Identifier
-
Mp00040:
Integer compositions
—to partition⟶
Integer partitions
St000318: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)
Values
[1] => [1] => 2
[1,1] => [1,1] => 2
[2] => [2] => 2
[1,1,1] => [1,1,1] => 2
[1,2] => [2,1] => 3
[2,1] => [2,1] => 3
[3] => [3] => 2
[1,1,1,1] => [1,1,1,1] => 2
[1,1,2] => [2,1,1] => 3
[1,2,1] => [2,1,1] => 3
[1,3] => [3,1] => 3
[2,1,1] => [2,1,1] => 3
[2,2] => [2,2] => 2
[3,1] => [3,1] => 3
[4] => [4] => 2
[1,1,1,1,1] => [1,1,1,1,1] => 2
[1,1,1,2] => [2,1,1,1] => 3
[1,1,2,1] => [2,1,1,1] => 3
[1,1,3] => [3,1,1] => 3
[1,2,1,1] => [2,1,1,1] => 3
[1,2,2] => [2,2,1] => 3
[1,3,1] => [3,1,1] => 3
[1,4] => [4,1] => 3
[2,1,1,1] => [2,1,1,1] => 3
[2,1,2] => [2,2,1] => 3
[2,2,1] => [2,2,1] => 3
[2,3] => [3,2] => 3
[3,1,1] => [3,1,1] => 3
[3,2] => [3,2] => 3
[4,1] => [4,1] => 3
[5] => [5] => 2
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 2
[1,1,1,1,2] => [2,1,1,1,1] => 3
[1,1,1,2,1] => [2,1,1,1,1] => 3
[1,1,1,3] => [3,1,1,1] => 3
[1,1,2,1,1] => [2,1,1,1,1] => 3
[1,1,2,2] => [2,2,1,1] => 3
[1,1,3,1] => [3,1,1,1] => 3
[1,1,4] => [4,1,1] => 3
[1,2,1,1,1] => [2,1,1,1,1] => 3
[1,2,1,2] => [2,2,1,1] => 3
[1,2,2,1] => [2,2,1,1] => 3
[1,2,3] => [3,2,1] => 4
[1,3,1,1] => [3,1,1,1] => 3
[1,3,2] => [3,2,1] => 4
[1,4,1] => [4,1,1] => 3
[1,5] => [5,1] => 3
[2,1,1,1,1] => [2,1,1,1,1] => 3
[2,1,1,2] => [2,2,1,1] => 3
[2,1,2,1] => [2,2,1,1] => 3
[2,1,3] => [3,2,1] => 4
[2,2,1,1] => [2,2,1,1] => 3
[2,2,2] => [2,2,2] => 2
[2,3,1] => [3,2,1] => 4
[2,4] => [4,2] => 3
[3,1,1,1] => [3,1,1,1] => 3
[3,1,2] => [3,2,1] => 4
[3,2,1] => [3,2,1] => 4
[3,3] => [3,3] => 2
[4,1,1] => [4,1,1] => 3
[4,2] => [4,2] => 3
[5,1] => [5,1] => 3
[6] => [6] => 2
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 2
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 3
[1,1,1,1,2,1] => [2,1,1,1,1,1] => 3
[1,1,1,1,3] => [3,1,1,1,1] => 3
[1,1,1,2,1,1] => [2,1,1,1,1,1] => 3
[1,1,1,2,2] => [2,2,1,1,1] => 3
[1,1,1,3,1] => [3,1,1,1,1] => 3
[1,1,1,4] => [4,1,1,1] => 3
[1,1,2,1,1,1] => [2,1,1,1,1,1] => 3
[1,1,2,1,2] => [2,2,1,1,1] => 3
[1,1,2,2,1] => [2,2,1,1,1] => 3
[1,1,2,3] => [3,2,1,1] => 4
[1,1,3,1,1] => [3,1,1,1,1] => 3
[1,1,3,2] => [3,2,1,1] => 4
[1,1,4,1] => [4,1,1,1] => 3
[1,1,5] => [5,1,1] => 3
[1,2,1,1,1,1] => [2,1,1,1,1,1] => 3
[1,2,1,1,2] => [2,2,1,1,1] => 3
[1,2,1,2,1] => [2,2,1,1,1] => 3
[1,2,1,3] => [3,2,1,1] => 4
[1,2,2,1,1] => [2,2,1,1,1] => 3
[1,2,2,2] => [2,2,2,1] => 3
[1,2,3,1] => [3,2,1,1] => 4
[1,2,4] => [4,2,1] => 4
[1,3,1,1,1] => [3,1,1,1,1] => 3
[1,3,1,2] => [3,2,1,1] => 4
[1,3,2,1] => [3,2,1,1] => 4
[1,3,3] => [3,3,1] => 3
[1,4,1,1] => [4,1,1,1] => 3
[1,4,2] => [4,2,1] => 4
[1,5,1] => [5,1,1] => 3
[1,6] => [6,1] => 3
[2,1,1,1,1,1] => [2,1,1,1,1,1] => 3
[2,1,1,1,2] => [2,2,1,1,1] => 3
[2,1,1,2,1] => [2,2,1,1,1] => 3
[2,1,1,3] => [3,2,1,1] => 4
[2,1,2,1,1] => [2,2,1,1,1] => 3
[2,1,2,2] => [2,2,2,1] => 3
>>> Load all 1192 entries. <<<[2,1,3,1] => [3,2,1,1] => 4
[2,1,4] => [4,2,1] => 4
[2,2,1,1,1] => [2,2,1,1,1] => 3
[2,2,1,2] => [2,2,2,1] => 3
[2,2,2,1] => [2,2,2,1] => 3
[2,2,3] => [3,2,2] => 3
[2,3,1,1] => [3,2,1,1] => 4
[2,3,2] => [3,2,2] => 3
[2,4,1] => [4,2,1] => 4
[2,5] => [5,2] => 3
[3,1,1,1,1] => [3,1,1,1,1] => 3
[3,1,1,2] => [3,2,1,1] => 4
[3,1,2,1] => [3,2,1,1] => 4
[3,1,3] => [3,3,1] => 3
[3,2,1,1] => [3,2,1,1] => 4
[3,2,2] => [3,2,2] => 3
[3,3,1] => [3,3,1] => 3
[3,4] => [4,3] => 3
[4,1,1,1] => [4,1,1,1] => 3
[4,1,2] => [4,2,1] => 4
[4,2,1] => [4,2,1] => 4
[4,3] => [4,3] => 3
[5,1,1] => [5,1,1] => 3
[5,2] => [5,2] => 3
[6,1] => [6,1] => 3
[7] => [7] => 2
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1] => 3
[1,1,1,1,1,3] => [3,1,1,1,1,1] => 3
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1] => 3
[1,1,1,1,2,2] => [2,2,1,1,1,1] => 3
[1,1,1,1,3,1] => [3,1,1,1,1,1] => 3
[1,1,1,1,4] => [4,1,1,1,1] => 3
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1] => 3
[1,1,1,2,1,2] => [2,2,1,1,1,1] => 3
[1,1,1,2,2,1] => [2,2,1,1,1,1] => 3
[1,1,1,2,3] => [3,2,1,1,1] => 4
[1,1,1,3,1,1] => [3,1,1,1,1,1] => 3
[1,1,1,3,2] => [3,2,1,1,1] => 4
[1,1,1,4,1] => [4,1,1,1,1] => 3
[1,1,1,5] => [5,1,1,1] => 3
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1] => 3
[1,1,2,1,1,2] => [2,2,1,1,1,1] => 3
[1,1,2,1,2,1] => [2,2,1,1,1,1] => 3
[1,1,2,1,3] => [3,2,1,1,1] => 4
[1,1,2,2,1,1] => [2,2,1,1,1,1] => 3
[1,1,2,2,2] => [2,2,2,1,1] => 3
[1,1,2,3,1] => [3,2,1,1,1] => 4
[1,1,2,4] => [4,2,1,1] => 4
[1,1,3,1,1,1] => [3,1,1,1,1,1] => 3
[1,1,3,1,2] => [3,2,1,1,1] => 4
[1,1,3,2,1] => [3,2,1,1,1] => 4
[1,1,3,3] => [3,3,1,1] => 3
[1,1,4,1,1] => [4,1,1,1,1] => 3
[1,1,4,2] => [4,2,1,1] => 4
[1,1,5,1] => [5,1,1,1] => 3
[1,1,6] => [6,1,1] => 3
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1] => 3
[1,2,1,1,1,2] => [2,2,1,1,1,1] => 3
[1,2,1,1,2,1] => [2,2,1,1,1,1] => 3
[1,2,1,1,3] => [3,2,1,1,1] => 4
[1,2,1,2,1,1] => [2,2,1,1,1,1] => 3
[1,2,1,2,2] => [2,2,2,1,1] => 3
[1,2,1,3,1] => [3,2,1,1,1] => 4
[1,2,1,4] => [4,2,1,1] => 4
[1,2,2,1,1,1] => [2,2,1,1,1,1] => 3
[1,2,2,1,2] => [2,2,2,1,1] => 3
[1,2,2,2,1] => [2,2,2,1,1] => 3
[1,2,2,3] => [3,2,2,1] => 4
[1,2,3,1,1] => [3,2,1,1,1] => 4
[1,2,3,2] => [3,2,2,1] => 4
[1,2,4,1] => [4,2,1,1] => 4
[1,2,5] => [5,2,1] => 4
[1,3,1,1,1,1] => [3,1,1,1,1,1] => 3
[1,3,1,1,2] => [3,2,1,1,1] => 4
[1,3,1,2,1] => [3,2,1,1,1] => 4
[1,3,1,3] => [3,3,1,1] => 3
[1,3,2,1,1] => [3,2,1,1,1] => 4
[1,3,2,2] => [3,2,2,1] => 4
[1,3,3,1] => [3,3,1,1] => 3
[1,3,4] => [4,3,1] => 4
[1,4,1,1,1] => [4,1,1,1,1] => 3
[1,4,1,2] => [4,2,1,1] => 4
[1,4,2,1] => [4,2,1,1] => 4
[1,4,3] => [4,3,1] => 4
[1,5,1,1] => [5,1,1,1] => 3
[1,5,2] => [5,2,1] => 4
[1,6,1] => [6,1,1] => 3
[1,7] => [7,1] => 3
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 3
[2,1,1,1,1,2] => [2,2,1,1,1,1] => 3
[2,1,1,1,2,1] => [2,2,1,1,1,1] => 3
[2,1,1,1,3] => [3,2,1,1,1] => 4
[2,1,1,2,1,1] => [2,2,1,1,1,1] => 3
[2,1,1,2,2] => [2,2,2,1,1] => 3
[2,1,1,3,1] => [3,2,1,1,1] => 4
[2,1,1,4] => [4,2,1,1] => 4
[2,1,2,1,1,1] => [2,2,1,1,1,1] => 3
[2,1,2,1,2] => [2,2,2,1,1] => 3
[2,1,2,2,1] => [2,2,2,1,1] => 3
[2,1,2,3] => [3,2,2,1] => 4
[2,1,3,1,1] => [3,2,1,1,1] => 4
[2,1,3,2] => [3,2,2,1] => 4
[2,1,4,1] => [4,2,1,1] => 4
[2,1,5] => [5,2,1] => 4
[2,2,1,1,1,1] => [2,2,1,1,1,1] => 3
[2,2,1,1,2] => [2,2,2,1,1] => 3
[2,2,1,2,1] => [2,2,2,1,1] => 3
[2,2,1,3] => [3,2,2,1] => 4
[2,2,2,1,1] => [2,2,2,1,1] => 3
[2,2,2,2] => [2,2,2,2] => 2
[2,2,3,1] => [3,2,2,1] => 4
[2,2,4] => [4,2,2] => 3
[2,3,1,1,1] => [3,2,1,1,1] => 4
[2,3,1,2] => [3,2,2,1] => 4
[2,3,2,1] => [3,2,2,1] => 4
[2,3,3] => [3,3,2] => 3
[2,4,1,1] => [4,2,1,1] => 4
[2,4,2] => [4,2,2] => 3
[2,5,1] => [5,2,1] => 4
[2,6] => [6,2] => 3
[3,1,1,1,1,1] => [3,1,1,1,1,1] => 3
[3,1,1,1,2] => [3,2,1,1,1] => 4
[3,1,1,2,1] => [3,2,1,1,1] => 4
[3,1,1,3] => [3,3,1,1] => 3
[3,1,2,1,1] => [3,2,1,1,1] => 4
[3,1,2,2] => [3,2,2,1] => 4
[3,1,3,1] => [3,3,1,1] => 3
[3,1,4] => [4,3,1] => 4
[3,2,1,1,1] => [3,2,1,1,1] => 4
[3,2,1,2] => [3,2,2,1] => 4
[3,2,2,1] => [3,2,2,1] => 4
[3,2,3] => [3,3,2] => 3
[3,3,1,1] => [3,3,1,1] => 3
[3,3,2] => [3,3,2] => 3
[3,4,1] => [4,3,1] => 4
[3,5] => [5,3] => 3
[4,1,1,1,1] => [4,1,1,1,1] => 3
[4,1,1,2] => [4,2,1,1] => 4
[4,1,2,1] => [4,2,1,1] => 4
[4,1,3] => [4,3,1] => 4
[4,2,1,1] => [4,2,1,1] => 4
[4,2,2] => [4,2,2] => 3
[4,3,1] => [4,3,1] => 4
[4,4] => [4,4] => 2
[5,1,1,1] => [5,1,1,1] => 3
[5,1,2] => [5,2,1] => 4
[5,2,1] => [5,2,1] => 4
[5,3] => [5,3] => 3
[6,1,1] => [6,1,1] => 3
[6,2] => [6,2] => 3
[7,1] => [7,1] => 3
[8] => [8] => 2
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => 3
[1,1,1,1,1,3,1] => [3,1,1,1,1,1,1] => 3
[1,1,1,1,1,4] => [4,1,1,1,1,1] => 3
[1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1] => 3
[1,1,1,1,2,1,2] => [2,2,1,1,1,1,1] => 3
[1,1,1,1,2,2,1] => [2,2,1,1,1,1,1] => 3
[1,1,1,1,2,3] => [3,2,1,1,1,1] => 4
[1,1,1,1,3,1,1] => [3,1,1,1,1,1,1] => 3
[1,1,1,1,3,2] => [3,2,1,1,1,1] => 4
[1,1,1,1,4,1] => [4,1,1,1,1,1] => 3
[1,1,1,1,5] => [5,1,1,1,1] => 3
[1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1] => 3
[1,1,1,2,1,1,2] => [2,2,1,1,1,1,1] => 3
[1,1,1,2,1,2,1] => [2,2,1,1,1,1,1] => 3
[1,1,1,2,1,3] => [3,2,1,1,1,1] => 4
[1,1,1,2,2,1,1] => [2,2,1,1,1,1,1] => 3
[1,1,1,2,2,2] => [2,2,2,1,1,1] => 3
[1,1,1,2,3,1] => [3,2,1,1,1,1] => 4
[1,1,1,2,4] => [4,2,1,1,1] => 4
[1,1,1,3,1,1,1] => [3,1,1,1,1,1,1] => 3
[1,1,1,3,1,2] => [3,2,1,1,1,1] => 4
[1,1,1,3,2,1] => [3,2,1,1,1,1] => 4
[1,1,1,3,3] => [3,3,1,1,1] => 3
[1,1,1,4,1,1] => [4,1,1,1,1,1] => 3
[1,1,1,4,2] => [4,2,1,1,1] => 4
[1,1,1,5,1] => [5,1,1,1,1] => 3
[1,1,1,6] => [6,1,1,1] => 3
[1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 3
[1,1,2,1,1,1,2] => [2,2,1,1,1,1,1] => 3
[1,1,2,1,1,2,1] => [2,2,1,1,1,1,1] => 3
[1,1,2,1,1,3] => [3,2,1,1,1,1] => 4
[1,1,2,1,2,1,1] => [2,2,1,1,1,1,1] => 3
[1,1,2,1,2,2] => [2,2,2,1,1,1] => 3
[1,1,2,1,3,1] => [3,2,1,1,1,1] => 4
[1,1,2,1,4] => [4,2,1,1,1] => 4
[1,1,2,2,1,1,1] => [2,2,1,1,1,1,1] => 3
[1,1,2,2,1,2] => [2,2,2,1,1,1] => 3
[1,1,2,2,2,1] => [2,2,2,1,1,1] => 3
[1,1,2,2,3] => [3,2,2,1,1] => 4
[1,1,2,3,1,1] => [3,2,1,1,1,1] => 4
[1,1,2,3,2] => [3,2,2,1,1] => 4
[1,1,2,4,1] => [4,2,1,1,1] => 4
[1,1,2,5] => [5,2,1,1] => 4
[1,1,3,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[1,1,3,1,1,2] => [3,2,1,1,1,1] => 4
[1,1,3,1,2,1] => [3,2,1,1,1,1] => 4
[1,1,3,1,3] => [3,3,1,1,1] => 3
[1,1,3,2,1,1] => [3,2,1,1,1,1] => 4
[1,1,3,2,2] => [3,2,2,1,1] => 4
[1,1,3,3,1] => [3,3,1,1,1] => 3
[1,1,3,4] => [4,3,1,1] => 4
[1,1,4,1,1,1] => [4,1,1,1,1,1] => 3
[1,1,4,1,2] => [4,2,1,1,1] => 4
[1,1,4,2,1] => [4,2,1,1,1] => 4
[1,1,4,3] => [4,3,1,1] => 4
[1,1,5,1,1] => [5,1,1,1,1] => 3
[1,1,5,2] => [5,2,1,1] => 4
[1,1,6,1] => [6,1,1,1] => 3
[1,1,7] => [7,1,1] => 3
[1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 3
[1,2,1,1,1,1,2] => [2,2,1,1,1,1,1] => 3
[1,2,1,1,1,2,1] => [2,2,1,1,1,1,1] => 3
[1,2,1,1,1,3] => [3,2,1,1,1,1] => 4
[1,2,1,1,2,1,1] => [2,2,1,1,1,1,1] => 3
[1,2,1,1,2,2] => [2,2,2,1,1,1] => 3
[1,2,1,1,3,1] => [3,2,1,1,1,1] => 4
[1,2,1,1,4] => [4,2,1,1,1] => 4
[1,2,1,2,1,1,1] => [2,2,1,1,1,1,1] => 3
[1,2,1,2,1,2] => [2,2,2,1,1,1] => 3
[1,2,1,2,2,1] => [2,2,2,1,1,1] => 3
[1,2,1,2,3] => [3,2,2,1,1] => 4
[1,2,1,3,1,1] => [3,2,1,1,1,1] => 4
[1,2,1,3,2] => [3,2,2,1,1] => 4
[1,2,1,4,1] => [4,2,1,1,1] => 4
[1,2,1,5] => [5,2,1,1] => 4
[1,2,2,1,1,1,1] => [2,2,1,1,1,1,1] => 3
[1,2,2,1,1,2] => [2,2,2,1,1,1] => 3
[1,2,2,1,2,1] => [2,2,2,1,1,1] => 3
[1,2,2,1,3] => [3,2,2,1,1] => 4
[1,2,2,2,1,1] => [2,2,2,1,1,1] => 3
[1,2,2,2,2] => [2,2,2,2,1] => 3
[1,2,2,3,1] => [3,2,2,1,1] => 4
[1,2,2,4] => [4,2,2,1] => 4
[1,2,3,1,1,1] => [3,2,1,1,1,1] => 4
[1,2,3,1,2] => [3,2,2,1,1] => 4
[1,2,3,2,1] => [3,2,2,1,1] => 4
[1,2,3,3] => [3,3,2,1] => 4
[1,2,4,1,1] => [4,2,1,1,1] => 4
[1,2,4,2] => [4,2,2,1] => 4
[1,2,5,1] => [5,2,1,1] => 4
[1,2,6] => [6,2,1] => 4
[1,3,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[1,3,1,1,1,2] => [3,2,1,1,1,1] => 4
[1,3,1,1,2,1] => [3,2,1,1,1,1] => 4
[1,3,1,1,3] => [3,3,1,1,1] => 3
[1,3,1,2,1,1] => [3,2,1,1,1,1] => 4
[1,3,1,2,2] => [3,2,2,1,1] => 4
[1,3,1,3,1] => [3,3,1,1,1] => 3
[1,3,1,4] => [4,3,1,1] => 4
[1,3,2,1,1,1] => [3,2,1,1,1,1] => 4
[1,3,2,1,2] => [3,2,2,1,1] => 4
[1,3,2,2,1] => [3,2,2,1,1] => 4
[1,3,2,3] => [3,3,2,1] => 4
[1,3,3,1,1] => [3,3,1,1,1] => 3
[1,3,3,2] => [3,3,2,1] => 4
[1,3,4,1] => [4,3,1,1] => 4
[1,3,5] => [5,3,1] => 4
[1,4,1,1,1,1] => [4,1,1,1,1,1] => 3
[1,4,1,1,2] => [4,2,1,1,1] => 4
[1,4,1,2,1] => [4,2,1,1,1] => 4
[1,4,1,3] => [4,3,1,1] => 4
[1,4,2,1,1] => [4,2,1,1,1] => 4
[1,4,2,2] => [4,2,2,1] => 4
[1,4,3,1] => [4,3,1,1] => 4
[1,4,4] => [4,4,1] => 3
[1,5,1,1,1] => [5,1,1,1,1] => 3
[1,5,1,2] => [5,2,1,1] => 4
[1,5,2,1] => [5,2,1,1] => 4
[1,5,3] => [5,3,1] => 4
[1,6,1,1] => [6,1,1,1] => 3
[1,6,2] => [6,2,1] => 4
[1,7,1] => [7,1,1] => 3
[1,8] => [8,1] => 3
[2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,2] => [2,2,1,1,1,1,1] => 3
[2,1,1,1,1,2,1] => [2,2,1,1,1,1,1] => 3
[2,1,1,1,1,3] => [3,2,1,1,1,1] => 4
[2,1,1,1,2,1,1] => [2,2,1,1,1,1,1] => 3
[2,1,1,1,2,2] => [2,2,2,1,1,1] => 3
[2,1,1,1,3,1] => [3,2,1,1,1,1] => 4
[2,1,1,1,4] => [4,2,1,1,1] => 4
[2,1,1,2,1,1,1] => [2,2,1,1,1,1,1] => 3
[2,1,1,2,1,2] => [2,2,2,1,1,1] => 3
[2,1,1,2,2,1] => [2,2,2,1,1,1] => 3
[2,1,1,2,3] => [3,2,2,1,1] => 4
[2,1,1,3,1,1] => [3,2,1,1,1,1] => 4
[2,1,1,3,2] => [3,2,2,1,1] => 4
[2,1,1,4,1] => [4,2,1,1,1] => 4
[2,1,1,5] => [5,2,1,1] => 4
[2,1,2,1,1,1,1] => [2,2,1,1,1,1,1] => 3
[2,1,2,1,1,2] => [2,2,2,1,1,1] => 3
[2,1,2,1,2,1] => [2,2,2,1,1,1] => 3
[2,1,2,1,3] => [3,2,2,1,1] => 4
[2,1,2,2,1,1] => [2,2,2,1,1,1] => 3
[2,1,2,2,2] => [2,2,2,2,1] => 3
[2,1,2,3,1] => [3,2,2,1,1] => 4
[2,1,2,4] => [4,2,2,1] => 4
[2,1,3,1,1,1] => [3,2,1,1,1,1] => 4
[2,1,3,1,2] => [3,2,2,1,1] => 4
[2,1,3,2,1] => [3,2,2,1,1] => 4
[2,1,3,3] => [3,3,2,1] => 4
[2,1,4,1,1] => [4,2,1,1,1] => 4
[2,1,4,2] => [4,2,2,1] => 4
[2,1,5,1] => [5,2,1,1] => 4
[2,1,6] => [6,2,1] => 4
[2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 3
[2,2,1,1,1,2] => [2,2,2,1,1,1] => 3
[2,2,1,1,2,1] => [2,2,2,1,1,1] => 3
[2,2,1,1,3] => [3,2,2,1,1] => 4
[2,2,1,2,1,1] => [2,2,2,1,1,1] => 3
[2,2,1,2,2] => [2,2,2,2,1] => 3
[2,2,1,3,1] => [3,2,2,1,1] => 4
[2,2,1,4] => [4,2,2,1] => 4
[2,2,2,1,1,1] => [2,2,2,1,1,1] => 3
[2,2,2,1,2] => [2,2,2,2,1] => 3
[2,2,2,2,1] => [2,2,2,2,1] => 3
[2,2,2,3] => [3,2,2,2] => 3
[2,2,3,1,1] => [3,2,2,1,1] => 4
[2,2,3,2] => [3,2,2,2] => 3
[2,2,4,1] => [4,2,2,1] => 4
[2,2,5] => [5,2,2] => 3
[2,3,1,1,1,1] => [3,2,1,1,1,1] => 4
[2,3,1,1,2] => [3,2,2,1,1] => 4
[2,3,1,2,1] => [3,2,2,1,1] => 4
[2,3,1,3] => [3,3,2,1] => 4
[2,3,2,1,1] => [3,2,2,1,1] => 4
[2,3,2,2] => [3,2,2,2] => 3
[2,3,3,1] => [3,3,2,1] => 4
[2,3,4] => [4,3,2] => 4
[2,4,1,1,1] => [4,2,1,1,1] => 4
[2,4,1,2] => [4,2,2,1] => 4
[2,4,2,1] => [4,2,2,1] => 4
[2,4,3] => [4,3,2] => 4
[2,5,1,1] => [5,2,1,1] => 4
[2,5,2] => [5,2,2] => 3
[2,6,1] => [6,2,1] => 4
[2,7] => [7,2] => 3
[3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[3,1,1,1,1,2] => [3,2,1,1,1,1] => 4
[3,1,1,1,2,1] => [3,2,1,1,1,1] => 4
[3,1,1,1,3] => [3,3,1,1,1] => 3
[3,1,1,2,1,1] => [3,2,1,1,1,1] => 4
[3,1,1,2,2] => [3,2,2,1,1] => 4
[3,1,1,3,1] => [3,3,1,1,1] => 3
[3,1,1,4] => [4,3,1,1] => 4
[3,1,2,1,1,1] => [3,2,1,1,1,1] => 4
[3,1,2,1,2] => [3,2,2,1,1] => 4
[3,1,2,2,1] => [3,2,2,1,1] => 4
[3,1,2,3] => [3,3,2,1] => 4
[3,1,3,1,1] => [3,3,1,1,1] => 3
[3,1,3,2] => [3,3,2,1] => 4
[3,1,4,1] => [4,3,1,1] => 4
[3,1,5] => [5,3,1] => 4
[3,2,1,1,1,1] => [3,2,1,1,1,1] => 4
[3,2,1,1,2] => [3,2,2,1,1] => 4
[3,2,1,2,1] => [3,2,2,1,1] => 4
[3,2,1,3] => [3,3,2,1] => 4
[3,2,2,1,1] => [3,2,2,1,1] => 4
[3,2,2,2] => [3,2,2,2] => 3
[3,2,3,1] => [3,3,2,1] => 4
[3,2,4] => [4,3,2] => 4
[3,3,1,1,1] => [3,3,1,1,1] => 3
[3,3,1,2] => [3,3,2,1] => 4
[3,3,2,1] => [3,3,2,1] => 4
[3,3,3] => [3,3,3] => 2
[3,4,1,1] => [4,3,1,1] => 4
[3,4,2] => [4,3,2] => 4
[3,5,1] => [5,3,1] => 4
[3,6] => [6,3] => 3
[4,1,1,1,1,1] => [4,1,1,1,1,1] => 3
[4,1,1,1,2] => [4,2,1,1,1] => 4
[4,1,1,2,1] => [4,2,1,1,1] => 4
[4,1,1,3] => [4,3,1,1] => 4
[4,1,2,1,1] => [4,2,1,1,1] => 4
[4,1,2,2] => [4,2,2,1] => 4
[4,1,3,1] => [4,3,1,1] => 4
[4,1,4] => [4,4,1] => 3
[4,2,1,1,1] => [4,2,1,1,1] => 4
[4,2,1,2] => [4,2,2,1] => 4
[4,2,2,1] => [4,2,2,1] => 4
[4,2,3] => [4,3,2] => 4
[4,3,1,1] => [4,3,1,1] => 4
[4,3,2] => [4,3,2] => 4
[4,4,1] => [4,4,1] => 3
[4,5] => [5,4] => 3
[5,1,1,1,1] => [5,1,1,1,1] => 3
[5,1,1,2] => [5,2,1,1] => 4
[5,1,2,1] => [5,2,1,1] => 4
[5,1,3] => [5,3,1] => 4
[5,2,1,1] => [5,2,1,1] => 4
[5,2,2] => [5,2,2] => 3
[5,3,1] => [5,3,1] => 4
[5,4] => [5,4] => 3
[6,1,1,1] => [6,1,1,1] => 3
[6,1,2] => [6,2,1] => 4
[6,2,1] => [6,2,1] => 4
[6,3] => [6,3] => 3
[7,1,1] => [7,1,1] => 3
[7,2] => [7,2] => 3
[8,1] => [8,1] => 3
[9] => [9] => 2
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,1,2] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,2,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,3] => [3,2,1,1,1,1,1] => 4
[1,1,1,1,1,3,1,1] => [3,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,3,2] => [3,2,1,1,1,1,1] => 4
[1,1,1,1,1,4,1] => [4,1,1,1,1,1,1] => 3
[1,1,1,1,1,5] => [5,1,1,1,1,1] => 3
[1,1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,2,1,2,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,2,1,3] => [3,2,1,1,1,1,1] => 4
[1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,2,2,2] => [2,2,2,1,1,1,1] => 3
[1,1,1,1,2,3,1] => [3,2,1,1,1,1,1] => 4
[1,1,1,1,2,4] => [4,2,1,1,1,1] => 4
[1,1,1,1,3,1,1,1] => [3,1,1,1,1,1,1,1] => 3
[1,1,1,1,3,1,2] => [3,2,1,1,1,1,1] => 4
[1,1,1,1,3,2,1] => [3,2,1,1,1,1,1] => 4
[1,1,1,1,3,3] => [3,3,1,1,1,1] => 3
[1,1,1,1,4,1,1] => [4,1,1,1,1,1,1] => 3
[1,1,1,1,4,2] => [4,2,1,1,1,1] => 4
[1,1,1,1,5,1] => [5,1,1,1,1,1] => 3
[1,1,1,1,6] => [6,1,1,1,1] => 3
[1,1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,1,2,1,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,2,1,1,2,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,2,1,1,3] => [3,2,1,1,1,1,1] => 4
[1,1,1,2,1,2,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,2,1,2,2] => [2,2,2,1,1,1,1] => 3
[1,1,1,2,1,3,1] => [3,2,1,1,1,1,1] => 4
[1,1,1,2,1,4] => [4,2,1,1,1,1] => 4
[1,1,1,2,2,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,2,2,1,2] => [2,2,2,1,1,1,1] => 3
[1,1,1,2,2,2,1] => [2,2,2,1,1,1,1] => 3
[1,1,1,2,2,3] => [3,2,2,1,1,1] => 4
[1,1,1,2,3,1,1] => [3,2,1,1,1,1,1] => 4
[1,1,1,2,3,2] => [3,2,2,1,1,1] => 4
[1,1,1,2,4,1] => [4,2,1,1,1,1] => 4
[1,1,1,2,5] => [5,2,1,1,1] => 4
[1,1,1,3,1,1,1,1] => [3,1,1,1,1,1,1,1] => 3
[1,1,1,3,1,1,2] => [3,2,1,1,1,1,1] => 4
[1,1,1,3,1,2,1] => [3,2,1,1,1,1,1] => 4
[1,1,1,3,1,3] => [3,3,1,1,1,1] => 3
[1,1,1,3,2,1,1] => [3,2,1,1,1,1,1] => 4
[1,1,1,3,2,2] => [3,2,2,1,1,1] => 4
[1,1,1,3,3,1] => [3,3,1,1,1,1] => 3
[1,1,1,3,4] => [4,3,1,1,1] => 4
[1,1,1,4,1,1,1] => [4,1,1,1,1,1,1] => 3
[1,1,1,4,1,2] => [4,2,1,1,1,1] => 4
[1,1,1,4,2,1] => [4,2,1,1,1,1] => 4
[1,1,1,4,3] => [4,3,1,1,1] => 4
[1,1,1,5,1,1] => [5,1,1,1,1,1] => 3
[1,1,1,5,2] => [5,2,1,1,1] => 4
[1,1,1,6,1] => [6,1,1,1,1] => 3
[1,1,1,7] => [7,1,1,1] => 3
[1,1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[1,1,2,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,2,1,1,1,3] => [3,2,1,1,1,1,1] => 4
[1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,2,1,1,2,2] => [2,2,2,1,1,1,1] => 3
[1,1,2,1,1,3,1] => [3,2,1,1,1,1,1] => 4
[1,1,2,1,1,4] => [4,2,1,1,1,1] => 4
[1,1,2,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,2,1,2,1,2] => [2,2,2,1,1,1,1] => 3
[1,1,2,1,2,2,1] => [2,2,2,1,1,1,1] => 3
[1,1,2,1,2,3] => [3,2,2,1,1,1] => 4
[1,1,2,1,3,1,1] => [3,2,1,1,1,1,1] => 4
[1,1,2,1,3,2] => [3,2,2,1,1,1] => 4
[1,1,2,1,4,1] => [4,2,1,1,1,1] => 4
[1,1,2,1,5] => [5,2,1,1,1] => 4
[1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,1,2,2,1,1,2] => [2,2,2,1,1,1,1] => 3
[1,1,2,2,1,2,1] => [2,2,2,1,1,1,1] => 3
[1,1,2,2,1,3] => [3,2,2,1,1,1] => 4
[1,1,2,2,2,1,1] => [2,2,2,1,1,1,1] => 3
[1,1,2,2,2,2] => [2,2,2,2,1,1] => 3
[1,1,2,2,3,1] => [3,2,2,1,1,1] => 4
[1,1,2,2,4] => [4,2,2,1,1] => 4
[1,1,2,3,1,1,1] => [3,2,1,1,1,1,1] => 4
[1,1,2,3,1,2] => [3,2,2,1,1,1] => 4
[1,1,2,3,2,1] => [3,2,2,1,1,1] => 4
[1,1,2,3,3] => [3,3,2,1,1] => 4
[1,1,2,4,1,1] => [4,2,1,1,1,1] => 4
[1,1,2,4,2] => [4,2,2,1,1] => 4
[1,1,2,5,1] => [5,2,1,1,1] => 4
[1,1,2,6] => [6,2,1,1] => 4
[1,1,3,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 3
[1,1,3,1,1,1,2] => [3,2,1,1,1,1,1] => 4
[1,1,3,1,1,2,1] => [3,2,1,1,1,1,1] => 4
[1,1,3,1,1,3] => [3,3,1,1,1,1] => 3
[1,1,3,1,2,1,1] => [3,2,1,1,1,1,1] => 4
[1,1,3,1,2,2] => [3,2,2,1,1,1] => 4
[1,1,3,1,3,1] => [3,3,1,1,1,1] => 3
[1,1,3,1,4] => [4,3,1,1,1] => 4
[1,1,3,2,1,1,1] => [3,2,1,1,1,1,1] => 4
[1,1,3,2,1,2] => [3,2,2,1,1,1] => 4
[1,1,3,2,2,1] => [3,2,2,1,1,1] => 4
[1,1,3,2,3] => [3,3,2,1,1] => 4
[1,1,3,3,1,1] => [3,3,1,1,1,1] => 3
[1,1,3,3,2] => [3,3,2,1,1] => 4
[1,1,3,4,1] => [4,3,1,1,1] => 4
[1,1,3,5] => [5,3,1,1] => 4
[1,1,4,1,1,1,1] => [4,1,1,1,1,1,1] => 3
[1,1,4,1,1,2] => [4,2,1,1,1,1] => 4
[1,1,4,1,2,1] => [4,2,1,1,1,1] => 4
[1,1,4,1,3] => [4,3,1,1,1] => 4
[1,1,4,2,1,1] => [4,2,1,1,1,1] => 4
[1,1,4,2,2] => [4,2,2,1,1] => 4
[1,1,4,3,1] => [4,3,1,1,1] => 4
[1,1,4,4] => [4,4,1,1] => 3
[1,1,5,1,1,1] => [5,1,1,1,1,1] => 3
[1,1,5,1,2] => [5,2,1,1,1] => 4
[1,1,5,2,1] => [5,2,1,1,1] => 4
[1,1,5,3] => [5,3,1,1] => 4
[1,1,6,1,1] => [6,1,1,1,1] => 3
[1,1,6,2] => [6,2,1,1] => 4
[1,1,7,1] => [7,1,1,1] => 3
[1,1,8] => [8,1,1] => 3
[1,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[1,2,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 3
[1,2,1,1,1,1,3] => [3,2,1,1,1,1,1] => 4
[1,2,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,2,1,1,1,2,2] => [2,2,2,1,1,1,1] => 3
[1,2,1,1,1,3,1] => [3,2,1,1,1,1,1] => 4
[1,2,1,1,1,4] => [4,2,1,1,1,1] => 4
[1,2,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,2,1,1,2,1,2] => [2,2,2,1,1,1,1] => 3
[1,2,1,1,2,2,1] => [2,2,2,1,1,1,1] => 3
[1,2,1,1,2,3] => [3,2,2,1,1,1] => 4
[1,2,1,1,3,1,1] => [3,2,1,1,1,1,1] => 4
[1,2,1,1,3,2] => [3,2,2,1,1,1] => 4
[1,2,1,1,4,1] => [4,2,1,1,1,1] => 4
[1,2,1,1,5] => [5,2,1,1,1] => 4
[1,2,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,2,1,2,1,1,2] => [2,2,2,1,1,1,1] => 3
[1,2,1,2,1,2,1] => [2,2,2,1,1,1,1] => 3
[1,2,1,2,1,3] => [3,2,2,1,1,1] => 4
[1,2,1,2,2,1,1] => [2,2,2,1,1,1,1] => 3
[1,2,1,2,2,2] => [2,2,2,2,1,1] => 3
[1,2,1,2,3,1] => [3,2,2,1,1,1] => 4
[1,2,1,2,4] => [4,2,2,1,1] => 4
[1,2,1,3,1,1,1] => [3,2,1,1,1,1,1] => 4
[1,2,1,3,1,2] => [3,2,2,1,1,1] => 4
[1,2,1,3,2,1] => [3,2,2,1,1,1] => 4
[1,2,1,3,3] => [3,3,2,1,1] => 4
[1,2,1,4,1,1] => [4,2,1,1,1,1] => 4
[1,2,1,4,2] => [4,2,2,1,1] => 4
[1,2,1,5,1] => [5,2,1,1,1] => 4
[1,2,1,6] => [6,2,1,1] => 4
[1,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[1,2,2,1,1,1,2] => [2,2,2,1,1,1,1] => 3
[1,2,2,1,1,2,1] => [2,2,2,1,1,1,1] => 3
[1,2,2,1,1,3] => [3,2,2,1,1,1] => 4
[1,2,2,1,2,1,1] => [2,2,2,1,1,1,1] => 3
[1,2,2,1,2,2] => [2,2,2,2,1,1] => 3
[1,2,2,1,3,1] => [3,2,2,1,1,1] => 4
[1,2,2,1,4] => [4,2,2,1,1] => 4
[1,2,2,2,1,1,1] => [2,2,2,1,1,1,1] => 3
[1,2,2,2,1,2] => [2,2,2,2,1,1] => 3
[1,2,2,2,2,1] => [2,2,2,2,1,1] => 3
[1,2,2,2,3] => [3,2,2,2,1] => 4
[1,2,2,3,1,1] => [3,2,2,1,1,1] => 4
[1,2,2,3,2] => [3,2,2,2,1] => 4
[1,2,2,4,1] => [4,2,2,1,1] => 4
[1,2,2,5] => [5,2,2,1] => 4
[1,2,3,1,1,1,1] => [3,2,1,1,1,1,1] => 4
[1,2,3,1,1,2] => [3,2,2,1,1,1] => 4
[1,2,3,1,2,1] => [3,2,2,1,1,1] => 4
[1,2,3,1,3] => [3,3,2,1,1] => 4
[1,2,3,2,1,1] => [3,2,2,1,1,1] => 4
[1,2,3,2,2] => [3,2,2,2,1] => 4
[1,2,3,3,1] => [3,3,2,1,1] => 4
[1,2,3,4] => [4,3,2,1] => 5
[1,2,4,1,1,1] => [4,2,1,1,1,1] => 4
[1,2,4,1,2] => [4,2,2,1,1] => 4
[1,2,4,2,1] => [4,2,2,1,1] => 4
[1,2,4,3] => [4,3,2,1] => 5
[1,2,5,1,1] => [5,2,1,1,1] => 4
[1,2,5,2] => [5,2,2,1] => 4
[1,2,6,1] => [6,2,1,1] => 4
[1,2,7] => [7,2,1] => 4
[1,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 3
[1,3,1,1,1,1,2] => [3,2,1,1,1,1,1] => 4
[1,3,1,1,1,2,1] => [3,2,1,1,1,1,1] => 4
[1,3,1,1,1,3] => [3,3,1,1,1,1] => 3
[1,3,1,1,2,1,1] => [3,2,1,1,1,1,1] => 4
[1,3,1,1,2,2] => [3,2,2,1,1,1] => 4
[1,3,1,1,3,1] => [3,3,1,1,1,1] => 3
[1,3,1,1,4] => [4,3,1,1,1] => 4
[1,3,1,2,1,1,1] => [3,2,1,1,1,1,1] => 4
[1,3,1,2,1,2] => [3,2,2,1,1,1] => 4
[1,3,1,2,2,1] => [3,2,2,1,1,1] => 4
[1,3,1,2,3] => [3,3,2,1,1] => 4
[1,3,1,3,1,1] => [3,3,1,1,1,1] => 3
[1,3,1,3,2] => [3,3,2,1,1] => 4
[1,3,1,4,1] => [4,3,1,1,1] => 4
[1,3,1,5] => [5,3,1,1] => 4
[1,3,2,1,1,1,1] => [3,2,1,1,1,1,1] => 4
[1,3,2,1,1,2] => [3,2,2,1,1,1] => 4
[1,3,2,1,2,1] => [3,2,2,1,1,1] => 4
[1,3,2,1,3] => [3,3,2,1,1] => 4
[1,3,2,2,1,1] => [3,2,2,1,1,1] => 4
[1,3,2,2,2] => [3,2,2,2,1] => 4
[1,3,2,3,1] => [3,3,2,1,1] => 4
[1,3,2,4] => [4,3,2,1] => 5
[1,3,3,1,1,1] => [3,3,1,1,1,1] => 3
[1,3,3,1,2] => [3,3,2,1,1] => 4
[1,3,3,2,1] => [3,3,2,1,1] => 4
[1,3,3,3] => [3,3,3,1] => 3
[1,3,4,1,1] => [4,3,1,1,1] => 4
[1,3,4,2] => [4,3,2,1] => 5
[1,3,5,1] => [5,3,1,1] => 4
[1,3,6] => [6,3,1] => 4
[1,4,1,1,1,1,1] => [4,1,1,1,1,1,1] => 3
[1,4,1,1,1,2] => [4,2,1,1,1,1] => 4
[1,4,1,1,2,1] => [4,2,1,1,1,1] => 4
[1,4,1,1,3] => [4,3,1,1,1] => 4
[1,4,1,2,1,1] => [4,2,1,1,1,1] => 4
[1,4,1,2,2] => [4,2,2,1,1] => 4
[1,4,1,3,1] => [4,3,1,1,1] => 4
[1,4,1,4] => [4,4,1,1] => 3
[1,4,2,1,1,1] => [4,2,1,1,1,1] => 4
[1,4,2,1,2] => [4,2,2,1,1] => 4
[1,4,2,2,1] => [4,2,2,1,1] => 4
[1,4,2,3] => [4,3,2,1] => 5
[1,4,3,1,1] => [4,3,1,1,1] => 4
[1,4,3,2] => [4,3,2,1] => 5
[1,4,4,1] => [4,4,1,1] => 3
[1,4,5] => [5,4,1] => 4
[1,5,1,1,1,1] => [5,1,1,1,1,1] => 3
[1,5,1,1,2] => [5,2,1,1,1] => 4
[1,5,1,2,1] => [5,2,1,1,1] => 4
[1,5,1,3] => [5,3,1,1] => 4
[1,5,2,1,1] => [5,2,1,1,1] => 4
[1,5,2,2] => [5,2,2,1] => 4
[1,5,3,1] => [5,3,1,1] => 4
[1,5,4] => [5,4,1] => 4
[1,6,1,1,1] => [6,1,1,1,1] => 3
[1,6,1,2] => [6,2,1,1] => 4
[1,6,2,1] => [6,2,1,1] => 4
[1,6,3] => [6,3,1] => 4
[1,7,1,1] => [7,1,1,1] => 3
[1,7,2] => [7,2,1] => 4
[1,8,1] => [8,1,1] => 3
[1,9] => [9,1] => 3
[2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,3] => [3,2,1,1,1,1,1] => 4
[2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 3
[2,1,1,1,1,2,2] => [2,2,2,1,1,1,1] => 3
[2,1,1,1,1,3,1] => [3,2,1,1,1,1,1] => 4
[2,1,1,1,1,4] => [4,2,1,1,1,1] => 4
[2,1,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[2,1,1,1,2,1,2] => [2,2,2,1,1,1,1] => 3
[2,1,1,1,2,2,1] => [2,2,2,1,1,1,1] => 3
[2,1,1,1,2,3] => [3,2,2,1,1,1] => 4
[2,1,1,1,3,1,1] => [3,2,1,1,1,1,1] => 4
[2,1,1,1,3,2] => [3,2,2,1,1,1] => 4
[2,1,1,1,4,1] => [4,2,1,1,1,1] => 4
[2,1,1,1,5] => [5,2,1,1,1] => 4
[2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[2,1,1,2,1,1,2] => [2,2,2,1,1,1,1] => 3
[2,1,1,2,1,2,1] => [2,2,2,1,1,1,1] => 3
[2,1,1,2,1,3] => [3,2,2,1,1,1] => 4
[2,1,1,2,2,1,1] => [2,2,2,1,1,1,1] => 3
[2,1,1,2,2,2] => [2,2,2,2,1,1] => 3
[2,1,1,2,3,1] => [3,2,2,1,1,1] => 4
[2,1,1,2,4] => [4,2,2,1,1] => 4
[2,1,1,3,1,1,1] => [3,2,1,1,1,1,1] => 4
[2,1,1,3,1,2] => [3,2,2,1,1,1] => 4
[2,1,1,3,2,1] => [3,2,2,1,1,1] => 4
[2,1,1,3,3] => [3,3,2,1,1] => 4
[2,1,1,4,1,1] => [4,2,1,1,1,1] => 4
[2,1,1,4,2] => [4,2,2,1,1] => 4
[2,1,1,5,1] => [5,2,1,1,1] => 4
[2,1,1,6] => [6,2,1,1] => 4
[2,1,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[2,1,2,1,1,1,2] => [2,2,2,1,1,1,1] => 3
[2,1,2,1,1,2,1] => [2,2,2,1,1,1,1] => 3
[2,1,2,1,1,3] => [3,2,2,1,1,1] => 4
[2,1,2,1,2,1,1] => [2,2,2,1,1,1,1] => 3
[2,1,2,1,2,2] => [2,2,2,2,1,1] => 3
[2,1,2,1,3,1] => [3,2,2,1,1,1] => 4
[2,1,2,1,4] => [4,2,2,1,1] => 4
[2,1,2,2,1,1,1] => [2,2,2,1,1,1,1] => 3
[2,1,2,2,1,2] => [2,2,2,2,1,1] => 3
[2,1,2,2,2,1] => [2,2,2,2,1,1] => 3
[2,1,2,2,3] => [3,2,2,2,1] => 4
[2,1,2,3,1,1] => [3,2,2,1,1,1] => 4
[2,1,2,3,2] => [3,2,2,2,1] => 4
[2,1,2,4,1] => [4,2,2,1,1] => 4
[2,1,2,5] => [5,2,2,1] => 4
[2,1,3,1,1,1,1] => [3,2,1,1,1,1,1] => 4
[2,1,3,1,1,2] => [3,2,2,1,1,1] => 4
[2,1,3,1,2,1] => [3,2,2,1,1,1] => 4
[2,1,3,1,3] => [3,3,2,1,1] => 4
[2,1,3,2,1,1] => [3,2,2,1,1,1] => 4
[2,1,3,2,2] => [3,2,2,2,1] => 4
[2,1,3,3,1] => [3,3,2,1,1] => 4
[2,1,3,4] => [4,3,2,1] => 5
[2,1,4,1,1,1] => [4,2,1,1,1,1] => 4
[2,1,4,1,2] => [4,2,2,1,1] => 4
[2,1,4,2,1] => [4,2,2,1,1] => 4
[2,1,4,3] => [4,3,2,1] => 5
[2,1,5,1,1] => [5,2,1,1,1] => 4
[2,1,5,2] => [5,2,2,1] => 4
[2,1,6,1] => [6,2,1,1] => 4
[2,1,7] => [7,2,1] => 4
[2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,2] => [2,2,2,1,1,1,1] => 3
[2,2,1,1,1,2,1] => [2,2,2,1,1,1,1] => 3
[2,2,1,1,1,3] => [3,2,2,1,1,1] => 4
[2,2,1,1,2,1,1] => [2,2,2,1,1,1,1] => 3
[2,2,1,1,2,2] => [2,2,2,2,1,1] => 3
[2,2,1,1,3,1] => [3,2,2,1,1,1] => 4
[2,2,1,1,4] => [4,2,2,1,1] => 4
[2,2,1,2,1,1,1] => [2,2,2,1,1,1,1] => 3
[2,2,1,2,1,2] => [2,2,2,2,1,1] => 3
[2,2,1,2,2,1] => [2,2,2,2,1,1] => 3
[2,2,1,2,3] => [3,2,2,2,1] => 4
[2,2,1,3,1,1] => [3,2,2,1,1,1] => 4
[2,2,1,3,2] => [3,2,2,2,1] => 4
[2,2,1,4,1] => [4,2,2,1,1] => 4
[2,2,1,5] => [5,2,2,1] => 4
[2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 3
[2,2,2,1,1,2] => [2,2,2,2,1,1] => 3
[2,2,2,1,2,1] => [2,2,2,2,1,1] => 3
[2,2,2,1,3] => [3,2,2,2,1] => 4
[2,2,2,2,1,1] => [2,2,2,2,1,1] => 3
[2,2,2,2,2] => [2,2,2,2,2] => 2
[2,2,2,3,1] => [3,2,2,2,1] => 4
[2,2,2,4] => [4,2,2,2] => 3
[2,2,3,1,1,1] => [3,2,2,1,1,1] => 4
[2,2,3,1,2] => [3,2,2,2,1] => 4
[2,2,3,2,1] => [3,2,2,2,1] => 4
[2,2,3,3] => [3,3,2,2] => 3
[2,2,4,1,1] => [4,2,2,1,1] => 4
[2,2,4,2] => [4,2,2,2] => 3
[2,2,5,1] => [5,2,2,1] => 4
[2,2,6] => [6,2,2] => 3
[2,3,1,1,1,1,1] => [3,2,1,1,1,1,1] => 4
[2,3,1,1,1,2] => [3,2,2,1,1,1] => 4
[2,3,1,1,2,1] => [3,2,2,1,1,1] => 4
[2,3,1,1,3] => [3,3,2,1,1] => 4
[2,3,1,2,1,1] => [3,2,2,1,1,1] => 4
[2,3,1,2,2] => [3,2,2,2,1] => 4
[2,3,1,3,1] => [3,3,2,1,1] => 4
[2,3,1,4] => [4,3,2,1] => 5
[2,3,2,1,1,1] => [3,2,2,1,1,1] => 4
[2,3,2,1,2] => [3,2,2,2,1] => 4
[2,3,2,2,1] => [3,2,2,2,1] => 4
[2,3,2,3] => [3,3,2,2] => 3
[2,3,3,1,1] => [3,3,2,1,1] => 4
[2,3,3,2] => [3,3,2,2] => 3
[2,3,4,1] => [4,3,2,1] => 5
[2,3,5] => [5,3,2] => 4
[2,4,1,1,1,1] => [4,2,1,1,1,1] => 4
[2,4,1,1,2] => [4,2,2,1,1] => 4
[2,4,1,2,1] => [4,2,2,1,1] => 4
[2,4,1,3] => [4,3,2,1] => 5
[2,4,2,1,1] => [4,2,2,1,1] => 4
[2,4,2,2] => [4,2,2,2] => 3
[2,4,3,1] => [4,3,2,1] => 5
[2,4,4] => [4,4,2] => 3
[2,5,1,1,1] => [5,2,1,1,1] => 4
[2,5,1,2] => [5,2,2,1] => 4
[2,5,2,1] => [5,2,2,1] => 4
[2,5,3] => [5,3,2] => 4
[2,6,1,1] => [6,2,1,1] => 4
[2,6,2] => [6,2,2] => 3
[2,7,1] => [7,2,1] => 4
[2,8] => [8,2] => 3
[3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,2] => [3,2,1,1,1,1,1] => 4
[3,1,1,1,1,2,1] => [3,2,1,1,1,1,1] => 4
[3,1,1,1,1,3] => [3,3,1,1,1,1] => 3
[3,1,1,1,2,1,1] => [3,2,1,1,1,1,1] => 4
[3,1,1,1,2,2] => [3,2,2,1,1,1] => 4
[3,1,1,1,3,1] => [3,3,1,1,1,1] => 3
[3,1,1,1,4] => [4,3,1,1,1] => 4
[3,1,1,2,1,1,1] => [3,2,1,1,1,1,1] => 4
[3,1,1,2,1,2] => [3,2,2,1,1,1] => 4
[3,1,1,2,2,1] => [3,2,2,1,1,1] => 4
[3,1,1,2,3] => [3,3,2,1,1] => 4
[3,1,1,3,1,1] => [3,3,1,1,1,1] => 3
[3,1,1,3,2] => [3,3,2,1,1] => 4
[3,1,1,4,1] => [4,3,1,1,1] => 4
[3,1,1,5] => [5,3,1,1] => 4
[3,1,2,1,1,1,1] => [3,2,1,1,1,1,1] => 4
[3,1,2,1,1,2] => [3,2,2,1,1,1] => 4
[3,1,2,1,2,1] => [3,2,2,1,1,1] => 4
[3,1,2,1,3] => [3,3,2,1,1] => 4
[3,1,2,2,1,1] => [3,2,2,1,1,1] => 4
[3,1,2,2,2] => [3,2,2,2,1] => 4
[3,1,2,3,1] => [3,3,2,1,1] => 4
[3,1,2,4] => [4,3,2,1] => 5
[3,1,3,1,1,1] => [3,3,1,1,1,1] => 3
[3,1,3,1,2] => [3,3,2,1,1] => 4
[3,1,3,2,1] => [3,3,2,1,1] => 4
[3,1,3,3] => [3,3,3,1] => 3
[3,1,4,1,1] => [4,3,1,1,1] => 4
[3,1,4,2] => [4,3,2,1] => 5
[3,1,5,1] => [5,3,1,1] => 4
[3,1,6] => [6,3,1] => 4
[3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 4
[3,2,1,1,1,2] => [3,2,2,1,1,1] => 4
[3,2,1,1,2,1] => [3,2,2,1,1,1] => 4
[3,2,1,1,3] => [3,3,2,1,1] => 4
[3,2,1,2,1,1] => [3,2,2,1,1,1] => 4
[3,2,1,2,2] => [3,2,2,2,1] => 4
[3,2,1,3,1] => [3,3,2,1,1] => 4
[3,2,1,4] => [4,3,2,1] => 5
[3,2,2,1,1,1] => [3,2,2,1,1,1] => 4
[3,2,2,1,2] => [3,2,2,2,1] => 4
[3,2,2,2,1] => [3,2,2,2,1] => 4
[3,2,2,3] => [3,3,2,2] => 3
[3,2,3,1,1] => [3,3,2,1,1] => 4
[3,2,3,2] => [3,3,2,2] => 3
[3,2,4,1] => [4,3,2,1] => 5
[3,2,5] => [5,3,2] => 4
[3,3,1,1,1,1] => [3,3,1,1,1,1] => 3
[3,3,1,1,2] => [3,3,2,1,1] => 4
[3,3,1,2,1] => [3,3,2,1,1] => 4
[3,3,1,3] => [3,3,3,1] => 3
[3,3,2,1,1] => [3,3,2,1,1] => 4
[3,3,2,2] => [3,3,2,2] => 3
[3,3,3,1] => [3,3,3,1] => 3
[3,3,4] => [4,3,3] => 3
[3,4,1,1,1] => [4,3,1,1,1] => 4
[3,4,1,2] => [4,3,2,1] => 5
[3,4,2,1] => [4,3,2,1] => 5
[3,4,3] => [4,3,3] => 3
[3,5,1,1] => [5,3,1,1] => 4
[3,5,2] => [5,3,2] => 4
[3,6,1] => [6,3,1] => 4
[3,7] => [7,3] => 3
[4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 3
[4,1,1,1,1,2] => [4,2,1,1,1,1] => 4
[4,1,1,1,2,1] => [4,2,1,1,1,1] => 4
[4,1,1,1,3] => [4,3,1,1,1] => 4
[4,1,1,2,1,1] => [4,2,1,1,1,1] => 4
[4,1,1,2,2] => [4,2,2,1,1] => 4
[4,1,1,3,1] => [4,3,1,1,1] => 4
[4,1,1,4] => [4,4,1,1] => 3
[4,1,2,1,1,1] => [4,2,1,1,1,1] => 4
[4,1,2,1,2] => [4,2,2,1,1] => 4
[4,1,2,2,1] => [4,2,2,1,1] => 4
[4,1,2,3] => [4,3,2,1] => 5
[4,1,3,1,1] => [4,3,1,1,1] => 4
[4,1,3,2] => [4,3,2,1] => 5
[4,1,4,1] => [4,4,1,1] => 3
[4,1,5] => [5,4,1] => 4
[4,2,1,1,1,1] => [4,2,1,1,1,1] => 4
[4,2,1,1,2] => [4,2,2,1,1] => 4
[4,2,1,2,1] => [4,2,2,1,1] => 4
[4,2,1,3] => [4,3,2,1] => 5
[4,2,2,1,1] => [4,2,2,1,1] => 4
[4,2,2,2] => [4,2,2,2] => 3
[4,2,3,1] => [4,3,2,1] => 5
[4,2,4] => [4,4,2] => 3
[4,3,1,1,1] => [4,3,1,1,1] => 4
[4,3,1,2] => [4,3,2,1] => 5
[4,3,2,1] => [4,3,2,1] => 5
[4,3,3] => [4,3,3] => 3
[4,4,1,1] => [4,4,1,1] => 3
[4,4,2] => [4,4,2] => 3
[4,5,1] => [5,4,1] => 4
[4,6] => [6,4] => 3
[5,1,1,1,1,1] => [5,1,1,1,1,1] => 3
[5,1,1,1,2] => [5,2,1,1,1] => 4
[5,1,1,2,1] => [5,2,1,1,1] => 4
[5,1,1,3] => [5,3,1,1] => 4
[5,1,2,1,1] => [5,2,1,1,1] => 4
[5,1,2,2] => [5,2,2,1] => 4
[5,1,3,1] => [5,3,1,1] => 4
[5,1,4] => [5,4,1] => 4
[5,2,1,1,1] => [5,2,1,1,1] => 4
[5,2,1,2] => [5,2,2,1] => 4
[5,2,2,1] => [5,2,2,1] => 4
[5,2,3] => [5,3,2] => 4
[5,3,1,1] => [5,3,1,1] => 4
[5,3,2] => [5,3,2] => 4
[5,4,1] => [5,4,1] => 4
[5,5] => [5,5] => 2
[6,1,1,1,1] => [6,1,1,1,1] => 3
[6,1,1,2] => [6,2,1,1] => 4
[6,1,2,1] => [6,2,1,1] => 4
[6,1,3] => [6,3,1] => 4
[6,2,1,1] => [6,2,1,1] => 4
[6,2,2] => [6,2,2] => 3
[6,3,1] => [6,3,1] => 4
[6,4] => [6,4] => 3
[7,1,1,1] => [7,1,1,1] => 3
[7,1,2] => [7,2,1] => 4
[7,2,1] => [7,2,1] => 4
[7,3] => [7,3] => 3
[8,1,1] => [8,1,1] => 3
[8,2] => [8,2] => 3
[9,1] => [9,1] => 3
[10] => [10] => 2
[1,5,4,1] => [5,4,1,1] => 4
[1,4,5,1] => [5,4,1,1] => 4
[1,1,2,2,3,3] => [3,3,2,2,1,1] => 4
[1,1,3,3,2,2] => [3,3,2,2,1,1] => 4
[1,1,3,2,2,3] => [3,3,2,2,1,1] => 4
[2,2,1,1,3,3] => [3,3,2,2,1,1] => 4
[2,2,3,3,1,1] => [3,3,2,2,1,1] => 4
[2,2,3,1,1,3] => [3,3,2,2,1,1] => 4
[2,2,4,4] => [4,4,2,2] => 3
[2,1,1,2,3,3] => [3,3,2,2,1,1] => 4
[2,1,3,3,1,2] => [3,3,2,2,1,1] => 4
[2,1,3,2,1,3] => [3,3,2,2,1,1] => 4
[2,1,4,5] => [5,4,2,1] => 5
[3,3,1,1,2,2] => [3,3,2,2,1,1] => 4
[3,3,2,2,1,1] => [3,3,2,2,1,1] => 4
[3,3,2,1,1,2] => [3,3,2,2,1,1] => 4
[3,2,1,1,2,3] => [3,3,2,2,1,1] => 4
[3,2,2,3,1,1] => [3,3,2,2,1,1] => 4
[3,2,2,1,1,3] => [3,3,2,2,1,1] => 4
[3,2,3,4] => [4,3,3,2] => 4
[3,1,1,3,2,2] => [3,3,2,2,1,1] => 4
[3,1,1,2,2,3] => [3,3,2,2,1,1] => 4
[3,1,2,3,1,2] => [3,3,2,2,1,1] => 4
[3,1,2,2,1,3] => [3,3,2,2,1,1] => 4
[3,1,3,5] => [5,3,3,1] => 4
[4,4,2,2] => [4,4,2,2] => 3
[4,3,2,3] => [4,3,3,2] => 4
[4,2,2,4] => [4,4,2,2] => 3
[4,1,2,5] => [5,4,2,1] => 5
[5,4,1,2] => [5,4,2,1] => 5
[5,3,1,3] => [5,3,3,1] => 4
[5,2,1,4] => [5,4,2,1] => 5
[1,2,2,3,3,1] => [3,3,2,2,1,1] => 4
[1,2,2,4,3] => [4,3,2,2,1] => 5
[1,2,3,3,2,1] => [3,3,2,2,1,1] => 4
[1,2,3,4,2] => [4,3,2,2,1] => 5
[1,2,3,2,3,1] => [3,3,2,2,1,1] => 4
[1,2,3,3,3] => [3,3,3,2,1] => 4
[1,2,4,4,1] => [4,4,2,1,1] => 4
[1,2,4,3,2] => [4,3,2,2,1] => 5
[1,2,4,2,3] => [4,3,2,2,1] => 5
[1,2,5,4] => [5,4,2,1] => 5
[1,3,3,2,2,1] => [3,3,2,2,1,1] => 4
[1,3,3,3,2] => [3,3,3,2,1] => 4
[1,3,4,3,1] => [4,3,3,1,1] => 4
[1,3,4,2,2] => [4,3,2,2,1] => 5
[1,3,5,3] => [5,3,3,1] => 4
[1,3,2,3,2,1] => [3,3,2,2,1,1] => 4
[1,3,2,4,2] => [4,3,2,2,1] => 5
[1,3,2,2,3,1] => [3,3,2,2,1,1] => 4
[1,3,2,3,3] => [3,3,3,2,1] => 4
[1,3,3,4,1] => [4,3,3,1,1] => 4
[1,3,3,2,3] => [3,3,3,2,1] => 4
[1,3,4,4] => [4,4,3,1] => 4
[1,4,4,2,1] => [4,4,2,1,1] => 4
[1,4,5,2] => [5,4,2,1] => 5
[1,4,3,3,1] => [4,3,3,1,1] => 4
[1,4,3,2,2] => [4,3,2,2,1] => 5
[1,4,4,3] => [4,4,3,1] => 4
[1,4,2,4,1] => [4,4,2,1,1] => 4
[1,4,2,3,2] => [4,3,2,2,1] => 5
[1,4,2,2,3] => [4,3,2,2,1] => 5
[1,4,3,4] => [4,4,3,1] => 4
[1,5,4,2] => [5,4,2,1] => 5
[1,5,3,3] => [5,3,3,1] => 4
[1,5,2,4] => [5,4,2,1] => 5
[2,3,3,3,1] => [3,3,3,2,1] => 4
[2,3,4,3] => [4,3,3,2] => 4
[2,4,3,2,1] => [4,3,2,2,1] => 5
[2,4,4,2] => [4,4,2,2] => 3
[2,4,2,3,1] => [4,3,2,2,1] => 5
[2,4,3,3] => [4,3,3,2] => 4
[2,5,4,1] => [5,4,2,1] => 5
[2,5,3,2] => [5,3,2,2] => 4
[2,5,2,3] => [5,3,2,2] => 4
[2,6,4] => [6,4,2] => 4
[2,2,4,3,1] => [4,3,2,2,1] => 5
[2,2,5,3] => [5,3,2,2] => 4
[2,2,3,4,1] => [4,3,2,2,1] => 5
[2,3,4,2,1] => [4,3,2,2,1] => 5
[2,3,5,2] => [5,3,2,2] => 4
[2,3,2,4,1] => [4,3,2,2,1] => 5
[2,3,3,4] => [4,3,3,2] => 4
[2,4,5,1] => [5,4,2,1] => 5
[2,4,2,4] => [4,4,2,2] => 3
[3,4,2,2,1] => [4,3,2,2,1] => 5
[3,4,3,2] => [4,3,3,2] => 4
[3,5,3,1] => [5,3,3,1] => 4
[3,5,2,2] => [5,3,2,2] => 4
[3,3,3,2,1] => [3,3,3,2,1] => 4
[3,3,4,2] => [4,3,3,2] => 4
[3,3,2,3,1] => [3,3,3,2,1] => 4
[3,4,4,1] => [4,4,3,1] => 4
[3,4,2,3] => [4,3,3,2] => 4
[3,5,4] => [5,4,3] => 4
[3,2,4,2,1] => [4,3,2,2,1] => 5
[3,2,5,2] => [5,3,2,2] => 4
[3,2,3,3,1] => [3,3,3,2,1] => 4
[3,2,4,3] => [4,3,3,2] => 4
[3,2,2,4,1] => [4,3,2,2,1] => 5
[3,3,5,1] => [5,3,3,1] => 4
[3,3,2,4] => [4,3,3,2] => 4
[3,4,5] => [5,4,3] => 4
[4,5,2,1] => [5,4,2,1] => 5
[4,6,2] => [6,4,2] => 4
[4,4,3,1] => [4,4,3,1] => 4
[4,5,3] => [5,4,3] => 4
[4,3,4,1] => [4,4,3,1] => 4
[4,3,3,2] => [4,3,3,2] => 4
[4,2,5,1] => [5,4,2,1] => 5
[4,2,4,2] => [4,4,2,2] => 3
[4,2,3,3] => [4,3,3,2] => 4
[4,3,5] => [5,4,3] => 4
[5,4,3] => [5,4,3] => 4
[5,3,4] => [5,4,3] => 4
[1,5,2,1,2] => [5,2,2,1,1] => 4
[1,2,2,2,5] => [5,2,2,2,1] => 4
[1,1,2,3,5] => [5,3,2,1,1] => 5
[3,1,2,1,5] => [5,3,2,1,1] => 5
[1,3,2,1,5] => [5,3,2,1,1] => 5
[1,1,4,1,5] => [5,4,1,1,1] => 4
[1,1,2,5,3] => [5,3,2,1,1] => 5
[3,1,2,3,3] => [3,3,3,2,1] => 4
[1,1,4,3,3] => [4,3,3,1,1] => 4
[5,1,2,1,3] => [5,3,2,1,1] => 5
[3,3,2,1,3] => [3,3,3,2,1] => 4
[1,5,2,1,3] => [5,3,2,1,1] => 5
[3,1,4,1,3] => [4,3,3,1,1] => 4
[1,3,4,1,3] => [4,3,3,1,1] => 4
[5,2,2,2,1] => [5,2,2,2,1] => 4
[3,1,2,5,1] => [5,3,2,1,1] => 5
[1,3,2,5,1] => [5,3,2,1,1] => 5
[1,1,4,5,1] => [5,4,1,1,1] => 4
[5,1,2,3,1] => [5,3,2,1,1] => 5
[1,5,2,3,1] => [5,3,2,1,1] => 5
[3,1,4,3,1] => [4,3,3,1,1] => 4
[5,3,2,1,1] => [5,3,2,1,1] => 5
[3,5,2,1,1] => [5,3,2,1,1] => 5
[5,1,4,1,1] => [5,4,1,1,1] => 4
[3,3,4,1,1] => [4,3,3,1,1] => 4
[1,5,4,1,1] => [5,4,1,1,1] => 4
[4,3,2,2,1] => [4,3,2,2,1] => 5
[4,2,3,2,1] => [4,3,2,2,1] => 5
[4,2,2,3,1] => [4,3,2,2,1] => 5
[5,4,2,1] => [5,4,2,1] => 5
[5,3,3,1] => [5,3,3,1] => 4
[5,3,2,2] => [5,3,2,2] => 4
[5,2,4,1] => [5,4,2,1] => 5
[5,2,3,2] => [5,3,2,2] => 4
[5,2,2,3] => [5,3,2,2] => 4
[6,4,2] => [6,4,2] => 4
[6,2,4] => [6,4,2] => 4
[2,1,2,5,1] => [5,2,2,1,1] => 4
[1,2,5,2,2] => [5,2,2,2,1] => 4
[1,3,5,2,1] => [5,3,2,1,1] => 5
[1,5,2,2,2] => [5,2,2,2,1] => 4
[1,5,3,2,1] => [5,3,2,1,1] => 5
[2,5,2,2,1] => [5,2,2,2,1] => 4
[2,2,5,2,1] => [5,2,2,2,1] => 4
[1,1,2,3,2,3] => [3,3,2,2,1,1] => 4
[1,3,1,5,1] => [5,3,1,1,1] => 4
[1,5,1,3,1] => [5,3,1,1,1] => 4
[1,4,1,4,1] => [4,4,1,1,1] => 3
[1,4,3,2,1] => [4,3,2,1,1] => 5
[1,2,3,4,1] => [4,3,2,1,1] => 5
[1,2,5,2,1] => [5,2,2,1,1] => 4
[1,4,3,4,1] => [4,4,3,1,1] => 4
[1,4,5,2,1] => [5,4,2,1,1] => 5
[1,2,5,4,1] => [5,4,2,1,1] => 5
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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:
2,2 3,5 2,14 4,22,6 2,44,18 4,68,56 3,107,146 4,172,312,24
$F_{1} = q^{2}$
$F_{2} = 2\ q^{2}$
$F_{3} = 2\ q^{2} + 2\ q^{3}$
$F_{4} = 3\ q^{2} + 5\ q^{3}$
$F_{5} = 2\ q^{2} + 14\ q^{3}$
$F_{6} = 4\ q^{2} + 22\ q^{3} + 6\ q^{4}$
$F_{7} = 2\ q^{2} + 44\ q^{3} + 18\ q^{4}$
$F_{8} = 4\ q^{2} + 68\ q^{3} + 56\ q^{4}$
$F_{9} = 3\ q^{2} + 107\ q^{3} + 146\ q^{4}$
$F_{10} = 4\ q^{2} + 172\ q^{3} + 312\ q^{4} + 24\ q^{5}$
Description
The number of addable cells of the Ferrers diagram of an integer partition.
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Description
Sends a composition to the partition obtained by sorting the entries.
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