- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000937: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

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

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

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 22

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 20

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1,1] => 22

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

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

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1] => 24

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

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

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 18

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

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

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

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

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

[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => 17

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

[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2,1] => 31

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

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

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

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

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

[1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 18

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

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

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

[1,1,0,1,1,0,0,0,1,1,0,0] => [4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1] => 22

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

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

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

[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 22

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

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

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

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

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

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

[1,1,1,0,0,1,0,0,1,0,1,0] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 18

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

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

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

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

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

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

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

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

[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 20

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 23

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

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

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

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

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

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

>>> Load all 492 entries. <<<

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

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

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

[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [3,3,3,2,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [6,3,2] => 18

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

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

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

[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => 23

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

[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [5,5,4,3,2] => [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

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

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

[1,1,0,0,1,1,0,1,1,0,1,0,0,0] => [4,3,3,2,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1,1,1] => 24

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

[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [4,4,2,2,2] => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2,1] => 22

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

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

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

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

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

[1,1,0,1,0,0,1,1,1,0,1,0,0,0] => [4,3,3,3,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1,1] => 26

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

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

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

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

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

[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1,1] => 22

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

[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [4,4,4,1,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [6,5,1] => 30

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

[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1] => 24

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

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

[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 18

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

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

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

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

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

[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1,1] => 25

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

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

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

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

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

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

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

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

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

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

[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => 17

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

[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2,1] => 31

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

[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [5,5,4,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1,1] => 19

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

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

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

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

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

[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 18

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

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

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

[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1] => 22

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

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

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

[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 22

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

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

[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1,1] => 19

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 18

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 20

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

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

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

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

[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1,1] => 11

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

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

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

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

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

[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 23

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1] => 18

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

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

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

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

[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [6,3,2] => 18

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

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

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

[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => 23

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

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

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

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

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

[1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0] => [5,5,4,3,2] => [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

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

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

[1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1,1,1] => 24

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

[1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2,2] => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2,1] => 22

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

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

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

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

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

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

[1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0] => [4,3,3,3,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1,1] => 26

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

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

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

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

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

[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1,1] => 22

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

[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0] => [4,4,4,1,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [6,5,1] => 30

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

[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1] => 24

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

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

[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 18

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

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

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

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

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

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

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

[1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0] => [5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1,1] => 25

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => 17

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

[1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2,1] => 31

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

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

[1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0] => [5,5,4,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1,1] => 19

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

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

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

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

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

[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 18

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

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

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

[1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1] => 22

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

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

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

[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 22

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

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

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

[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0] => [6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1,1] => 11

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

[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0] => [6,6,4] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => 11

[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0] => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1,1] => 19

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

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

[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0] => [6,6,3] => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,1] => 23

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

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

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

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

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

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

[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 18

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 20

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

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

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

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

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

[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1,1] => 11

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

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

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

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

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

[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 23

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1] => 18

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

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

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

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

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

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

[1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => 23

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

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

[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0] => [5,5,4,3,2] => [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1,1] => 22

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

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

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

[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 18

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

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

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

[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 22

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

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

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

[1,1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0,0] => [6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1,1] => 11

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

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

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

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

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

[1,1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 18

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 20

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,3,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0] => [7,2,1] => 11

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

[1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0] => [5,5,5,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0] => [7,1] => 11

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

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

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

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

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

[1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => [4,3,3,3,3,3] => [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] => 20

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0,0] => [5,5,4,3,2] => [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => 18

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

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

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

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

[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] => [1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 18

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

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

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

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

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

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

[1,1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [4,3,3,3,3,3] => [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] => 20

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

Description

The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation

$S^{(2,2)}$ are

$2$ on the conjugacy classes

$(4)$ and

$(2,2)$ ,

$0$ on the conjugacy classes

$(3,1)$ and

$(1,1,1,1)$ , and

$-1$ on the conjugacy class

$(2,1,1)$ . Therefore, the statistic on the partition

$(2,2)$ is

$2$ .

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

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