- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤ (values match St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 9

[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 9

[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

>>> Load all 626 entries. <<<

[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 7

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

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

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

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

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

[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 9

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

[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 8

[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 11

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

[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 7

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

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

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

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

[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 10

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

[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9

[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,4,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0] => 5

[7,2,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0] => 5

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

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

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

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

[6,4,2] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => 8

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

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

[6,3,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,1,0,0] => 8

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

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

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

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

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

[5,5,2] => [1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => 7

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

[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 9

[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 12

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

[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 11

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

[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 9

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

[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 7

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

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

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

[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 10

[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 9

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

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

[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 5

[7,5,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0] => 6

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

[7,2,2,2] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0] => 5

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

[7,2,1,1,1,1] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0] => 5

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

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

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

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

[6,4,3] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 9

[6,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => 10

[6,4,1,1,1] => [1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0] => 8

[6,3,3,1] => [1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,1,0,0,1,0,0] => 9

[6,3,2,2] => [1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0] => 8

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

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

[6,2,2,2,1] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 7

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

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

[5,5,3] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 8

[5,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 9

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

[5,4,4] => [1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 7

[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 13

[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 12

[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 11

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

[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 11

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

[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 9

[5,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 8

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

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

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

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

[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 10

[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,5,1,1] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0] => 6

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

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

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

[7,2,2,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0] => 5

[7,2,2,1,1,1] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0] => 5

[7,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0] => 5

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

[6,5,3] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => 9

[6,5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0] => 8

[6,5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0] => 8

[6,4,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => 8

[6,4,3,1] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0] => 12

[6,4,2,2] => [1,1,1,0,0,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 11

[6,4,2,1,1] => [1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => 10

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

[6,3,3,2] => [1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0] => 10

[6,3,3,1,1] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0] => 9

[6,3,2,2,1] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0] => 8

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

[6,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 7

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

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

[5,5,4] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0] => 7

[5,5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0] => 11

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

[5,5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,1,0,0] => 9

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

[5,4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0] => 10

[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 14

[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 13

[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 12

[5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,1,0,0] => 9

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

[5,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 9

[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 11

[5,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,1,0,0] => 8

[5,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0] => 8

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

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

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

[4,4,4,2] => [1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => 9

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

[4,4,3,3] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0] => 8

[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0] => 7

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

[7,5,3] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0] => 8

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

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

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

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

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

[7,2,2,2,2] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0] => 6

[7,2,2,2,1,1] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0] => 5

[7,2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0] => 5

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

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

[6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 9

[6,5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,1,0,0,0] => 11

[6,5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0] => 10

[6,5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0] => 9

[6,5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 8

[6,4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,1,0,0,0] => 10

[6,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => 14

[6,4,3,1,1] => [1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 13

[6,4,2,2,1] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0] => 12

[6,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0] => 9

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

[6,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => 9

[6,3,3,2,1] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 11

[6,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0] => 8

[6,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0] => 8

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

[6,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0] => 7

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

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

[5,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,1,0,0,0] => 9

[5,5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 13

[5,5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 12

[5,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0] => 11

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

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

[5,4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0] => 12

[5,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,1,0,0] => 11

[5,4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 11

[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15

[5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,1,0,0] => 10

[5,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 10

[5,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => 9

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

[5,3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => 9

[5,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0] => 8

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

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

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

[4,4,4,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0] => 8

[4,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 10

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

[4,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0] => 9

[4,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,5,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 9

[7,5,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0] => 8

[7,5,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0] => 8

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

[7,3,3,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0] => 8

[7,3,2,2,2] => [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0] => 7

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

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

[7,2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0] => 5

[6,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 8

[6,6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0] => 7

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

[6,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 7

[6,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0] => 10

[6,5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 14

[6,5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0] => 13

[6,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 12

[6,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,1,0,0,0,0] => 9

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

[6,4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0] => 13

[6,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0] => 12

[6,4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,1,0,1,0,0,0] => 12

[6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 16

[6,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,1,0,0,0] => 11

[6,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0] => 11

[6,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => 10

[6,3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,1,0,1,0,0,1,0,0,0] => 11

[6,3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0] => 10

[6,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => 9

[6,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0] => 8

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

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

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

[5,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,1,1,0,0,1,0,0,0] => 12

[5,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0] => 11

[5,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 11

[5,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 15

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

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

[5,5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0] => 9

[5,4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,1,1,0,1,0,0,0] => 10

[5,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 14

[5,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,1,0,1,1,0,0,0] => 9

[5,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0] => 13

[5,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 12

[5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 11

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

[5,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => 9

[5,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => 8

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

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

[4,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0] => 10

[4,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 9

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

[4,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

[7,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0] => 9

[7,6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0] => 7

[7,6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0] => 8

[7,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0] => 8

[7,5,4,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0] => 10

[7,5,2,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0] => 10

[7,5,2,1,1,1] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0] => 9

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

[7,3,3,3,1] => [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0] => 9

[7,3,3,2,2] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0] => 8

[7,3,3,2,1,1] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0] => 7

[7,3,2,2,2,1] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0] => 8

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

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

[7,2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0] => 6

[6,6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0] => 7

[6,6,4,1] => [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0] => 9

[6,6,2,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 9

[6,6,2,1,1,1] => [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0] => 8

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

[6,5,5,1] => [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0] => 8

[6,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0] => 14

[6,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 13

[6,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,1,0,1,0,0,0,0] => 13

[6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 17

[6,5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,1,1,0,0,0,0] => 12

[6,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 12

[6,5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => 11

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

[6,4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0] => 12

[6,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0] => 16

[6,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0] => 11

[6,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0] => 15

[6,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0] => 14

[6,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 13

[6,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => 10

[6,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 11

[6,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 10

[6,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => 9

[6,3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 8

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

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

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

[5,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0] => 11

[5,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => 15

[5,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0] => 10

[5,5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0] => 14

[5,5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0] => 13

[5,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,1,0,0,0] => 12

[5,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => 9

[5,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0] => 13

[5,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0] => 12

[5,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,1,0,0,0] => 11

[5,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,1,0,1,0,0,0] => 11

[5,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => 10

[5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 9

[5,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0] => 8

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

[4,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0] => 10

[4,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0] => 9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 21

[5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 15

[6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 16

[5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 11

[4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 10

[6,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 17

[5,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0] => 12

[6,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 13

[5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 9

[6,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => 11

[6,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 18

[5,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0] => 13

[6,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 14

[5,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0] => 10

[6,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 15

[5,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,1,0,0,0,0] => 11

[6,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 12

[6,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 12

[6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 19

[5,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0] => 14

[6,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0] => 15

[5,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0] => 11

[6,5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0] => 16

[5,5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 12

[6,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0] => 13

[6,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 17

[5,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0] => 13

[6,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => 14

[6,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 15

[6,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => 13

[6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 20

[5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0] => 15

[6,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 16

[5,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,1,0,1,0,0,0,0] => 12

[6,5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0] => 17

[5,5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13

[6,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => 14

[6,5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 18

[5,5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 14

[6,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 15

[6,5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 16

[6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 19

[5,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 15

[6,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0] => 16

[6,5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 17

[6,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 18

[6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 14

[6,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 21

[6,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 16

[5,5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 15

[5,5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 11

[7,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 16

[5,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => 11

[4,4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 10

[6,3,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0] => 8

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

[5,5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 9

[7,6,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 18

[6,5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0] => 8

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

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

[5,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 11

[7,6,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 19

[7,6,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => 14

[6,5,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0] => 10

[6,6,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0] => 7

[7,6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 8

[5,5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0] => 15

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

[4,4,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 8

[7,6,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 18

[6,6,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0] => 10

[4,4,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0] => 10

[5,5,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0] => 15

[4,4,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 10

[5,5,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 15

[7,6,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 18

[7,6,5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 19

[7,6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 20

[7,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0] => 9

[4,4,4,4,2] => [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0] => 9

[7,6,5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => 15

[7,6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 14

[6,5,5,4] => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0] => 10

[7,6,5,3] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0] => 14

[7,6,4,3] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0] => [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 14

[7,5,4,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0] => 14

[7,6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 9

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + q^{2} + q^{3}$

$F_{4} = q + 2\ q^{2} + q^{3} + q^{4}$

$F_{5} = q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{6} = q + 3\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{7} = q + 3\ q^{2} + 4\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

Description

The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with area sequences

$(a_1,\ldots,a_n)$ such that

$a_1 = 0, a_{k+1} \leq a_k + 1$ .
2. The generating function

$\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence

$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of

$q,t$ -Catalan numbers.

Map

zeta map

Description

The zeta map on Dyck paths.
The zeta map

$\zeta$ is a bijection on Dyck paths of semilength

$n$ .
It was defined in [1, Theorem 1], see also [2, Theorem 3.15] and sends the bistatistic (area, dinv) to the bistatistic (bounce, area). It is defined by sending a Dyck path

$D$ with corresponding area sequence

$a=(a_1,\ldots,a_n)$ to a Dyck path as follows:

First, build an intermediate Dyck path consisting of $d_1$ north steps, followed by $d_1$ east steps, followed by $d_2$ north steps and $d_2$ east steps, and so on, where $d_i$ is the number of $i-1$ 's within the sequence $a$ .
For example, given $a=(0,1,2,2,2,3,1,2)$ , we build the path
$NE\ NNEE\ NNNNEEEE\ NE.$
Next, the rectangles between two consecutive peaks are filled. Observe that such the rectangle between the $k$ th and the $(k+1)$ st peak must be filled by $d_k$ east steps and $d_{k+1}$ north steps. In the above example, the rectangle between the second and the third peak must be filled by $2$ east and $4$ north steps, the $2$ being the number of $1$ 's in $a$ , and $4$ being the number of $2$ 's. To fill such a rectangle, scan through the sequence a from left to right, and add east or north steps whenever you see a $k-1$ or $k$ , respectively. So to fill the $2\times 4$ rectangle, we look for $1$ 's and $2$ 's in the sequence and see $122212$ , so this rectangle gets filled with $ENNNEN$ .
The complete path we obtain in thus
$NENNENNNENEEENEE.$

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.