Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
St000013: Dyck paths ⟶ ℤ (values match St000442The maximal area to the right of an up step of a Dyck path.)
Values
[1] => [1,0,1,0] => 1
[2] => [1,1,0,0,1,0] => 2
[1,1] => [1,0,1,1,0,0] => 2
[3] => [1,1,1,0,0,0,1,0] => 3
[2,1] => [1,0,1,0,1,0] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => 3
[4] => [1,1,1,1,0,0,0,0,1,0] => 4
[3,1] => [1,1,0,1,0,0,1,0] => 2
[2,2] => [1,1,0,0,1,1,0,0] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[4,1] => [1,1,1,0,1,0,0,0,1,0] => 3
[3,2] => [1,1,0,0,1,0,1,0] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 3
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 4
[4,2] => [1,1,1,0,0,1,0,0,1,0] => 3
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 3
[3,3] => [1,1,1,0,0,0,1,1,0,0] => 3
[3,2,1] => [1,0,1,0,1,0,1,0] => 1
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 3
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 4
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 7
[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => 5
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => 4
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 4
[4,3] => [1,1,1,0,0,0,1,0,1,0] => 3
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 3
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 2
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 2
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 3
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 4
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 5
[1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 8
[7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => 6
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => 5
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 5
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => 4
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => 2
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 4
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => 2
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => 2
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 5
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => 4
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => 5
[2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6
[1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 8
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => 9
[8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => 7
[7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => 6
[7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => 6
[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => 5
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => 4
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => 5
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => 3
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => 3
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => 3
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 5
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => 2
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => 3
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => 3
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 4
[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 6
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 5
[2,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 6
[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => 7
[1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 9
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => 10
[9,1] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => 8
[8,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => 7
[8,1,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => 7
[7,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => 6
>>> Load all 829 entries. <<<[7,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => 5
[7,1,1,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => 6
[6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => 5
[6,3,1] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0] => 4
[6,2,2] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => 4
[6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0] => 4
[6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => 5
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 5
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => 3
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => 3
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => 3
[5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 5
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => 3
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 3
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 1
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 3
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 3
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => 3
[4,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,1,0,0,0] => 4
[4,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 6
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 3
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 3
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => 3
[3,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => 4
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
[3,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4
[3,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 5
[3,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => 7
[2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => 5
[2,2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 6
[2,2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => 7
[2,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => 8
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 10
[11] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0] => 11
[10,1] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0] => 9
[9,1,1] => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0] => 8
[8,3] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0] => 7
[8,1,1,1] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0] => 7
[7,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => 6
[7,2,2] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => 5
[7,1,1,1,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0] => 6
[6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 5
[6,4,1] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0] => 4
[6,3,2] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0] => 4
[6,3,1,1] => [1,1,1,0,1,1,0,0,1,0,0,0,1,0] => 4
[6,2,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0] => 4
[6,2,1,1,1] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0] => 4
[6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 5
[5,5,1] => [1,1,1,1,0,1,0,0,0,0,1,1,0,0] => 4
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => 3
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => 3
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => 2
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => 3
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => 3
[5,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 4
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => 2
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => 2
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => 2
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 2
[4,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0] => 4
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 3
[4,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0] => 4
[4,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 5
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => 3
[3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 4
[3,2,2,2,2] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0] => 4
[3,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0] => 4
[3,2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => 6
[3,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => 8
[2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
[12] => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0] => 12
[10,2] => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0] => 9
[8,2,2] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0] => 6
[7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => 6
[7,2,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0] => 5
[7,1,1,1,1,1] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
[6,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 6
[6,5,1] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0] => 4
[6,4,2] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0] => 4
[6,4,1,1] => [1,1,1,0,1,1,0,0,0,1,0,0,1,0] => 4
[6,3,3] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0] => 4
[6,3,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0] => 3
[6,3,1,1,1] => [1,1,0,1,1,1,0,0,1,0,0,0,1,0] => 4
[6,2,2,2] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0] => 4
[6,2,2,1,1] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0] => 4
[6,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0,1,0] => 4
[5,5,2] => [1,1,1,1,0,0,1,0,0,0,1,1,0,0] => 4
[5,5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => 4
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => 2
[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => 2
[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => 2
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => 2
[5,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0] => 4
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[5,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 4
[4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 4
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => 2
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => 2
[4,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => 4
[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => 2
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => 2
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
[4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3
[4,3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 5
[4,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0] => 4
[4,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => 4
[3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[3,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => 4
[3,3,2,2,2] => [1,1,0,0,1,1,1,0,1,1,0,0,0,0] => 4
[3,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0] => 4
[3,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 4
[3,2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => 7
[2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[13] => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] => 13
[8,3,2] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0] => 6
[7,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => 6
[7,3,3] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0] => 5
[7,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
[6,6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0] => 5
[6,5,2] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0] => 4
[6,5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0,1,0] => 4
[6,4,3] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0] => 4
[6,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0] => 3
[6,4,1,1,1] => [1,1,0,1,1,1,0,0,0,1,0,0,1,0] => 4
[6,3,3,1] => [1,1,1,0,1,0,0,1,1,0,0,0,1,0] => 3
[6,3,2,2] => [1,1,1,0,0,1,1,0,1,0,0,0,1,0] => 3
[6,3,2,1,1] => [1,1,0,1,1,0,1,0,1,0,0,0,1,0] => 3
[6,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0] => 4
[6,2,2,2,1] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0] => 4
[6,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0,1,0] => 4
[5,5,3] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 4
[5,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,1,0,0] => 3
[5,5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0] => 4
[5,4,4] => [1,1,1,1,0,0,0,0,1,1,0,1,0,0] => 4
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => 2
[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => 2
[5,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 4
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => 2
[5,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 3
[5,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0] => 4
[5,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0] => 4
[4,4,4,1] => [1,1,1,0,1,0,0,0,1,1,1,0,0,0] => 3
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2
[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[4,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0] => 3
[4,3,3,3] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0] => 3
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[4,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0] => 3
[4,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0] => 3
[4,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => 3
[4,3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 4
[4,3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => 6
[4,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0] => 4
[4,2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 5
[3,3,3,3,1] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,2,2] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0] => 4
[3,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0] => 4
[3,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 4
[3,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => 5
[2,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[14] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] => 14
[8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0] => 7
[8,1,1,1,1,1,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 7
[7,7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => 7
[7,5,2] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0] => 5
[7,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0] => 5
[6,6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0] => 5
[6,6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0] => 5
[6,5,3] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0] => 4
[6,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0] => 3
[6,5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0] => 4
[6,4,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 4
[6,4,3,1] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0] => 3
[6,4,2,2] => [1,1,1,0,0,1,1,0,0,1,0,0,1,0] => 3
[6,4,2,1,1] => [1,1,0,1,1,0,1,0,0,1,0,0,1,0] => 3
[6,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0] => 4
[6,3,3,2] => [1,1,1,0,0,1,0,1,1,0,0,0,1,0] => 3
[6,3,3,1,1] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0] => 3
[6,3,2,2,1] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0] => 3
[6,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0] => 3
[6,3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 5
[6,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 4
[6,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0,1,0] => 4
[5,5,4] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 4
[5,5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,1,0,0] => 3
[5,5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0] => 3
[5,5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,1,0,0] => 3
[5,5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
[5,4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,1,0,0] => 3
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 3
[5,4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0] => 5
[5,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => 3
[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
[5,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => 3
[5,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,1,0,0] => 3
[5,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0] => 3
[5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 4
[4,4,4,2] => [1,1,1,0,0,1,0,0,1,1,1,0,0,0] => 3
[4,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,4,3,3] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => 3
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[4,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0] => 3
[4,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,1,0,0,0] => 3
[4,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0] => 3
[4,3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0] => 3
[4,3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,1,0,0,0] => 3
[4,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0] => 3
[4,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 3
[4,3,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => 5
[4,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0] => 5
[3,3,3,3,2] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0] => 4
[3,3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 5
[3,3,2,2,2,2] => [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => 5
[3,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 5
[2,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
[15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] => 15
[8,7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => 7
[8,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 7
[7,7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0] => 6
[7,6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0] => 5
[7,2,2,2,2] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0] => 5
[6,6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0] => 5
[6,6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0] => 4
[6,6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0] => 5
[6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 4
[6,5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0] => 3
[6,5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0,1,0] => 3
[6,5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0,1,0] => 3
[6,5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 4
[6,4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0,1,0] => 3
[6,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0] => 3
[6,4,3,1,1] => [1,1,0,1,1,0,0,1,0,1,0,0,1,0] => 3
[6,4,2,2,1] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0] => 3
[6,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0,1,0] => 3
[6,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 3
[6,3,3,2,1] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0] => 3
[6,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => 3
[6,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0,1,0] => 3
[6,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0,1,0] => 3
[6,3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 4
[6,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 4
[6,2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 5
[5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => 5
[5,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,1,0,0] => 3
[5,5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,1,0,0] => 3
[5,5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,1,0,0] => 3
[5,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0] => 3
[5,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0] => 3
[5,4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,1,0,0] => 3
[5,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,1,0,0] => 3
[5,4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,1,0,0] => 3
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
[5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3
[5,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0] => 3
[5,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0] => 3
[5,4,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0] => 4
[5,3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0] => 3
[5,3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0] => 3
[5,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => 3
[5,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 3
[5,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0] => 5
[4,4,4,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 3
[4,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0] => 3
[4,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0] => 3
[4,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,1,0,0,0] => 3
[4,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0] => 3
[4,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0] => 3
[4,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,1,0,0,0] => 3
[4,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,1,0,0,0] => 3
[4,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0] => 3
[4,3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 4
[4,3,2,2,2,2] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0] => 4
[4,3,2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => 6
[4,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 5
[3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,2,2,2] => [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0] => 5
[3,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6
[2,2,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
[12,4] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] => 11
[10,6] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0] => 9
[8,8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => 8
[7,7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0] => 6
[7,7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,1,0,0] => 6
[7,6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0] => 5
[7,5,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0] => 5
[7,5,3,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0] => 4
[7,5,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0] => 4
[7,4,2,1,1,1] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0] => 4
[7,3,2,2,1,1] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0] => 4
[7,2,2,2,2,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0] => 5
[6,6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0] => 4
[6,6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0] => 5
[6,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0] => 5
[6,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0] => 3
[6,5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0] => 3
[6,5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0] => 3
[6,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0] => 3
[6,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0] => 3
[6,5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 5
[6,4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0,1,0] => 3
[6,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0,1,0] => 3
[6,4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0,1,0] => 3
[6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => 2
[6,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0,1,0] => 3
[6,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0,1,0] => 3
[6,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0] => 3
[6,4,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0] => 4
[6,3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0] => 3
[6,3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0,1,0] => 3
[6,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0,1,0] => 3
[6,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => 3
[6,3,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4
[6,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0] => 5
[5,5,5,1] => [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0] => 4
[5,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,1,0,0] => 3
[5,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,1,0,0] => 3
[5,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 3
[5,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0] => 2
[5,5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0] => 3
[5,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 3
[5,5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0] => 3
[5,4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,1,0,0] => 3
[5,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0] => 2
[5,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0] => 3
[5,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0] => 2
[5,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => 2
[5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2
[5,4,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0] => 4
[5,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 3
[5,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0] => 3
[5,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0] => 3
[5,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => 3
[5,3,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0] => 4
[4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => 4
[4,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0] => 3
[4,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 3
[4,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,1,0,0,0] => 3
[4,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,1,0,0,0] => 3
[4,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0] => 3
[4,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0] => 3
[4,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0] => 4
[4,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 3
[4,3,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0] => 4
[4,3,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 4
[4,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 6
[3,3,3,3,3,1] => [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,3,3,3,2,2] => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0] => 5
[3,3,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 6
[2,2,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 8
[9,8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => 8
[9,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 8
[8,8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0] => 7
[8,7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,1,0] => 6
[7,7,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,1,0,0] => 5
[7,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0] => 5
[7,4,3,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => 3
[7,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => 5
[6,6,3,1,1] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0] => 4
[6,6,2,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0] => 4
[6,6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 5
[6,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => 3
[6,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0] => 3
[6,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 3
[6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0] => 2
[6,5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0] => 3
[6,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 3
[6,5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0] => 3
[6,4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 3
[6,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0] => 2
[6,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 3
[6,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0] => 2
[6,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0,1,0] => 2
[6,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0] => 2
[6,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0] => 3
[6,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 3
[6,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 3
[6,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0] => 3
[6,3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 4
[6,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 5
[5,5,5,2] => [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0] => 4
[5,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 3
[5,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0] => 2
[5,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 3
[5,5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0] => 2
[5,5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,1,0,0] => 2
[5,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0] => 2
[5,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
[5,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0] => 4
[5,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0] => 2
[5,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0] => 2
[5,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0] => 2
[5,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,1,0,0] => 2
[5,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0] => 2
[5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 2
[5,4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 3
[5,4,3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => 5
[5,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 3
[5,3,3,2,2,2] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0] => 4
[4,4,4,4,1] => [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[4,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 3
[4,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 3
[4,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 3
[4,4,3,3,3] => [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0] => 4
[4,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0] => 3
[4,3,3,3,3,1] => [1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0] => 4
[4,3,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4
[3,3,3,3,3,2] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,3,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,2,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => 7
[2,2,2,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 8
[] => [] => 0
[6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
[5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
[5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[6,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 2
[5,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[6,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0] => 2
[5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 2
[6,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 3
[6,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 2
[5,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
[6,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 2
[5,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0] => 2
[6,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0] => 2
[5,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0] => 2
[6,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0] => 2
[6,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 3
[6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 2
[5,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
[6,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 2
[5,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0] => 2
[6,5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 2
[5,5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[6,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0] => 2
[6,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0] => 2
[5,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0] => 2
[6,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0,1,0] => 2
[6,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0] => 2
[6,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 3
[6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 2
[5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 2
[5,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0] => 2
[6,5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 2
[5,5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 2
[6,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0] => 2
[6,5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 2
[5,5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 2
[6,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 2
[6,5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0] => 2
[6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => 2
[5,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,1,0,0] => 2
[6,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0] => 2
[6,5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0] => 2
[6,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0] => 2
[6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 3
[7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
[6,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[7,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
[6,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[5,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[7,6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 2
[6,6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[7,5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0] => 2
[6,5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 2
[5,5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0] => 3
[7,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 3
[6,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 3
[5,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 3
[4,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[6,6,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
[6,5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 2
[6,5,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 3
[6,4,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 3
[6,3,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 4
[3,3,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
[6,6,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
[6,5,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0] => 2
[6,5,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0] => 2
[6,5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2
[5,5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0] => 3
[7,6,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 3
[6,5,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3
[6,5,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 3
[6,4,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 3
[6,5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 4
[7,2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 5
[7,6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 2
[6,6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,5,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0] => 2
[6,6,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0] => 2
[6,5,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => 2
[4,4,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0] => 4
[6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[5,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0] => 3
[7,6,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 3
[6,5,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0] => 3
[6,5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 3
[7,6,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0] => 3
[5,5,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0] => 3
[7,5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0] => 3
[6,5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 3
[7,6,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0] => 3
[7,4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0] => 3
[6,4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 3
[6,5,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 4
[6,6,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0] => 4
[7,6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 5
[7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 2
[6,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[7,5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
[6,5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[5,5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[7,6,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => 2
[6,5,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0] => 2
[6,5,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0] => 2
[5,4,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0] => 3
[4,3,3,3,3,2] => [1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0] => 4
[6,5,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2
[4,4,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0] => 4
[7,6,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => 3
[6,5,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3
[5,4,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3
[6,6,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
[7,6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => 2
[6,6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,5,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => 2
[4,4,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0] => 4
[7,6,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => 2
[5,5,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0] => 3
[4,4,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0] => 4
[7,6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => 2
[6,6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0] => 2
[5,5,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0] => 3
[7,6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => 2
[6,6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => 2
[7,5,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => 2
[7,6,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0] => 3
[7,6,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0] => 3
[7,6,3,2,1,1] => [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0] => 3
[7,4,3,2,1,1] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0] => 3
[7,6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 3
[6,6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => 3
[6,5,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0] => 3
[5,4,4,4,3] => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0] => 3
[5,4,3,3,3] => [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0] => 3
[7,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => 4
[7,6,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0] => 3
[4,4,4,4,2] => [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0] => 4
[7,6,5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0] => 3
[7,6,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0] => 3
[7,5,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0] => 3
[7,6,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0] => 3
[7,5,4,3,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0] => 3
[7,6,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0] => 3
[6,6,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0] => 3
[7,6,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0] => 3
[6,6,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0] => 3
[7,5,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0] => 3
[7,6,3,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0] => 3
[6,6,3,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0] => 3
[7,6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => 4
[6,5,5,4] => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0] => 4
[7,6,5,3] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0] => 4
[7,6,4,3] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0] => 4
[7,5,4,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0] => 4
[7,6,5,2] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0] => 4
[7,6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => 5
[2,2,2,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 9
[3,3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[3,3,3,3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 8
[3,3,3,3,3,3,1] => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[3,3,3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
[3,3,3,3,3,2,2] => [1,1,0,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 6
[4,3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => 5
[4,4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,4,4,3,3,3] => [1,1,1,0,0,0,1,1,1,0,1,1,1,0,0,0,0,0] => 5
[4,4,4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[5,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => 5
[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] => 16
[6,6,6,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => 6
[6,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 6
[6,6,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => 6
[7,6,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0] => 6
[10,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0] => 8
[10,10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0] => 10
[8,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
[10,10,10,10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => 10
[8,7,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => 6
[9,8,7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0] => 7
[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] => 24
[5,5,5,5,5,5,5,5,5,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0] => 12
[15,15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0] => 15
[5,5,5,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[15,5,5] => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => 13
[7,7,7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0] => 7
[9,7,5,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => 5
[11,9,7,5,3,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => 6
[8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
[7,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[7,6,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[7,6,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 2
[7,6,5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 2
[7,6,5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2
[7,6,5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[7,6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[9,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
[8,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,6,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[8,7,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[7,5,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 3
[8,7,6,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 2
[7,6,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 3
[8,7,6,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 2
[7,6,5,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3
[8,7,6,5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2
[7,6,5,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 3
[8,7,6,5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[7,6,5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => 3
[8,7,6,5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[8,7,6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => 2
[8,7,5,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => 2
[8,6,5,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => 2
[9,8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
[8,7,6,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 3
[9,7,6,5,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => 2
[5,5,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[9,9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0] => 9
[7,7,1,1,1,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 6
[8,5,4,3,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => 3
[8,7,4,3,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0] => 3
[7,7,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 6
[5,4,3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => 4
[6,5,4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => 3
[5,4,3,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => 5
[6,5,4,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0] => 4
[6,5,4,3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => 4
[7,6,5,4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => 3
[8,6,5,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => 3
[7,5,4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => 3
[7,6,4,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 3
[7,6,5,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0] => 2
[7,6,6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[7,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,6,6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0] => 3
[5,5,5,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0] => 4
[6,5,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 3
[5,4,4,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0] => 4
[5,4,3,3,3,3,1] => [1,1,0,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0] => 4
[5,4,3,2,2,2,2] => [1,1,0,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 4
[6,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,1,0,0,0] => 5
[5,4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0] => 4
[8,7,6,3,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0] => 3
[8,7,6,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0] => 3
[8,7,6,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 3
[7,6,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[9,8,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[8,7,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[7,6,6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[7,7,6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[7,7,6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[8,8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[9,9,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[8,7,6,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 2
[9,8,7,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 2
[9,8,6,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0] => 2
[9,7,6,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0] => 2
[7,7,6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0] => 2
[8,7,6,5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 2
[8,7,6,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 3
[9,8,7,6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 3
[10,9,8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 2
[8,6,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
[8,7,6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => 5
[8,7,6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => 4
[8,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0] => 6
[9,8,7,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0] => 6
[8,7,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0] => 5
[9,8,7,6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => 5
[8,7,6,5,3] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0] => 4
[9,8,7,6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 4
[8,7,6,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => 3
[4,4,4,4,4,1] => [1,1,1,0,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[5,5,5,5,1] => [1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0] => 4
[6,6,6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,1,0,0,0] => 5
[7,7,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[7,7,5,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => 2
[6,6,6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,1,0,0,0] => 5
[7,7,4,3,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0] => 3
[6,5,4,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 3
[8,5,4,3,2,1,1] => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => 3
[6,6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => 3
[7,6,5,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,0,2 3,0,2 3,2,0,2 1,0,6,2,0,2 4,3,4,2,0,2 6,2,6,4,2,0,2 4,9,5,4,4,2,0,2 1,0,15,6,8,4,4,2,0,2
$F_{1} = q$
$F_{2} = 2\ q^{2}$
$F_{3} = q + 2\ q^{3}$
$F_{4} = 3\ q^{2} + 2\ q^{4}$
$F_{5} = 3\ q^{2} + 2\ q^{3} + 2\ q^{5}$
$F_{6} = q + 6\ q^{3} + 2\ q^{4} + 2\ q^{6}$
$F_{7} = 4\ q^{2} + 3\ q^{3} + 4\ q^{4} + 2\ q^{5} + 2\ q^{7}$
$F_{8} = 6\ q^{2} + 2\ q^{3} + 6\ q^{4} + 4\ q^{5} + 2\ q^{6} + 2\ q^{8}$
$F_{9} = 4\ q^{2} + 9\ q^{3} + 5\ q^{4} + 4\ q^{5} + 4\ q^{6} + 2\ q^{7} + 2\ q^{9}$
$F_{10} = q + 15\ q^{3} + 6\ q^{4} + 8\ q^{5} + 4\ q^{6} + 4\ q^{7} + 2\ q^{8} + 2\ q^{10}$
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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