => 2
[1,4,1,1,2] => 1
[1,4,1,2,1] => 1
[1,4,1,3] => 1
[1,4,2,1,1] => 3
[1,4,2,2] => 2
[1,4,3,1] => 1
[1,4,4] => 0
[1,5,1,1,1] => 2
[1,5,1,2] => 1
[1,5,2,1] => 1
[1,5,3] => 1
[1,6,1,1] => 1
[1,6,2] => 0
[1,7,1] => 4
[1,8] => 0
[2,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,2] => 4
[2,1,1,1,1,2,1] => 2
[2,1,1,1,1,3] => 2
[2,1,1,1,2,1,1] => 2
[2,1,1,1,2,2] => 3
[2,1,1,1,3,1] => 2
[2,1,1,1,4] => 1
[2,1,1,2,1,1,1] => 2
[2,1,1,2,1,2] => 3
[2,1,1,2,2,1] => 1
[2,1,1,2,3] => 1
[2,1,1,3,1,1] => 2
[2,1,1,3,2] => 3
[2,1,1,4,1] => 1
[2,1,1,5] => 1
[2,1,2,1,1,1,1] => 2
[2,1,2,1,1,2] => 3
[2,1,2,1,2,1] => 1
[2,1,2,1,3] => 1
[2,1,2,2,1,1] => 1
[2,1,2,2,2] => 2
[2,1,2,3,1] => 1
[2,1,2,4] => 0
[2,1,3,1,1,1] => 3
[2,1,3,1,2] => 4
[2,1,3,2,1] => 2
[2,1,3,3] => 2
[2,1,4,1,1] => 1
[2,1,4,2] => 2
[2,1,5,1] => 1
[2,1,6] => 0
[2,2,1,1,1,1,1] => 2
[2,2,1,1,1,2] => 3
[2,2,1,1,2,1] => 1
[2,2,1,1,3] => 1
[2,2,1,2,1,1] => 3
[2,2,1,2,2] => 4
[2,2,1,3,1] => 1
[2,2,1,4] => 2
[2,2,2,1,1,1] => 1
[2,2,2,1,2] => 2
[2,2,2,2,1] => 0
[2,2,2,3] => 0
[2,2,3,1,1] => 1
[2,2,3,2] => 2
[2,2,4,1] => 2
[2,2,5] => 0
[2,3,1,1,1,1] => 2
[2,3,1,1,2] => 3
[2,3,1,2,1] => 1
[2,3,1,3] => 1
[2,3,2,1,1] => 1
[2,3,2,2] => 2
[2,3,3,1] => 1
[2,3,4] => 0
[2,4,1,1,1] => 1
[2,4,1,2] => 2
[2,4,2,1] => 0
[2,4,3] => 0
[2,5,1,1] => 3
[2,5,2] => 4
[2,6,1] => 0
[2,7] => 0
[3,1,1,1,1,1,1] => 3
[3,1,1,1,1,2] => 2
[3,1,1,1,2,1] => 2
[3,1,1,1,3] => 4
[3,1,1,2,1,1] => 2
[3,1,1,2,2] => 1
[3,1,1,3,1] => 2
[3,1,1,4] => 1
[3,1,2,1,1,1] => 2
[3,1,2,1,2] => 1
[3,1,2,2,1] => 1
[3,1,2,3] => 3
[3,1,3,1,1] => 2
[3,1,3,2] => 1
[3,1,4,1] => 1
[3,1,5] => 1
[3,2,1,1,1,1] => 2
[3,2,1,1,2] => 1
[3,2,1,2,1] => 1
[3,2,1,3] => 3
[3,2,2,1,1] => 1
[3,2,2,2] => 0
[3,2,3,1] => 1
[3,2,4] => 0
[3,3,1,1,1] => 3
[3,3,1,2] => 2
[3,3,2,1] => 2
[3,3,3] => 4
[3,4,1,1] => 1
[3,4,2] => 0
[3,5,1] => 1
[3,6] => 2
[4,1,1,1,1,1] => 2
[4,1,1,1,2] => 1
[4,1,1,2,1] => 3
[4,1,1,3] => 1
[4,1,2,1,1] => 1
[4,1,2,2] => 2
[4,1,3,1] => 1
[4,1,4] => 4
[4,2,1,1,1] => 1
[4,2,1,2] => 0
[4,2,2,1] => 2
[4,2,3] => 0
[4,3,1,1] => 1
[4,3,2] => 0
[4,4,1] => 0
[4,5] => 0
[5,1,1,1,1] => 2
[5,1,1,2] => 1
[5,1,2,1] => 1
[5,1,3] => 1
[5,2,1,1] => 1
[5,2,2] => 0
[5,3,1] => 3
[5,4] => 0
[6,1,1,1] => 1
[6,1,2] => 0
[6,2,1] => 0
[6,3] => 2
[7,1,1] => 1
[7,2] => 0
[8,1] => 0
[9] => 4
[1,1,1,1,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1,2] => 3
[1,1,1,1,1,1,1,2,1] => 3
[1,1,1,1,1,1,1,3] => 3
[1,1,1,1,1,1,2,1,1] => 3
[1,1,1,1,1,1,2,2] => 2
[1,1,1,1,1,1,3,1] => 3
[1,1,1,1,1,1,4] => 2
[1,1,1,1,1,2,1,1,1] => 3
[1,1,1,1,1,2,1,2] => 2
[1,1,1,1,1,2,2,1] => 2
[1,1,1,1,1,2,3] => 2
[1,1,1,1,1,3,1,1] => 3
[1,1,1,1,1,3,2] => 2
[1,1,1,1,1,4,1] => 2
[1,1,1,1,1,5] => 2
[1,1,1,1,2,1,1,1,1] => 4
[1,1,1,1,2,1,1,2] => 3
[1,1,1,1,2,1,2,1] => 3
[1,1,1,1,2,1,3] => 3
[1,1,1,1,2,2,1,1] => 3
[1,1,1,1,2,2,2] => 2
[1,1,1,1,2,3,1] => 3
[1,1,1,1,2,4] => 2
[1,1,1,1,3,1,1,1] => 3
[1,1,1,1,3,1,2] => 2
[1,1,1,1,3,2,1] => 2
[1,1,1,1,3,3] => 2
[1,1,1,1,4,1,1] => 2
[1,1,1,1,4,2] => 1
[1,1,1,1,5,1] => 2
[1,1,1,1,6] => 1
[1,1,1,2,1,1,1,1,1] => 3
[1,1,1,2,1,1,1,2] => 2
[1,1,1,2,1,1,2,1] => 2
[1,1,1,2,1,1,3] => 2
[1,1,1,2,1,2,1,1] => 2
[1,1,1,2,1,2,2] => 1
[1,1,1,2,1,3,1] => 2
[1,1,1,2,1,4] => 1
[1,1,1,2,2,1,1,1] => 4
[1,1,1,2,2,1,2] => 3
[1,1,1,2,2,2,1] => 3
[1,1,1,2,2,3] => 3
[1,1,1,2,3,1,1] => 2
[1,1,1,2,3,2] => 1
[1,1,1,2,4,1] => 1
[1,1,1,2,5] => 1
[1,1,1,3,1,1,1,1] => 3
[1,1,1,3,1,1,2] => 2
[1,1,1,3,1,2,1] => 2
[1,1,1,3,1,3] => 2
[1,1,1,3,2,1,1] => 2
[1,1,1,3,2,2] => 1
[1,1,1,3,3,1] => 2
[1,1,1,3,4] => 1
[1,1,1,4,1,1,1] => 4
[1,1,1,4,1,2] => 3
[1,1,1,4,2,1] => 3
[1,1,1,4,3] => 3
[1,1,1,5,1,1] => 2
[1,1,1,5,2] => 1
[1,1,1,6,1] => 1
[1,1,1,7] => 1
[1,1,2,1,1,1,1,1,1] => 3
[1,1,2,1,1,1,1,2] => 2
[1,1,2,1,1,1,2,1] => 2
[1,1,2,1,1,1,3] => 2
[1,1,2,1,1,2,1,1] => 4
[1,1,2,1,1,2,2] => 3
[1,1,2,1,1,3,1] => 2
[1,1,2,1,1,4] => 1
[1,1,2,1,2,1,1,1] => 2
[1,1,2,1,2,1,2] => 1
[1,1,2,1,2,2,1] => 1
[1,1,2,1,2,3] => 1
[1,1,2,1,3,1,1] => 2
[1,1,2,1,3,2] => 1
[1,1,2,1,4,1] => 3
[1,1,2,1,5] => 1
[1,1,2,2,1,1,1,1] => 3
[1,1,2,2,1,1,2] => 2
[1,1,2,2,1,2,1] => 2
[1,1,2,2,1,3] => 2
[1,1,2,2,2,1,1] => 4
[1,1,2,2,2,2] => 3
[1,1,2,2,3,1] => 2
[1,1,2,2,4] => 1
[1,1,2,3,1,1,1] => 2
[1,1,2,3,1,2] => 1
[1,1,2,3,2,1] => 1
[1,1,2,3,3] => 1
[1,1,2,4,1,1] => 3
[1,1,2,4,2] => 2
[1,1,2,5,1] => 1
[1,1,2,6] => 2
[1,1,3,1,1,1,1,1] => 3
[1,1,3,1,1,1,2] => 2
[1,1,3,1,1,2,1] => 2
[1,1,3,1,1,3] => 2
[1,1,3,1,2,1,1] => 2
[1,1,3,1,2,2] => 1
[1,1,3,1,3,1] => 2
[1,1,3,1,4] => 1
[1,1,3,2,1,1,1] => 2
[1,1,3,2,1,2] => 1
[1,1,3,2,2,1] => 1
[1,1,3,2,3] => 1
[1,1,3,3,1,1] => 4
[1,1,3,3,2] => 3
[1,1,3,4,1] => 1
[1,1,3,5] => 1
[1,1,4,1,1,1,1] => 2
[1,1,4,1,1,2] => 1
[1,1,4,1,2,1] => 1
[1,1,4,1,3] => 1
[1,1,4,2,1,1] => 3
[1,1,4,2,2] => 2
[1,1,4,3,1] => 1
[1,1,4,4] => 0
[1,1,5,1,1,1] => 2
[1,1,5,1,2] => 1
[1,1,5,2,1] => 1
[1,1,5,3] => 1
[1,1,6,1,1] => 4
[1,1,6,2] => 3
[1,1,7,1] => 1
[1,1,8] => 0
[1,2,1,1,1,1,1,1,1] => 3
[1,2,1,1,1,1,1,2] => 2
[1,2,1,1,1,1,2,1] => 4
[1,2,1,1,1,1,3] => 2
[1,2,1,1,1,2,1,1] => 2
[1,2,1,1,1,2,2] => 1
[1,2,1,1,1,3,1] => 2
[1,2,1,1,1,4] => 3
[1,2,1,1,2,1,1,1] => 2
[1,2,1,1,2,1,2] => 1
[1,2,1,1,2,2,1] => 3
[1,2,1,1,2,3] => 1
[1,2,1,1,3,1,1] => 2
[1,2,1,1,3,2] => 1
[1,2,1,1,4,1] => 1
[1,2,1,1,5] => 1
[1,2,1,2,1,1,1,1] => 3
[1,2,1,2,1,1,2] => 2
[1,2,1,2,1,2,1] => 4
[1,2,1,2,1,3] => 2
[1,2,1,2,2,1,1] => 2
[1,2,1,2,2,2] => 1
[1,2,1,2,3,1] => 2
[1,2,1,2,4] => 3
[1,2,1,3,1,1,1] => 2
[1,2,1,3,1,2] => 1
[1,2,1,3,2,1] => 3
[1,2,1,3,3] => 1
[1,2,1,4,1,1] => 1
[1,2,1,4,2] => 0
[1,2,1,5,1] => 3
[1,2,1,6] => 0
[1,2,2,1,1,1,1,1] => 2
[1,2,2,1,1,1,2] => 1
[1,2,2,1,1,2,1] => 3
[1,2,2,1,1,3] => 1
[1,2,2,1,2,1,1] => 1
[1,2,2,1,2,2] => 0
[1,2,2,1,3,1] => 1
[1,2,2,1,4] => 2
[1,2,2,2,1,1,1] => 3
[1,2,2,2,1,2] => 2
[1,2,2,2,2,1] => 4
[1,2,2,2,3] => 2
[1,2,2,3,1,1] => 1
[1,2,2,3,2] => 0
[1,2,2,4,1] => 2
[1,2,2,5] => 0
[1,2,3,1,1,1,1] => 2
[1,2,3,1,1,2] => 1
[1,2,3,1,2,1] => 3
[1,2,3,1,3] => 1
[1,2,3,2,1,1] => 1
[1,2,3,2,2] => 0
[1,2,3,3,1] => 1
[1,2,3,4] => 2
[1,2,4,1,1,1] => 3
[1,2,4,1,2] => 2
[1,2,4,2,1] => 4
[1,2,4,3] => 2
[1,2,5,1,1] => 1
[1,2,5,2] => 0
[1,2,6,1] => 2
[1,2,7] => 2
[1,3,1,1,1,1,1,1] => 3
[1,3,1,1,1,1,2] => 2
[1,3,1,1,1,2,1] => 2
[1,3,1,1,1,3] => 2
[1,3,1,1,2,1,1] => 2
[1,3,1,1,2,2] => 1
[1,3,1,1,3,1] => 4
[1,3,1,1,4] => 1
[1,3,1,2,1,1,1] => 2
[1,3,1,2,1,2] => 1
[1,3,1,2,2,1] => 1
[1,3,1,2,3] => 1
[1,3,1,3,1,1] => 2
[1,3,1,3,2] => 1
[1,3,1,4,1] => 1
[1,3,1,5] => 3
[1,3,2,1,1,1,1] => 3
[1,3,2,1,1,2] => 2
[1,3,2,1,2,1] => 2
[1,3,2,1,3] => 2
[1,3,2,2,1,1] => 2
[1,3,2,2,2] => 1
[1,3,2,3,1] => 4
[1,3,2,4] => 1
[1,3,3,1,1,1] => 2
[1,3,3,1,2] => 1
[1,3,3,2,1] => 1
[1,3,3,3] => 1
[1,3,4,1,1] => 1
[1,3,4,2] => 0
[1,3,5,1] => 1
[1,3,6] => 0
[1,4,1,1,1,1,1] => 2
[1,4,1,1,1,2] => 1
[1,4,1,1,2,1] => 1
[1,4,1,1,3] => 1
[1,4,1,2,1,1] => 3
[1,4,1,2,2] => 2
[1,4,1,3,1] => 1
[1,4,1,4] => 0
[1,4,2,1,1,1] => 1
[1,4,2,1,2] => 0
[1,4,2,2,1] => 2
[1,4,2,3] => 0
[1,4,3,1,1] => 1
[1,4,3,2] => 0
[1,4,4,1] => 4
[1,4,5] => 0
[1,5,1,1,1,1] => 2
[1,5,1,1,2] => 1
[1,5,1,2,1] => 3
[1,5,1,3] => 1
[1,5,2,1,1] => 1
[1,5,2,2] => 0
[1,5,3,1] => 1
[1,5,4] => 2
[1,6,1,1,1] => 1
[1,6,1,2] => 0
[1,6,2,1] => 2
[1,6,3] => 0
[1,7,1,1] => 1
[1,7,2] => 0
[1,8,1] => 4
[1,9] => 0
[2,1,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,2] => 4
[2,1,1,1,1,1,2,1] => 2
[2,1,1,1,1,1,3] => 2
[2,1,1,1,1,2,1,1] => 2
[2,1,1,1,1,2,2] => 3
[2,1,1,1,1,3,1] => 2
[2,1,1,1,1,4] => 1
[2,1,1,1,2,1,1,1] => 2
[2,1,1,1,2,1,2] => 3
[2,1,1,1,2,2,1] => 1
[2,1,1,1,2,3] => 1
[2,1,1,1,3,1,1] => 2
[2,1,1,1,3,2] => 3
[2,1,1,1,4,1] => 1
[2,1,1,1,5] => 1
[2,1,1,2,1,1,1,1] => 3
[2,1,1,2,1,1,2] => 4
[2,1,1,2,1,2,1] => 2
[2,1,1,2,1,3] => 2
[2,1,1,2,2,1,1] => 2
[2,1,1,2,2,2] => 3
[2,1,1,2,3,1] => 2
[2,1,1,2,4] => 1
[2,1,1,3,1,1,1] => 2
[2,1,1,3,1,2] => 3
[2,1,1,3,2,1] => 1
[2,1,1,3,3] => 1
[2,1,1,4,1,1] => 1
[2,1,1,4,2] => 2
[2,1,1,5,1] => 1
[2,1,1,6] => 2
[2,1,2,1,1,1,1,1] => 2
[2,1,2,1,1,1,2] => 3
[2,1,2,1,1,2,1] => 1
[2,1,2,1,1,3] => 1
[2,1,2,1,2,1,1] => 1
[2,1,2,1,2,2] => 2
[2,1,2,1,3,1] => 1
[2,1,2,1,4] => 0
[2,1,2,2,1,1,1] => 3
[2,1,2,2,1,2] => 4
[2,1,2,2,2,1] => 2
[2,1,2,2,3] => 2
[2,1,2,3,1,1] => 1
[2,1,2,3,2] => 2
[2,1,2,4,1] => 0
[2,1,2,5] => 2
[2,1,3,1,1,1,1] => 2
[2,1,3,1,1,2] => 3
[2,1,3,1,2,1] => 1
[2,1,3,1,3] => 1
[2,1,3,2,1,1] => 1
[2,1,3,2,2] => 2
[2,1,3,3,1] => 1
[2,1,3,4] => 0
[2,1,4,1,1,1] => 3
[2,1,4,1,2] => 4
[2,1,4,2,1] => 2
[2,1,4,3] => 2
[2,1,5,1,1] => 1
[2,1,5,2] => 2
[2,1,6,1] => 0
[2,1,7] => 0
[2,2,1,1,1,1,1,1] => 2
[2,2,1,1,1,1,2] => 3
[2,2,1,1,1,2,1] => 1
[2,2,1,1,1,3] => 1
[2,2,1,1,2,1,1] => 3
[2,2,1,1,2,2] => 4
[2,2,1,1,3,1] => 1
[2,2,1,1,4] => 2
[2,2,1,2,1,1,1] => 1
[2,2,1,2,1,2] => 2
[2,2,1,2,2,1] => 0
[2,2,1,2,3] => 0
[2,2,1,3,1,1] => 1
[2,2,1,3,2] => 2
[2,2,1,4,1] => 2
[2,2,1,5] => 0
[2,2,2,1,1,1,1] => 2
[2,2,2,1,1,2] => 3
[2,2,2,1,2,1] => 1
[2,2,2,1,3] => 1
[2,2,2,2,1,1] => 3
[2,2,2,2,2] => 4
[2,2,2,3,1] => 1
[2,2,2,4] => 2
[2,2,3,1,1,1] => 1
[2,2,3,1,2] => 2
[2,2,3,2,1] => 0
[2,2,3,3] => 0
[2,2,4,1,1] => 2
[2,2,4,2] => 3
[2,2,5,1] => 0
[2,2,6] => 3
[2,3,1,1,1,1,1] => 2
[2,3,1,1,1,2] => 3
[2,3,1,1,2,1] => 1
[2,3,1,1,3] => 1
[2,3,1,2,1,1] => 1
[2,3,1,2,2] => 2
[2,3,1,3,1] => 1
[2,3,1,4] => 0
[2,3,2,1,1,1] => 1
[2,3,2,1,2] => 2
[2,3,2,2,1] => 0
[2,3,2,3] => 0
[2,3,3,1,1] => 3
[2,3,3,2] => 4
[2,3,4,1] => 0
[2,3,5] => 0
[2,4,1,1,1,1] => 1
[2,4,1,1,2] => 2
[2,4,1,2,1] => 0
[2,4,1,3] => 0
[2,4,2,1,1] => 2
[2,4,2,2] => 3
[2,4,3,1] => 0
[2,4,4] => 1
[2,5,1,1,1] => 1
[2,5,1,2] => 2
[2,5,2,1] => 0
[2,5,3] => 0
[2,6,1,1] => 3
[2,6,2] => 4
[2,7,1] => 0
[2,8] => 1
[3,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,2] => 2
[3,1,1,1,1,2,1] => 2
[3,1,1,1,1,3] => 4
[3,1,1,1,2,1,1] => 2
[3,1,1,1,2,2] => 1
[3,1,1,1,3,1] => 2
[3,1,1,1,4] => 1
[3,1,1,2,1,1,1] => 2
[3,1,1,2,1,2] => 1
[3,1,1,2,2,1] => 1
[3,1,1,2,3] => 3
[3,1,1,3,1,1] => 2
[3,1,1,3,2] => 1
[3,1,1,4,1] => 1
[3,1,1,5] => 1
[3,1,2,1,1,1,1] => 3
[3,1,2,1,1,2] => 2
[3,1,2,1,2,1] => 2
[3,1,2,1,3] => 4
[3,1,2,2,1,1] => 2
[3,1,2,2,2] => 1
[3,1,2,3,1] => 2
[3,1,2,4] => 1
[3,1,3,1,1,1] => 2
[3,1,3,1,2] => 1
[3,1,3,2,1] => 1
[3,1,3,3] => 3
[3,1,4,1,1] => 1
[3,1,4,2] => 0
[3,1,5,1] => 1
[3,1,6] => 0
[3,2,1,1,1,1,1] => 2
[3,2,1,1,1,2] => 1
[3,2,1,1,2,1] => 1
[3,2,1,1,3] => 3
[3,2,1,2,1,1] => 1
[3,2,1,2,2] => 0
[3,2,1,3,1] => 1
[3,2,1,4] => 0
[3,2,2,1,1,1] => 3
[3,2,2,1,2] => 2
[3,2,2,2,1] => 2
[3,2,2,3] => 4
[3,2,3,1,1] => 1
[3,2,3,2] => 0
[3,2,4,1] => 0
[3,2,5] => 0
[3,3,1,1,1,1] => 2
[3,3,1,1,2] => 1
[3,3,1,2,1] => 1
[3,3,1,3] => 3
[3,3,2,1,1] => 1
[3,3,2,2] => 0
[3,3,3,1] => 1
[3,3,4] => 0
[3,4,1,1,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 2
[3,4,3] => 4
[3,5,1,1] => 1
[3,5,2] => 0
[3,6,1] => 0
[3,7] => 0
[4,1,1,1,1,1,1] => 2
[4,1,1,1,1,2] => 1
[4,1,1,1,2,1] => 3
[4,1,1,1,3] => 1
[4,1,1,2,1,1] => 1
[4,1,1,2,2] => 2
[4,1,1,3,1] => 1
[4,1,1,4] => 4
[4,1,2,1,1,1] => 1
[4,1,2,1,2] => 0
[4,1,2,2,1] => 2
[4,1,2,3] => 0
[4,1,3,1,1] => 1
[4,1,3,2] => 0
[4,1,4,1] => 0
[4,1,5] => 0
[4,2,1,1,1,1] => 2
[4,2,1,1,2] => 1
[4,2,1,2,1] => 3
[4,2,1,3] => 1
[4,2,2,1,1] => 1
[4,2,2,2] => 2
[4,2,3,1] => 1
[4,2,4] => 4
[4,3,1,1,1] => 1
[4,3,1,2] => 0
[4,3,2,1] => 2
[4,3,3] => 0
[4,4,1,1] => 0
[4,4,2] => 1
[4,5,1] => 2
[4,6] => 1
[5,1,1,1,1,1] => 2
[5,1,1,1,2] => 1
[5,1,1,2,1] => 1
[5,1,1,3] => 1
[5,1,2,1,1] => 1
[5,1,2,2] => 0
[5,1,3,1] => 3
[5,1,4] => 0
[5,2,1,1,1] => 1
[5,2,1,2] => 2
[5,2,2,1] => 0
[5,2,3] => 0
[5,3,1,1] => 1
[5,3,2] => 0
[5,4,1] => 0
[5,5] => 4
[6,1,1,1,1] => 1
[6,1,1,2] => 2
[6,1,2,1] => 0
[6,1,3] => 0
[6,2,1,1] => 2
[6,2,2] => 3
[6,3,1] => 0
[6,4] => 1
[7,1,1,1] => 1
[7,1,2] => 0
[7,2,1] => 2
[7,3] => 0
[8,1,1] => 0
[8,2] => 1
[9,1] => 0
[10] => 4
click to show generating function       
Description
The index of the maximal parabolic seaweed algebra associated with the composition.
Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows:
* place $n$ dots on a horizontal line
* subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line
* subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line
By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is
$$ \operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1, $$
where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander.
This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.
References
[1] Dergachev, V., Kirillov, A. Index of Lie algebras of seaweed type MathSciNet:1774864
[2] Coll, V., Mayers, A., Mayers, N. Statistics on integer partitions arising from seaweed algebras arXiv:1809.09271
Code
def to_meander(la, mu):
    """
    Return the meander associated with two partitions as a graph.
    """
    la = Composition(la)
    mu = Composition(mu)
    n = la.size()
    assert mu.size() == n
    M = Graph(n)
    offset = 0
    for part in la:
        M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)])
        offset += part
    offset = 0
    for part in mu:
        M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)])
        offset += part
    return M

def seaweed_index(la, mu):
    M = to_meander(la, mu)
    index = -1
    for G in M.connected_components_subgraphs():
        if G.is_cycle():
            index += 2
        else:
            index += 1
    return index

def statistic(la):
    return seaweed_index(la, la.size())
Created
Sep 26, 2018 at 04:49 by Martin Rubey
Updated
Sep 26, 2018 at 10:46 by Martin Rubey