- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤ

Values

[1,0,1,0,1,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,0,1,1,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,0,0,1,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,0,1,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,0,1,1,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,0,1,1,0,0,1,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,0,1,1,0,1,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,0,1,1,1,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,0,0,1,0,1,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,0,0,1,1,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,0,1,0,0,1,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,0,1,0,1,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,0,1,1,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,0,0,0,1,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,0,0,1,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => [3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 0

[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,0,0,0,1,0,1,0] => [4,3] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,0,0,1,1,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,0,1,0,1,0,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,0,0,1,1,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,0,0,1,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,0,0,1,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => [5,4,3,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => [6,4,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [4,3,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => [6,5,2] => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => [6,3,2] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => [5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => [6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 0

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0

[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => [5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => [6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => [5,4,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => [4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 2

[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,3,1] => [6,5,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 3

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => [6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,0,1,0,1,1,0,0,0,1,0] => [5,2,2,1] => [5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,0,0,1,0,0,1,0] => [5,3,1,1] => [5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,0,1,1,0,0,1,0,1,0,0] => [4,3,1,1] => [3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,0,1,1,0,1,0,0,0,1,0] => [5,2,1,1] => [5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 0

[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3

[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,0,0,1,0,1,0,0,1,0] => [5,3,2] => [5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,0,0,1,1,0,0,0,1,0] => [5,2,2] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,0,1,0,0,1,0,0,1,0] => [5,3,1] => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

>>> Load all 608 entries. <<<

[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => [5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,2,2,2,1] => [5,4,4,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 6

[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2,1] => [4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1

[1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,2,1,1] => [4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 3

[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2,1,1] => [5,4,3,2] => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 4

[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1,1] => [6,4,2] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1,1] => [4,3,2,1,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 1

[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1,1] => [4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1,1,1] => [4,3,3,2,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 3

[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => [5,4,2,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 2

[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => [4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1,1,1] => [3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 5

[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1,1] => [5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1,1] => [6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1,1] => [3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1,1] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1,1] => [3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4

[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1,1,1] => [5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0

[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [5,3,3,2,2] => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 7

[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [3,3,3,2,2] => [6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [6,2,2,2,2] => [4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1

[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [5,2,2,2,2] => [5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => [4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,2,1] => [4,3,3,3,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 0

[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2,1] => [5,4,3,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1

[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1] => [6,4,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1] => [4,3,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [5,3,3,1,1] => [6,5,2] => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3

[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1] => [6,3,2] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1,1] => [4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1] => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1,1] => [3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 3

[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1] => [5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1] => [6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1] => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1,1] => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1] => [2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 0

[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0

[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [5,3,3,3] => [6,5,3] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 2

[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => [6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 4

[1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [6,5,2,2] => [5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1

[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [6,3,2,2] => [4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0

[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2] => [5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2] => [6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [6,2,2,2] => [4,3,3,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 0

[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [5,2,2,2] => [5,4,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2] => [4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [5,3,3,1] => [6,5,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 3

[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1] => [6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1] => [4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1] => [5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,0,0,0,1,0,1,0] => [6,5,1,1] => [5,3,3,2] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1] => [5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1] => [3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1] => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1] => [3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 1

[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1] => [5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1] => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [6,5,3] => [5,3,3,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0

[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [6,3,3] => [6,3,3] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [5,3,3] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3

[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [3,3,3] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [6,3,2] => [3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [5,3,2] => [5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [3,3,2] => [6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [6,2,2] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [5,2,2] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,2,2] => [4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [6,5,1] => [5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [6,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [5,3,1] => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [4,3,1] => [3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [6,2,1] => [3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [5,2,1] => [5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,1,1] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => [5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [4,4] => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [6,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,3] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [4,3] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [6,2] => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,2] => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [4,2] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [6,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => [4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 6

[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2,2,1,1] => [5,4,4,2] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 8

[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2,1,1] => [4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 4

[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1,1] => [4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 2

[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1,1,1,1] => [4,3,3,2,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 5

[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1,1,1] => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 5

[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1,1] => [4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => [4,2,2,1,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1,1] => [3,2,2,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => [4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1,1] => [3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 6

[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1,1,1] => [5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 3

[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1,1] => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 0

[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2,2,2] => [5,4,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 6

[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,2,2] => [6,4,4] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2] => [4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2,2,1] => [5,4,4,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 6

[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2,1] => [4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1

[1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0] => [6,3,2,2,1,1] => [4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 3

[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2,1,1] => [5,4,3,2] => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 4

[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1,1] => [6,4,2] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1] => [4,3,2,1,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 1

[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1] => [4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1,1,1] => [4,3,3,2,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 3

[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1,1] => [5,4,2,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 2

[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1] => [4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1,1,1] => [3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 5

[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1,1,1] => [5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1,1] => [6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1] => [3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1] => [3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4

[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1,1] => [5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0

[1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0] => [5,3,3,2,2] => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 7

[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,2] => [6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0] => [6,2,2,2,2] => [4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1

[1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2,2] => [5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2] => [4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0] => [6,3,2,2,1] => [4,3,3,3,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 0

[1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2,1] => [5,4,3,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1

[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1] => [6,4,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => [4,3,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0] => [5,3,3,1,1] => [6,5,2] => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3

[1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1] => [6,3,2] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1,1] => [4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1] => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1,1] => [3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 3

[1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1,1] => [5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1] => [6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1] => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1] => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1] => [2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 0

[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0

[1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0] => [5,3,3,3] => [6,5,3] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 2

[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3] => [6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 4

[1,1,1,1,0,0,1,1,0,0,0,1,0,1,0,0] => [6,5,2,2] => [5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1

[1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0] => [6,3,2,2] => [4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0

[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2] => [5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2] => [6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0] => [6,2,2,2] => [4,3,3,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2] => [5,4,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2] => [4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0] => [5,3,3,1] => [6,5,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 3

[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1] => [6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1] => [4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1] => [5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0] => [6,5,1,1] => [5,3,3,2] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1] => [5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1] => [3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1] => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1] => [3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 1

[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1] => [5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1] => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0] => [6,5,3] => [5,3,3,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0

[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0] => [7,3,3] => [7,6] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 4

[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [6,3,3] => [6,3,3] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0] => [5,3,3] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3

[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0] => [3,3,3] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [6,3,2] => [3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [5,3,2] => [5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0] => [3,3,2] => [6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => [7,2,2] => [7,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0] => [6,2,2] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0] => [5,2,2] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0] => [4,2,2] => [4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [6,5,1] => [5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [6,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [5,3,1] => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [4,3,1] => [3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0] => [6,2,1] => [3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [5,2,1] => [5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [7,1,1] => [7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [6,1,1] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0] => [5,1,1] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => [7,5] => [7,5] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3

[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [6,5] => [5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0] => [4,4] => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [7,3] => [7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [6,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0] => [5,3] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [4,3] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [6,2] => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [5,2] => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [4,2] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [7,1] => [7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [6,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [6] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 3

[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1] => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3

[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10] => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3

[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 3

[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1] => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 3

[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1,1] => [2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 3

[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1] => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => 5

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1] => [5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 3

[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1,1,1] => [3,2,2,2,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 3

[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1,1] => [3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 5

[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1,1] => [4,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 3

[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1,1] => [5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3

[1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1,1,1,1] => [4,3,2,2,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4

[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1,1,1] => [3,2,2,2,2,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 5

[1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1,1,1,1] => [4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 5

[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1,1,1] => [4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 5

[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,1,1,1,1,1,1] => [4,3,2,2,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 6

[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2,1,1] => [4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 8

[1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,2,2,2] => [4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 6

[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2,2,2] => [4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0

[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => [4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 6

[1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1,1,1] => [4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => [4,2,2,1,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1,1] => [3,2,2,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => [4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1,1,1] => [3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 6

[1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1,1] => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2,2,2] => [6,4,4] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2] => [4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1,1] => [4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

[1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => [5,4,2,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 2

[1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1] => [4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1] => [3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1,1] => [3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4

[1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1,1] => [5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0

[1,1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0,0] => [6,2,2,2,2] => [4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1

[1,1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,2] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2] => [4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0,0] => [5,3,2,2,1] => [5,4,3,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1

[1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,1] => [4,3,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2

[1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1] => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1,1,1] => [3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 3

[1,1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1,1] => [6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1] => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => 4

[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1] => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0

[1,1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2] => [6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0,0] => [5,2,2,2] => [5,4,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 2

[1,1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3,1] => [6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0,0] => [6,5,1,1] => [5,3,3,2] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 3

[1,1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1] => [3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1] => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 1

[1,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1] => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0,0] => [6,3,3] => [6,3,3] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2] => [5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1

[1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2] => [6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0,0] => [7,2,2] => [7,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2] => [4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0] => [6,5,1] => [5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0] => [5,3,1] => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1] => [3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0,0] => [6,2,1] => [3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1

[1,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1] => [5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1] => [7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0] => [7,5] => [7,5] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3

[1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0] => [4,4] => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0] => [7,3] => [7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0] => [4,3] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0] => [4,2] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0] => [7,1] => [7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0] => [6,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0] => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [7] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [6] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,1] => [1,1,0,0] => 0

[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 3

[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0

[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0] => [7] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0] => [6] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1,1] => [3,2,2,2,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 3

[1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1,1] => [2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 3

[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0] => [3,3,3] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1] => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3

[1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 3

[1,1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1] => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 3

[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1] => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => 5

[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [5,1,1,1,1,1] => [5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1] => [3,2,2,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1,1] => [3,2,2,2,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 3

[1,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => [4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 6

[1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [5,1,1,1,1,1,1] => [5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 3

[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [6,1,1,1,1,1] => [3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4

[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0] => [4,2] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0] => [6,5,1] => [5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0] => [4,3,1] => [3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0

[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2] => [4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 6

[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0] => [7,1] => [7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0] => [6,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0] => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0

[1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,2,2,2,2] => [4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 6

[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0] => [3,3] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0] => [3] => [3] => [1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => [2] => [1,0,1,0] => 0

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0] => [7,1] => [7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0] => [6,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0] => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0] => [4,4] => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

[1,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0] => [5,3,1] => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0] => [3,3,2] => [6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 0

search for individual values

searching the database for the individual values of this statistic

Description

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path

$111011010000$ has three peaks in positions

$03, 15, 26$ . The boxes below

$03$ are

$01,02,\textbf{12}$ , the boxes below

$15$ are

$\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$ , and the boxes below

$26$ are

$\textbf{23},\textbf{24},25,\textbf{34},35,45$ .
We thus obtain the four boxes in positions

$12,23,24,34$ that are below at least two peaks.

Map

Glaisher-Franklin inverse

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition

$\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition

$(n-1,\ldots,2,1)$ when cutting out

$D$ considered as a path from

$(0,0)$ to

$(n,n)$ .
In other words,

$\lambda_{i}$ is the number of down-steps before the

$(n+1-i)$ -th up-step of

$D$ .
This map is a bijection between Dyck paths of size

$n$ and partitions inside the staircase partition

$(n-1,\ldots,2,1)$ .