Identifier
Identifier
Values
[1] => 1
[1,1] => 2
[2] => 2
[1,1,1] => 2
[1,2] => 3
[2,1] => 3
[3] => 2
[1,1,1,1] => 2
[1,1,2] => 3
[1,2,1] => 4
[1,3] => 3
[2,1,1] => 3
[2,2] => 4
[3,1] => 3
[4] => 2
[1,1,1,1,1] => 2
[1,1,1,2] => 3
[1,1,2,1] => 4
[1,1,3] => 3
[1,2,1,1] => 4
[1,2,2] => 5
[1,3,1] => 4
[1,4] => 3
[2,1,1,1] => 3
[2,1,2] => 4
[2,2,1] => 5
[2,3] => 4
[3,1,1] => 3
[3,2] => 4
[4,1] => 3
[5] => 2
[1,1,1,1,1,1] => 2
[1,1,1,1,2] => 3
[1,1,1,2,1] => 4
[1,1,1,3] => 3
[1,1,2,1,1] => 4
[1,1,2,2] => 5
[1,1,3,1] => 4
[1,1,4] => 3
[1,2,1,1,1] => 4
[1,2,1,2] => 5
[1,2,2,1] => 6
[1,2,3] => 5
[1,3,1,1] => 4
[1,3,2] => 5
[1,4,1] => 4
[1,5] => 3
[2,1,1,1,1] => 3
[2,1,1,2] => 4
[2,1,2,1] => 5
[2,1,3] => 4
[2,2,1,1] => 5
[2,2,2] => 6
[2,3,1] => 5
[2,4] => 4
[3,1,1,1] => 3
[3,1,2] => 4
[3,2,1] => 5
[3,3] => 4
[4,1,1] => 3
[4,2] => 4
[5,1] => 3
[6] => 2
[1,1,1,1,1,1,1] => 2
[1,1,1,1,1,2] => 3
[1,1,1,1,2,1] => 4
[1,1,1,1,3] => 3
[1,1,1,2,1,1] => 4
[1,1,1,2,2] => 5
[1,1,1,3,1] => 4
[1,1,1,4] => 3
[1,1,2,1,1,1] => 4
[1,1,2,1,2] => 5
[1,1,2,2,1] => 6
[1,1,2,3] => 5
[1,1,3,1,1] => 4
[1,1,3,2] => 5
[1,1,4,1] => 4
[1,1,5] => 3
[1,2,1,1,1,1] => 4
[1,2,1,1,2] => 5
[1,2,1,2,1] => 6
[1,2,1,3] => 5
[1,2,2,1,1] => 6
[1,2,2,2] => 7
[1,2,3,1] => 6
[1,2,4] => 5
[1,3,1,1,1] => 4
[1,3,1,2] => 5
[1,3,2,1] => 6
[1,3,3] => 5
[1,4,1,1] => 4
[1,4,2] => 5
[1,5,1] => 4
[1,6] => 3
[2,1,1,1,1,1] => 3
[2,1,1,1,2] => 4
[2,1,1,2,1] => 5
[2,1,1,3] => 4
[2,1,2,1,1] => 5
[2,1,2,2] => 6
[2,1,3,1] => 5
[2,1,4] => 4
[2,2,1,1,1] => 5
[2,2,1,2] => 6
[2,2,2,1] => 7
[2,2,3] => 6
[2,3,1,1] => 5
[2,3,2] => 6
[2,4,1] => 5
[2,5] => 4
[3,1,1,1,1] => 3
[3,1,1,2] => 4
[3,1,2,1] => 5
[3,1,3] => 4
[3,2,1,1] => 5
[3,2,2] => 6
[3,3,1] => 5
[3,4] => 4
[4,1,1,1] => 3
[4,1,2] => 4
[4,2,1] => 5
[4,3] => 4
[5,1,1] => 3
[5,2] => 4
[6,1] => 3
[7] => 2
[1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,2] => 3
[1,1,1,1,1,2,1] => 4
[1,1,1,1,1,3] => 3
[1,1,1,1,2,1,1] => 4
[1,1,1,1,2,2] => 5
[1,1,1,1,3,1] => 4
[1,1,1,1,4] => 3
[1,1,1,2,1,1,1] => 4
[1,1,1,2,1,2] => 5
[1,1,1,2,2,1] => 6
[1,1,1,2,3] => 5
[1,1,1,3,1,1] => 4
[1,1,1,3,2] => 5
[1,1,1,4,1] => 4
[1,1,1,5] => 3
[1,1,2,1,1,1,1] => 4
[1,1,2,1,1,2] => 5
[1,1,2,1,2,1] => 6
[1,1,2,1,3] => 5
[1,1,2,2,1,1] => 6
[1,1,2,2,2] => 7
[1,1,2,3,1] => 6
[1,1,2,4] => 5
[1,1,3,1,1,1] => 4
[1,1,3,1,2] => 5
[1,1,3,2,1] => 6
[1,1,3,3] => 5
[1,1,4,1,1] => 4
[1,1,4,2] => 5
[1,1,5,1] => 4
[1,1,6] => 3
[1,2,1,1,1,1,1] => 4
[1,2,1,1,1,2] => 5
[1,2,1,1,2,1] => 6
[1,2,1,1,3] => 5
[1,2,1,2,1,1] => 6
[1,2,1,2,2] => 7
[1,2,1,3,1] => 6
[1,2,1,4] => 5
[1,2,2,1,1,1] => 6
[1,2,2,1,2] => 7
[1,2,2,2,1] => 8
[1,2,2,3] => 7
[1,2,3,1,1] => 6
[1,2,3,2] => 7
[1,2,4,1] => 6
[1,2,5] => 5
[1,3,1,1,1,1] => 4
[1,3,1,1,2] => 5
[1,3,1,2,1] => 6
[1,3,1,3] => 5
[1,3,2,1,1] => 6
[1,3,2,2] => 7
[1,3,3,1] => 6
[1,3,4] => 5
[1,4,1,1,1] => 4
[1,4,1,2] => 5
[1,4,2,1] => 6
[1,4,3] => 5
[1,5,1,1] => 4
[1,5,2] => 5
[1,6,1] => 4
[1,7] => 3
[2,1,1,1,1,1,1] => 3
[2,1,1,1,1,2] => 4
[2,1,1,1,2,1] => 5
[2,1,1,1,3] => 4
[2,1,1,2,1,1] => 5
[2,1,1,2,2] => 6
[2,1,1,3,1] => 5
[2,1,1,4] => 4
[2,1,2,1,1,1] => 5
[2,1,2,1,2] => 6
[2,1,2,2,1] => 7
[2,1,2,3] => 6
[2,1,3,1,1] => 5
[2,1,3,2] => 6
[2,1,4,1] => 5
[2,1,5] => 4
[2,2,1,1,1,1] => 5
[2,2,1,1,2] => 6
[2,2,1,2,1] => 7
[2,2,1,3] => 6
[2,2,2,1,1] => 7
[2,2,2,2] => 8
[2,2,3,1] => 7
[2,2,4] => 6
[2,3,1,1,1] => 5
[2,3,1,2] => 6
[2,3,2,1] => 7
[2,3,3] => 6
[2,4,1,1] => 5
[2,4,2] => 6
[2,5,1] => 5
[2,6] => 4
[3,1,1,1,1,1] => 3
[3,1,1,1,2] => 4
[3,1,1,2,1] => 5
[3,1,1,3] => 4
[3,1,2,1,1] => 5
[3,1,2,2] => 6
[3,1,3,1] => 5
[3,1,4] => 4
[3,2,1,1,1] => 5
[3,2,1,2] => 6
[3,2,2,1] => 7
[3,2,3] => 6
[3,3,1,1] => 5
[3,3,2] => 6
[3,4,1] => 5
[3,5] => 4
[4,1,1,1,1] => 3
[4,1,1,2] => 4
[4,1,2,1] => 5
[4,1,3] => 4
[4,2,1,1] => 5
[4,2,2] => 6
[4,3,1] => 5
[4,4] => 4
[5,1,1,1] => 3
[5,1,2] => 4
[5,2,1] => 5
[5,3] => 4
[6,1,1] => 3
[6,2] => 4
[7,1] => 3
[8] => 2
[1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,2] => 3
[1,1,1,1,1,1,2,1] => 4
[1,1,1,1,1,1,3] => 3
[1,1,1,1,1,2,1,1] => 4
[1,1,1,1,1,2,2] => 5
[1,1,1,1,1,3,1] => 4
[1,1,1,1,1,4] => 3
[1,1,1,1,2,1,1,1] => 4
[1,1,1,1,2,1,2] => 5
[1,1,1,1,2,2,1] => 6
[1,1,1,1,2,3] => 5
[1,1,1,1,3,1,1] => 4
[1,1,1,1,3,2] => 5
[1,1,1,1,4,1] => 4
[1,1,1,1,5] => 3
[1,1,1,2,1,1,1,1] => 4
[1,1,1,2,1,1,2] => 5
[1,1,1,2,1,2,1] => 6
[1,1,1,2,1,3] => 5
[1,1,1,2,2,1,1] => 6
[1,1,1,2,2,2] => 7
[1,1,1,2,3,1] => 6
[1,1,1,2,4] => 5
[1,1,1,3,1,1,1] => 4
[1,1,1,3,1,2] => 5
[1,1,1,3,2,1] => 6
[1,1,1,3,3] => 5
[1,1,1,4,1,1] => 4
[1,1,1,4,2] => 5
[1,1,1,5,1] => 4
[1,1,1,6] => 3
[1,1,2,1,1,1,1,1] => 4
[1,1,2,1,1,1,2] => 5
[1,1,2,1,1,2,1] => 6
[1,1,2,1,1,3] => 5
[1,1,2,1,2,1,1] => 6
[1,1,2,1,2,2] => 7
[1,1,2,1,3,1] => 6
[1,1,2,1,4] => 5
[1,1,2,2,1,1,1] => 6
[1,1,2,2,1,2] => 7
[1,1,2,2,2,1] => 8
[1,1,2,2,3] => 7
[1,1,2,3,1,1] => 6
[1,1,2,3,2] => 7
[1,1,2,4,1] => 6
[1,1,2,5] => 5
[1,1,3,1,1,1,1] => 4
[1,1,3,1,1,2] => 5
[1,1,3,1,2,1] => 6
[1,1,3,1,3] => 5
[1,1,3,2,1,1] => 6
[1,1,3,2,2] => 7
[1,1,3,3,1] => 6
[1,1,3,4] => 5
[1,1,4,1,1,1] => 4
[1,1,4,1,2] => 5
[1,1,4,2,1] => 6
[1,1,4,3] => 5
[1,1,5,1,1] => 4
[1,1,5,2] => 5
[1,1,6,1] => 4
[1,1,7] => 3
[1,2,1,1,1,1,1,1] => 4
[1,2,1,1,1,1,2] => 5
[1,2,1,1,1,2,1] => 6
[1,2,1,1,1,3] => 5
[1,2,1,1,2,1,1] => 6
[1,2,1,1,2,2] => 7
[1,2,1,1,3,1] => 6
[1,2,1,1,4] => 5
[1,2,1,2,1,1,1] => 6
[1,2,1,2,1,2] => 7
[1,2,1,2,2,1] => 8
[1,2,1,2,3] => 7
[1,2,1,3,1,1] => 6
[1,2,1,3,2] => 7
[1,2,1,4,1] => 6
[1,2,1,5] => 5
[1,2,2,1,1,1,1] => 6
[1,2,2,1,1,2] => 7
[1,2,2,1,2,1] => 8
[1,2,2,1,3] => 7
[1,2,2,2,1,1] => 8
[1,2,2,2,2] => 9
[1,2,2,3,1] => 8
[1,2,2,4] => 7
[1,2,3,1,1,1] => 6
[1,2,3,1,2] => 7
[1,2,3,2,1] => 8
[1,2,3,3] => 7
[1,2,4,1,1] => 6
[1,2,4,2] => 7
[1,2,5,1] => 6
[1,2,6] => 5
[1,3,1,1,1,1,1] => 4
[1,3,1,1,1,2] => 5
[1,3,1,1,2,1] => 6
[1,3,1,1,3] => 5
[1,3,1,2,1,1] => 6
[1,3,1,2,2] => 7
[1,3,1,3,1] => 6
[1,3,1,4] => 5
[1,3,2,1,1,1] => 6
[1,3,2,1,2] => 7
[1,3,2,2,1] => 8
[1,3,2,3] => 7
[1,3,3,1,1] => 6
[1,3,3,2] => 7
[1,3,4,1] => 6
[1,3,5] => 5
[1,4,1,1,1,1] => 4
[1,4,1,1,2] => 5
[1,4,1,2,1] => 6
[1,4,1,3] => 5
[1,4,2,1,1] => 6
[1,4,2,2] => 7
[1,4,3,1] => 6
[1,4,4] => 5
[1,5,1,1,1] => 4
[1,5,1,2] => 5
[1,5,2,1] => 6
[1,5,3] => 5
[1,6,1,1] => 4
[1,6,2] => 5
[1,7,1] => 4
[1,8] => 3
[2,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,2] => 4
[2,1,1,1,1,2,1] => 5
[2,1,1,1,1,3] => 4
[2,1,1,1,2,1,1] => 5
[2,1,1,1,2,2] => 6
[2,1,1,1,3,1] => 5
[2,1,1,1,4] => 4
[2,1,1,2,1,1,1] => 5
[2,1,1,2,1,2] => 6
[2,1,1,2,2,1] => 7
[2,1,1,2,3] => 6
[2,1,1,3,1,1] => 5
[2,1,1,3,2] => 6
[2,1,1,4,1] => 5
[2,1,1,5] => 4
[2,1,2,1,1,1,1] => 5
[2,1,2,1,1,2] => 6
[2,1,2,1,2,1] => 7
[2,1,2,1,3] => 6
[2,1,2,2,1,1] => 7
[2,1,2,2,2] => 8
[2,1,2,3,1] => 7
[2,1,2,4] => 6
[2,1,3,1,1,1] => 5
[2,1,3,1,2] => 6
[2,1,3,2,1] => 7
[2,1,3,3] => 6
[2,1,4,1,1] => 5
[2,1,4,2] => 6
[2,1,5,1] => 5
[2,1,6] => 4
[2,2,1,1,1,1,1] => 5
[2,2,1,1,1,2] => 6
[2,2,1,1,2,1] => 7
[2,2,1,1,3] => 6
[2,2,1,2,1,1] => 7
[2,2,1,2,2] => 8
[2,2,1,3,1] => 7
[2,2,1,4] => 6
[2,2,2,1,1,1] => 7
[2,2,2,1,2] => 8
[2,2,2,2,1] => 9
[2,2,2,3] => 8
[2,2,3,1,1] => 7
[2,2,3,2] => 8
[2,2,4,1] => 7
[2,2,5] => 6
[2,3,1,1,1,1] => 5
[2,3,1,1,2] => 6
[2,3,1,2,1] => 7
[2,3,1,3] => 6
[2,3,2,1,1] => 7
[2,3,2,2] => 8
[2,3,3,1] => 7
[2,3,4] => 6
[2,4,1,1,1] => 5
[2,4,1,2] => 6
[2,4,2,1] => 7
[2,4,3] => 6
[2,5,1,1] => 5
[2,5,2] => 6
[2,6,1] => 5
[2,7] => 4
[3,1,1,1,1,1,1] => 3
[3,1,1,1,1,2] => 4
[3,1,1,1,2,1] => 5
[3,1,1,1,3] => 4
[3,1,1,2,1,1] => 5
[3,1,1,2,2] => 6
[3,1,1,3,1] => 5
[3,1,1,4] => 4
[3,1,2,1,1,1] => 5
[3,1,2,1,2] => 6
[3,1,2,2,1] => 7
[3,1,2,3] => 6
[3,1,3,1,1] => 5
[3,1,3,2] => 6
[3,1,4,1] => 5
[3,1,5] => 4
[3,2,1,1,1,1] => 5
[3,2,1,1,2] => 6
[3,2,1,2,1] => 7
[3,2,1,3] => 6
[3,2,2,1,1] => 7
[3,2,2,2] => 8
[3,2,3,1] => 7
[3,2,4] => 6
[3,3,1,1,1] => 5
[3,3,1,2] => 6
[3,3,2,1] => 7
[3,3,3] => 6
[3,4,1,1] => 5
[3,4,2] => 6
[3,5,1] => 5
[3,6] => 4
[4,1,1,1,1,1] => 3
[4,1,1,1,2] => 4
[4,1,1,2,1] => 5
[4,1,1,3] => 4
[4,1,2,1,1] => 5
[4,1,2,2] => 6
[4,1,3,1] => 5
[4,1,4] => 4
[4,2,1,1,1] => 5
[4,2,1,2] => 6
[4,2,2,1] => 7
[4,2,3] => 6
[4,3,1,1] => 5
[4,3,2] => 6
[4,4,1] => 5
[4,5] => 4
[5,1,1,1,1] => 3
[5,1,1,2] => 4
[5,1,2,1] => 5
[5,1,3] => 4
[5,2,1,1] => 5
[5,2,2] => 6
[5,3,1] => 5
[5,4] => 4
[6,1,1,1] => 3
[6,1,2] => 4
[6,2,1] => 5
[6,3] => 4
[7,1,1] => 3
[7,2] => 4
[8,1] => 3
[9] => 2
[1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,2] => 3
[1,1,1,1,1,1,1,2,1] => 4
[1,1,1,1,1,1,1,3] => 3
[1,1,1,1,1,1,2,1,1] => 4
[1,1,1,1,1,1,2,2] => 5
[1,1,1,1,1,1,3,1] => 4
[1,1,1,1,1,1,4] => 3
[1,1,1,1,1,2,1,1,1] => 4
[1,1,1,1,1,2,1,2] => 5
[1,1,1,1,1,2,2,1] => 6
[1,1,1,1,1,2,3] => 5
[1,1,1,1,1,3,1,1] => 4
[1,1,1,1,1,3,2] => 5
[1,1,1,1,1,4,1] => 4
[1,1,1,1,1,5] => 3
[1,1,1,1,2,1,1,1,1] => 4
[1,1,1,1,2,1,1,2] => 5
[1,1,1,1,2,1,2,1] => 6
[1,1,1,1,2,1,3] => 5
[1,1,1,1,2,2,1,1] => 6
[1,1,1,1,2,2,2] => 7
[1,1,1,1,2,3,1] => 6
[1,1,1,1,2,4] => 5
[1,1,1,1,3,1,1,1] => 4
[1,1,1,1,3,1,2] => 5
[1,1,1,1,3,2,1] => 6
[1,1,1,1,3,3] => 5
[1,1,1,1,4,1,1] => 4
[1,1,1,1,4,2] => 5
[1,1,1,1,5,1] => 4
[1,1,1,1,6] => 3
[1,1,1,2,1,1,1,1,1] => 4
[1,1,1,2,1,1,1,2] => 5
[1,1,1,2,1,1,2,1] => 6
[1,1,1,2,1,1,3] => 5
[1,1,1,2,1,2,1,1] => 6
[1,1,1,2,1,2,2] => 7
[1,1,1,2,1,3,1] => 6
[1,1,1,2,1,4] => 5
[1,1,1,2,2,1,1,1] => 6
[1,1,1,2,2,1,2] => 7
[1,1,1,2,2,2,1] => 8
[1,1,1,2,2,3] => 7
[1,1,1,2,3,1,1] => 6
[1,1,1,2,3,2] => 7
[1,1,1,2,4,1] => 6
[1,1,1,2,5] => 5
[1,1,1,3,1,1,1,1] => 4
[1,1,1,3,1,1,2] => 5
[1,1,1,3,1,2,1] => 6
[1,1,1,3,1,3] => 5
[1,1,1,3,2,1,1] => 6
[1,1,1,3,2,2] => 7
[1,1,1,3,3,1] => 6
[1,1,1,3,4] => 5
[1,1,1,4,1,1,1] => 4
[1,1,1,4,1,2] => 5
[1,1,1,4,2,1] => 6
[1,1,1,4,3] => 5
[1,1,1,5,1,1] => 4
[1,1,1,5,2] => 5
[1,1,1,6,1] => 4
[1,1,1,7] => 3
[1,1,2,1,1,1,1,1,1] => 4
[1,1,2,1,1,1,1,2] => 5
[1,1,2,1,1,1,2,1] => 6
[1,1,2,1,1,1,3] => 5
[1,1,2,1,1,2,1,1] => 6
[1,1,2,1,1,2,2] => 7
[1,1,2,1,1,3,1] => 6
[1,1,2,1,1,4] => 5
[1,1,2,1,2,1,1,1] => 6
[1,1,2,1,2,1,2] => 7
[1,1,2,1,2,2,1] => 8
[1,1,2,1,2,3] => 7
[1,1,2,1,3,1,1] => 6
[1,1,2,1,3,2] => 7
[1,1,2,1,4,1] => 6
[1,1,2,1,5] => 5
[1,1,2,2,1,1,1,1] => 6
[1,1,2,2,1,1,2] => 7
[1,1,2,2,1,2,1] => 8
[1,1,2,2,1,3] => 7
[1,1,2,2,2,1,1] => 8
[1,1,2,2,2,2] => 9
[1,1,2,2,3,1] => 8
[1,1,2,2,4] => 7
[1,1,2,3,1,1,1] => 6
[1,1,2,3,1,2] => 7
[1,1,2,3,2,1] => 8
[1,1,2,3,3] => 7
[1,1,2,4,1,1] => 6
[1,1,2,4,2] => 7
[1,1,2,5,1] => 6
[1,1,2,6] => 5
[1,1,3,1,1,1,1,1] => 4
[1,1,3,1,1,1,2] => 5
[1,1,3,1,1,2,1] => 6
[1,1,3,1,1,3] => 5
[1,1,3,1,2,1,1] => 6
[1,1,3,1,2,2] => 7
[1,1,3,1,3,1] => 6
[1,1,3,1,4] => 5
[1,1,3,2,1,1,1] => 6
[1,1,3,2,1,2] => 7
[1,1,3,2,2,1] => 8
[1,1,3,2,3] => 7
[1,1,3,3,1,1] => 6
[1,1,3,3,2] => 7
[1,1,3,4,1] => 6
[1,1,3,5] => 5
[1,1,4,1,1,1,1] => 4
[1,1,4,1,1,2] => 5
[1,1,4,1,2,1] => 6
[1,1,4,1,3] => 5
[1,1,4,2,1,1] => 6
[1,1,4,2,2] => 7
[1,1,4,3,1] => 6
[1,1,4,4] => 5
[1,1,5,1,1,1] => 4
[1,1,5,1,2] => 5
[1,1,5,2,1] => 6
[1,1,5,3] => 5
[1,1,6,1,1] => 4
[1,1,6,2] => 5
[1,1,7,1] => 4
[1,1,8] => 3
[1,2,1,1,1,1,1,1,1] => 4
[1,2,1,1,1,1,1,2] => 5
[1,2,1,1,1,1,2,1] => 6
[1,2,1,1,1,1,3] => 5
[1,2,1,1,1,2,1,1] => 6
[1,2,1,1,1,2,2] => 7
[1,2,1,1,1,3,1] => 6
[1,2,1,1,1,4] => 5
[1,2,1,1,2,1,1,1] => 6
[1,2,1,1,2,1,2] => 7
[1,2,1,1,2,2,1] => 8
[1,2,1,1,2,3] => 7
[1,2,1,1,3,1,1] => 6
[1,2,1,1,3,2] => 7
[1,2,1,1,4,1] => 6
[1,2,1,1,5] => 5
[1,2,1,2,1,1,1,1] => 6
[1,2,1,2,1,1,2] => 7
[1,2,1,2,1,2,1] => 8
[1,2,1,2,1,3] => 7
[1,2,1,2,2,1,1] => 8
[1,2,1,2,2,2] => 9
[1,2,1,2,3,1] => 8
[1,2,1,2,4] => 7
[1,2,1,3,1,1,1] => 6
[1,2,1,3,1,2] => 7
[1,2,1,3,2,1] => 8
[1,2,1,3,3] => 7
[1,2,1,4,1,1] => 6
[1,2,1,4,2] => 7
[1,2,1,5,1] => 6
[1,2,1,6] => 5
[1,2,2,1,1,1,1,1] => 6
[1,2,2,1,1,1,2] => 7
[1,2,2,1,1,2,1] => 8
[1,2,2,1,1,3] => 7
[1,2,2,1,2,1,1] => 8
[1,2,2,1,2,2] => 9
[1,2,2,1,3,1] => 8
[1,2,2,1,4] => 7
[1,2,2,2,1,1,1] => 8
[1,2,2,2,1,2] => 9
[1,2,2,2,2,1] => 10
[1,2,2,2,3] => 9
[1,2,2,3,1,1] => 8
[1,2,2,3,2] => 9
[1,2,2,4,1] => 8
[1,2,2,5] => 7
[1,2,3,1,1,1,1] => 6
[1,2,3,1,1,2] => 7
[1,2,3,1,2,1] => 8
[1,2,3,1,3] => 7
[1,2,3,2,1,1] => 8
[1,2,3,2,2] => 9
[1,2,3,3,1] => 8
[1,2,3,4] => 7
[1,2,4,1,1,1] => 6
[1,2,4,1,2] => 7
[1,2,4,2,1] => 8
[1,2,4,3] => 7
[1,2,5,1,1] => 6
[1,2,5,2] => 7
[1,2,6,1] => 6
[1,2,7] => 5
[1,3,1,1,1,1,1,1] => 4
[1,3,1,1,1,1,2] => 5
[1,3,1,1,1,2,1] => 6
[1,3,1,1,1,3] => 5
[1,3,1,1,2,1,1] => 6
[1,3,1,1,2,2] => 7
[1,3,1,1,3,1] => 6
[1,3,1,1,4] => 5
[1,3,1,2,1,1,1] => 6
[1,3,1,2,1,2] => 7
[1,3,1,2,2,1] => 8
[1,3,1,2,3] => 7
[1,3,1,3,1,1] => 6
[1,3,1,3,2] => 7
[1,3,1,4,1] => 6
[1,3,1,5] => 5
[1,3,2,1,1,1,1] => 6
[1,3,2,1,1,2] => 7
[1,3,2,1,2,1] => 8
[1,3,2,1,3] => 7
[1,3,2,2,1,1] => 8
[1,3,2,2,2] => 9
[1,3,2,3,1] => 8
[1,3,2,4] => 7
[1,3,3,1,1,1] => 6
[1,3,3,1,2] => 7
[1,3,3,2,1] => 8
[1,3,3,3] => 7
[1,3,4,1,1] => 6
[1,3,4,2] => 7
[1,3,5,1] => 6
[1,3,6] => 5
[1,4,1,1,1,1,1] => 4
[1,4,1,1,1,2] => 5
[1,4,1,1,2,1] => 6
[1,4,1,1,3] => 5
[1,4,1,2,1,1] => 6
[1,4,1,2,2] => 7
[1,4,1,3,1] => 6
[1,4,1,4] => 5
[1,4,2,1,1,1] => 6
[1,4,2,1,2] => 7
[1,4,2,2,1] => 8
[1,4,2,3] => 7
[1,4,3,1,1] => 6
[1,4,3,2] => 7
[1,4,4,1] => 6
[1,4,5] => 5
[1,5,1,1,1,1] => 4
[1,5,1,1,2] => 5
[1,5,1,2,1] => 6
[1,5,1,3] => 5
[1,5,2,1,1] => 6
[1,5,2,2] => 7
[1,5,3,1] => 6
[1,5,4] => 5
[1,6,1,1,1] => 4
[1,6,1,2] => 5
[1,6,2,1] => 6
[1,6,3] => 5
[1,7,1,1] => 4
[1,7,2] => 5
[1,8,1] => 4
[1,9] => 3
[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] => 5
[2,1,1,1,1,1,3] => 4
[2,1,1,1,1,2,1,1] => 5
[2,1,1,1,1,2,2] => 6
[2,1,1,1,1,3,1] => 5
[2,1,1,1,1,4] => 4
[2,1,1,1,2,1,1,1] => 5
[2,1,1,1,2,1,2] => 6
[2,1,1,1,2,2,1] => 7
[2,1,1,1,2,3] => 6
[2,1,1,1,3,1,1] => 5
[2,1,1,1,3,2] => 6
[2,1,1,1,4,1] => 5
[2,1,1,1,5] => 4
[2,1,1,2,1,1,1,1] => 5
[2,1,1,2,1,1,2] => 6
[2,1,1,2,1,2,1] => 7
[2,1,1,2,1,3] => 6
[2,1,1,2,2,1,1] => 7
[2,1,1,2,2,2] => 8
[2,1,1,2,3,1] => 7
[2,1,1,2,4] => 6
[2,1,1,3,1,1,1] => 5
[2,1,1,3,1,2] => 6
[2,1,1,3,2,1] => 7
[2,1,1,3,3] => 6
[2,1,1,4,1,1] => 5
[2,1,1,4,2] => 6
[2,1,1,5,1] => 5
[2,1,1,6] => 4
[2,1,2,1,1,1,1,1] => 5
[2,1,2,1,1,1,2] => 6
[2,1,2,1,1,2,1] => 7
[2,1,2,1,1,3] => 6
[2,1,2,1,2,1,1] => 7
[2,1,2,1,2,2] => 8
[2,1,2,1,3,1] => 7
[2,1,2,1,4] => 6
[2,1,2,2,1,1,1] => 7
[2,1,2,2,1,2] => 8
[2,1,2,2,2,1] => 9
[2,1,2,2,3] => 8
[2,1,2,3,1,1] => 7
[2,1,2,3,2] => 8
[2,1,2,4,1] => 7
[2,1,2,5] => 6
[2,1,3,1,1,1,1] => 5
[2,1,3,1,1,2] => 6
[2,1,3,1,2,1] => 7
[2,1,3,1,3] => 6
[2,1,3,2,1,1] => 7
[2,1,3,2,2] => 8
[2,1,3,3,1] => 7
[2,1,3,4] => 6
[2,1,4,1,1,1] => 5
[2,1,4,1,2] => 6
[2,1,4,2,1] => 7
[2,1,4,3] => 6
[2,1,5,1,1] => 5
[2,1,5,2] => 6
[2,1,6,1] => 5
[2,1,7] => 4
[2,2,1,1,1,1,1,1] => 5
[2,2,1,1,1,1,2] => 6
[2,2,1,1,1,2,1] => 7
[2,2,1,1,1,3] => 6
[2,2,1,1,2,1,1] => 7
[2,2,1,1,2,2] => 8
[2,2,1,1,3,1] => 7
[2,2,1,1,4] => 6
[2,2,1,2,1,1,1] => 7
[2,2,1,2,1,2] => 8
[2,2,1,2,2,1] => 9
[2,2,1,2,3] => 8
[2,2,1,3,1,1] => 7
[2,2,1,3,2] => 8
[2,2,1,4,1] => 7
[2,2,1,5] => 6
[2,2,2,1,1,1,1] => 7
[2,2,2,1,1,2] => 8
[2,2,2,1,2,1] => 9
[2,2,2,1,3] => 8
[2,2,2,2,1,1] => 9
[2,2,2,2,2] => 10
[2,2,2,3,1] => 9
[2,2,2,4] => 8
[2,2,3,1,1,1] => 7
[2,2,3,1,2] => 8
[2,2,3,2,1] => 9
[2,2,3,3] => 8
[2,2,4,1,1] => 7
[2,2,4,2] => 8
[2,2,5,1] => 7
[2,2,6] => 6
[2,3,1,1,1,1,1] => 5
[2,3,1,1,1,2] => 6
[2,3,1,1,2,1] => 7
[2,3,1,1,3] => 6
[2,3,1,2,1,1] => 7
[2,3,1,2,2] => 8
[2,3,1,3,1] => 7
[2,3,1,4] => 6
[2,3,2,1,1,1] => 7
[2,3,2,1,2] => 8
[2,3,2,2,1] => 9
[2,3,2,3] => 8
[2,3,3,1,1] => 7
[2,3,3,2] => 8
[2,3,4,1] => 7
[2,3,5] => 6
[2,4,1,1,1,1] => 5
[2,4,1,1,2] => 6
[2,4,1,2,1] => 7
[2,4,1,3] => 6
[2,4,2,1,1] => 7
[2,4,2,2] => 8
[2,4,3,1] => 7
[2,4,4] => 6
[2,5,1,1,1] => 5
[2,5,1,2] => 6
[2,5,2,1] => 7
[2,5,3] => 6
[2,6,1,1] => 5
[2,6,2] => 6
[2,7,1] => 5
[2,8] => 4
[3,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,2] => 4
[3,1,1,1,1,2,1] => 5
[3,1,1,1,1,3] => 4
[3,1,1,1,2,1,1] => 5
[3,1,1,1,2,2] => 6
[3,1,1,1,3,1] => 5
[3,1,1,1,4] => 4
[3,1,1,2,1,1,1] => 5
[3,1,1,2,1,2] => 6
[3,1,1,2,2,1] => 7
[3,1,1,2,3] => 6
[3,1,1,3,1,1] => 5
[3,1,1,3,2] => 6
[3,1,1,4,1] => 5
[3,1,1,5] => 4
[3,1,2,1,1,1,1] => 5
[3,1,2,1,1,2] => 6
[3,1,2,1,2,1] => 7
[3,1,2,1,3] => 6
[3,1,2,2,1,1] => 7
[3,1,2,2,2] => 8
[3,1,2,3,1] => 7
[3,1,2,4] => 6
[3,1,3,1,1,1] => 5
[3,1,3,1,2] => 6
[3,1,3,2,1] => 7
[3,1,3,3] => 6
[3,1,4,1,1] => 5
[3,1,4,2] => 6
[3,1,5,1] => 5
[3,1,6] => 4
[3,2,1,1,1,1,1] => 5
[3,2,1,1,1,2] => 6
[3,2,1,1,2,1] => 7
[3,2,1,1,3] => 6
[3,2,1,2,1,1] => 7
[3,2,1,2,2] => 8
[3,2,1,3,1] => 7
[3,2,1,4] => 6
[3,2,2,1,1,1] => 7
[3,2,2,1,2] => 8
[3,2,2,2,1] => 9
[3,2,2,3] => 8
[3,2,3,1,1] => 7
[3,2,3,2] => 8
[3,2,4,1] => 7
[3,2,5] => 6
[3,3,1,1,1,1] => 5
[3,3,1,1,2] => 6
[3,3,1,2,1] => 7
[3,3,1,3] => 6
[3,3,2,1,1] => 7
[3,3,2,2] => 8
[3,3,3,1] => 7
[3,3,4] => 6
[3,4,1,1,1] => 5
[3,4,1,2] => 6
[3,4,2,1] => 7
[3,4,3] => 6
[3,5,1,1] => 5
[3,5,2] => 6
[3,6,1] => 5
[3,7] => 4
[4,1,1,1,1,1,1] => 3
[4,1,1,1,1,2] => 4
[4,1,1,1,2,1] => 5
[4,1,1,1,3] => 4
[4,1,1,2,1,1] => 5
[4,1,1,2,2] => 6
[4,1,1,3,1] => 5
[4,1,1,4] => 4
[4,1,2,1,1,1] => 5
[4,1,2,1,2] => 6
[4,1,2,2,1] => 7
[4,1,2,3] => 6
[4,1,3,1,1] => 5
[4,1,3,2] => 6
[4,1,4,1] => 5
[4,1,5] => 4
[4,2,1,1,1,1] => 5
[4,2,1,1,2] => 6
[4,2,1,2,1] => 7
[4,2,1,3] => 6
[4,2,2,1,1] => 7
[4,2,2,2] => 8
[4,2,3,1] => 7
[4,2,4] => 6
[4,3,1,1,1] => 5
[4,3,1,2] => 6
[4,3,2,1] => 7
[4,3,3] => 6
[4,4,1,1] => 5
[4,4,2] => 6
[4,5,1] => 5
[4,6] => 4
[5,1,1,1,1,1] => 3
[5,1,1,1,2] => 4
[5,1,1,2,1] => 5
[5,1,1,3] => 4
[5,1,2,1,1] => 5
[5,1,2,2] => 6
[5,1,3,1] => 5
[5,1,4] => 4
[5,2,1,1,1] => 5
[5,2,1,2] => 6
[5,2,2,1] => 7
[5,2,3] => 6
[5,3,1,1] => 5
[5,3,2] => 6
[5,4,1] => 5
[5,5] => 4
[6,1,1,1,1] => 3
[6,1,1,2] => 4
[6,1,2,1] => 5
[6,1,3] => 4
[6,2,1,1] => 5
[6,2,2] => 6
[6,3,1] => 5
[6,4] => 4
[7,1,1,1] => 3
[7,1,2] => 4
[7,2,1] => 5
[7,3] => 4
[8,1,1] => 3
[8,2] => 4
[9,1] => 3
[10] => 2
click to show generating function       
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Code
def statistic(c):
    if c.size() == 1:
        return 1
    p = c.to_skew_partition()
    return len(p.inner_corners()) + len(p.outer_corners())

Created
Oct 28, 2019 at 18:32 by Martin Rubey
Updated
Oct 28, 2019 at 18:32 by Martin Rubey