Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
St000278: Integer partitions ⟶ ℤ
Values
[1,0] => [] => 1
[1,0,1,0] => [1] => 1
[1,1,0,0] => [] => 1
[1,0,1,0,1,0] => [2,1] => 2
[1,0,1,1,0,0] => [1,1] => 1
[1,1,0,0,1,0] => [2] => 1
[1,1,0,1,0,0] => [1] => 1
[1,1,1,0,0,0] => [] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => 6
[1,0,1,0,1,1,0,0] => [2,2,1] => 3
[1,0,1,1,0,0,1,0] => [3,1,1] => 3
[1,0,1,1,0,1,0,0] => [2,1,1] => 3
[1,0,1,1,1,0,0,0] => [1,1,1] => 1
[1,1,0,0,1,0,1,0] => [3,2] => 2
[1,1,0,0,1,1,0,0] => [2,2] => 1
[1,1,0,1,0,0,1,0] => [3,1] => 2
[1,1,0,1,0,1,0,0] => [2,1] => 2
[1,1,0,1,1,0,0,0] => [1,1] => 1
[1,1,1,0,0,0,1,0] => [3] => 1
[1,1,1,0,0,1,0,0] => [2] => 1
[1,1,1,0,1,0,0,0] => [1] => 1
[1,1,1,1,0,0,0,0] => [] => 1
[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 24
[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 12
[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 12
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 12
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 4
[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 12
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 6
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 12
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 12
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 6
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 4
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 4
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 6
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 3
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 6
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 3
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 6
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 6
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 3
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,2] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,2] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [2,1] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => 1
[1,1,1,1,1,0,0,0,0,0] => [] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 20
[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 60
[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 30
[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 20
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 20
[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 20
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 5
[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 30
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 10
[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 30
[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 60
[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 60
[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 30
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 30
[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 30
[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 30
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 10
[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 20
[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 10
[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 20
[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 20
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 10
[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 20
[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 20
[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => 20
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => 10
[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 5
[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => 5
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => 5
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => 5
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [4,3,3,2] => 12
[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 6
[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => 12
[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => 12
[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => 6
>>> Load all 875 entries. <<<[1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => 4
[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => 4
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [4,4,3,1] => 12
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,3,1] => 12
[1,1,0,1,0,0,1,1,0,1,0,0] => [4,3,3,1] => 12
[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => 4
[1,1,0,1,0,1,0,0,1,0,1,0] => [5,4,2,1] => 24
[1,1,0,1,0,1,0,0,1,1,0,0] => [4,4,2,1] => 12
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,3,2,1] => 24
[1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => 24
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,3,2,1] => 12
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,2,2,1] => 12
[1,1,0,1,0,1,1,0,0,1,0,0] => [4,2,2,1] => 12
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => 12
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => 4
[1,1,0,1,1,0,0,0,1,0,1,0] => [5,4,1,1] => 12
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,4,1,1] => 6
[1,1,0,1,1,0,0,1,0,0,1,0] => [5,3,1,1] => 12
[1,1,0,1,1,0,0,1,0,1,0,0] => [4,3,1,1] => 12
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => 6
[1,1,0,1,1,0,1,0,0,0,1,0] => [5,2,1,1] => 12
[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => 12
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => 12
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => 6
[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => 4
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => 4
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => 4
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => 4
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 6
[1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 3
[1,1,1,0,0,0,1,1,0,1,0,0] => [4,3,3] => 3
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [5,4,2] => 6
[1,1,1,0,0,1,0,0,1,1,0,0] => [4,4,2] => 3
[1,1,1,0,0,1,0,1,0,0,1,0] => [5,3,2] => 6
[1,1,1,0,0,1,0,1,0,1,0,0] => [4,3,2] => 6
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [5,2,2] => 3
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => 3
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => 3
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [5,4,1] => 6
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,4,1] => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => [5,3,1] => 6
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => 6
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 3
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => 6
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 6
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => 6
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => 3
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 3
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 3
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => 3
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [] => 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2,1] => 30
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => 6
[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1,1] => 90
[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2,1,1] => 60
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1,1] => 60
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1,1] => 15
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1,1] => 20
[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1,1] => 120
[1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 60
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 60
[1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1,1,1] => 60
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => 60
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => 20
[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1,1,1] => 15
[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1,1] => 30
[1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1,1,1] => 30
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1,1] => 15
[1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1,1,1] => 30
[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1,1] => 30
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1,1,1] => 30
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1,1] => 30
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1,1] => 15
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1,1] => 6
[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1,1,1] => 6
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1,1] => 6
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1,1] => 6
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [3,3,2,2,2] => 10
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [4,2,2,2,2] => 5
[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2] => 5
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2,1] => 20
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2,1] => 60
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1] => 30
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [5,2,2,2,1] => 20
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2,1] => 20
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1] => 20
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1] => 5
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [4,3,3,1,1] => 30
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1] => 10
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [4,4,2,1,1] => 30
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,1] => 60
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1] => 60
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1] => 30
[1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1,1] => 30
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1] => 30
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1,1] => 30
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1] => 30
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1] => 10
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1,1] => 20
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1,1] => 10
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1,1] => 20
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1] => 20
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1,1] => 20
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1] => 10
[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1,1] => 20
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1] => 20
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1,1] => 20
[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1] => 20
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => 10
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1] => 5
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1,1] => 5
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1] => 5
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => 5
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [4,3,3,2] => 12
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2] => 4
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [4,4,2,2] => 6
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2] => 12
[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2] => 12
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2] => 6
[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [6,2,2,2] => 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [5,2,2,2] => 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2] => 4
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2] => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2] => 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [4,4,3,1] => 12
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [5,3,3,1] => 12
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [4,3,3,1] => 12
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1] => 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1] => 24
[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [4,4,2,1] => 12
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1] => 24
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1] => 24
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1] => 24
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1] => 12
[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1] => 12
[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1] => 12
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1] => 12
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1] => 12
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => 4
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [5,5,1,1] => 6
[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [6,4,1,1] => 12
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1] => 12
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1] => 6
[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1] => 12
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1] => 12
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1] => 12
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1] => 6
[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1] => 12
[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1] => 12
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1] => 12
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => 12
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => 6
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1] => 4
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1] => 4
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => 4
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => 4
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,4,4] => 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => 6
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [4,4,3] => 3
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [6,3,3] => 3
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [5,3,3] => 3
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,3] => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [3,3,3] => 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [5,5,2] => 3
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [6,4,2] => 6
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [5,4,2] => 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,4,2] => 3
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [6,3,2] => 6
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [5,3,2] => 6
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [4,3,2] => 6
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [3,3,2] => 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [6,2,2] => 3
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [5,2,2] => 3
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,2,2] => 3
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [6,5,1] => 6
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [5,5,1] => 3
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [6,4,1] => 6
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [5,4,1] => 6
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [4,4,1] => 3
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [6,3,1] => 6
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [5,3,1] => 6
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [4,3,1] => 6
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [6,2,1] => 6
[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [5,2,1] => 6
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 6
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => 6
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => 3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,1,1] => 3
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 3
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 3
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => 3
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [6,4] => 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,4] => 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [4,4] => 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [6,3] => 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,3] => 2
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [4,3] => 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [6,2] => 2
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,2] => 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [4,2] => 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [6,1] => 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => 2
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,1,1] => 21
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1,1] => 140
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1,1] => 35
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1,1,1] => 105
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1,1,1] => 105
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1,1] => 105
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => 35
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1,1,1] => 42
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1,1,1] => 21
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1,1,1,1] => 42
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1,1,1] => 42
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1,1] => 42
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => 21
[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1,1] => 7
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1,1,1] => 7
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1,1,1] => 7
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1,1] => 7
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 7
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2] => 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2,1] => 30
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,1] => 6
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1,1] => 90
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,1,1] => 60
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1] => 60
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1] => 15
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1,1] => 20
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1,1] => 120
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1,1] => 60
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1,1] => 60
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1,1] => 60
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1] => 60
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1] => 20
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1,1,1] => 15
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1,1,1] => 30
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1,1] => 30
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1,1] => 15
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1,1,1] => 30
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1,1,1] => 30
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1,1] => 30
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1] => 30
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1] => 15
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,1,1,1,1] => 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1] => 6
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1,1] => 6
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1,1] => 6
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1] => 6
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1] => 6
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,2] => 10
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,2] => 5
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2] => 5
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2] => 1
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,1] => 20
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2,1] => 60
[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1] => 30
[1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2,1] => 20
[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,1] => 20
[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => 20
[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1] => 5
[1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0] => [4,3,3,1,1] => 30
[1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1] => 10
[1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0] => [4,4,2,1,1] => 30
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0] => [5,3,2,1,1] => 60
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1] => 60
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1] => 30
[1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1,1] => 30
[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1] => 30
[1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1] => 30
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1] => 30
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1] => 10
[1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0] => [5,4,1,1,1] => 20
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1,1] => 10
[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1,1] => 20
[1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1,1] => 20
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1] => 20
[1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1] => 10
[1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,1,1] => 20
[1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1,1] => 20
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1,1] => 20
[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1] => 20
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1] => 20
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1] => 10
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,1,1,1] => 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1] => 5
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1] => 5
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1] => 5
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1] => 5
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => 5
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3] => 1
[1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2] => 12
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2] => 4
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2] => 6
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2] => 12
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2] => 12
[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2] => 6
[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0] => [6,2,2,2] => 4
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2] => 4
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2] => 4
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2] => 4
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2] => 1
[1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0] => [4,4,3,1] => 12
[1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0] => [5,3,3,1] => 12
[1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => [4,3,3,1] => 12
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1] => 4
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0] => [5,4,2,1] => 24
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0] => [4,4,2,1] => 12
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0] => [6,3,2,1] => 24
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0] => [5,3,2,1] => 24
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1] => 24
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1] => 12
[1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0] => [7,2,2,1] => 12
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1] => 12
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1] => 12
[1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1] => 12
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1] => 12
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1] => 4
[1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0] => [5,5,1,1] => 6
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0] => [6,4,1,1] => 12
[1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0] => [5,4,1,1] => 12
[1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1] => 6
[1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0] => [7,3,1,1] => 12
[1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1] => 12
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1] => 12
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1] => 12
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1] => 6
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,1] => 12
[1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1] => 12
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1] => 12
[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1] => 12
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1] => 12
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1] => 6
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [7,1,1,1] => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1] => 4
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1] => 4
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1] => 4
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => 4
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => 4
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0] => [4,4,4] => 1
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0] => [5,4,3] => 6
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0] => [4,4,3] => 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [6,3,3] => 3
[1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0] => [5,3,3] => 3
[1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0] => [4,3,3] => 3
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0] => [3,3,3] => 1
[1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0] => [5,5,2] => 3
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0] => [6,4,2] => 6
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0] => [5,4,2] => 6
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0] => [4,4,2] => 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0] => [7,3,2] => 6
[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [6,3,2] => 6
[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [5,3,2] => 6
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [4,3,2] => 6
[1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0] => [3,3,2] => 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => [7,2,2] => 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0] => [6,2,2] => 3
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0] => [5,2,2] => 3
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0] => [4,2,2] => 3
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [3,2,2] => 3
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [6,5,1] => 6
[1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0] => [5,5,1] => 3
[1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0] => [7,4,1] => 6
[1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0] => [6,4,1] => 6
[1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0] => [5,4,1] => 6
[1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0] => [4,4,1] => 3
[1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0] => [7,3,1] => 6
[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [6,3,1] => 6
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [5,3,1] => 6
[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [4,3,1] => 6
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [7,2,1] => 6
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0] => [6,2,1] => 6
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [5,2,1] => 6
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => 6
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => 6
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [7,1,1] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [6,1,1] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0] => [5,1,1] => 3
[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => 3
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => 3
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => 3
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [6,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => [7,5] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [6,5] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0] => [5,5] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [7,4] => 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0] => [6,4] => 2
[1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0] => [5,4] => 2
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0] => [4,4] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [7,3] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [6,3] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0] => [5,3] => 2
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [4,3] => 2
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [7,2] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [6,2] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [5,2] => 2
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [4,2] => 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [7,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [6,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [5,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [6] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8] => 1
[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] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [9] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [8,1] => 2
[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] => 8
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [9,1] => 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [8,2] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [8,1,1] => 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] => 8
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1] => 28
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1,1] => 9
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0] => [11] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1] => 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0] => [9,2] => 2
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,1] => 3
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0] => [8,3] => 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0] => [8,2,1] => 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,1,1] => 4
[1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1,1,1,1,1] => 8
[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] => 56
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1,1] => 9
[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] => 56
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1,1] => 36
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0] => [11,1] => 2
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0] => [10,2] => 2
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1,1] => 3
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0] => [9,3] => 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0] => [9,2,1] => 6
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,1,1] => 4
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0] => [8,4] => 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0] => [8,3,1] => 6
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0] => [8,2,2] => 3
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0] => [8,2,1,1] => 12
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,1,1,1] => 5
[1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1,1,1,1] => 8
[1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1,1,1,1,1,1] => 56
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [4,1,1,1,1,1,1,1,1] => 9
[1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1,1,1,1,1] => 28
[1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1,1,1,1] => 168
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1,1,1] => 72
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1,1,1] => 10
[1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1,1,1,1] => 70
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,1,1,1,1,1,1] => 84
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1,1,1] => 45
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1,1,1,1] => 11
[] => [] => 1
[1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,1,1] => 21
[1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1,1,1] => 105
[1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => 35
[1,1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1,1,1,1] => 42
[1,1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1,1,1,1] => 21
[1,1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1,1,1,1,1] => 42
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1,1] => 42
[1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => 21
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1,1,1] => 7
[1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1,1,1,1] => 7
[1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1,1] => 7
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 1
[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
[1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,1] => 6
[1,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1,1] => 15
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1,1,1] => 120
[1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2,1,1,1] => 60
[1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => 60
[1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1] => 20
[1,1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1,1,1,1] => 15
[1,1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1,1,1] => 30
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [6,2,1,1,1,1] => 30
[1,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1] => 30
[1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1] => 15
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1,1,1,1] => 6
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1,1] => 6
[1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1,1] => 6
[1,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1,1,1] => 6
[1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1] => 6
[1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => 6
[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
[1,1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0,0] => [4,2,2,2,2] => 5
[1,1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,2] => 5
[1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2] => 1
[1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0] => [3,3,3,2,1] => 20
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0] => [4,3,2,2,1] => 60
[1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,1] => 20
[1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1] => 5
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1,1] => 60
[1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1] => 30
[1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1] => 10
[1,1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0,0] => [5,4,1,1,1] => 20
[1,1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1,1,1] => 10
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1,1,1] => 20
[1,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1,1] => 20
[1,1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1,1] => 10
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0,0] => [7,2,1,1,1] => 20
[1,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1,1,1] => 20
[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1] => 20
[1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1] => 10
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1,1,1] => 5
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1] => 5
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1] => 5
[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1] => 5
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0] => [4,4,2,2] => 6
[1,1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2] => 6
[1,1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0,0] => [5,2,2,2] => 4
[1,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0] => [4,3,3,1] => 12
[1,1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3,1] => 4
[1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2,1] => 24
[1,1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0,0] => [4,4,2,1] => 12
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0] => [6,3,2,1] => 24
[1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2,1] => 24
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1] => 24
[1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2,1] => 12
[1,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2,1] => 12
[1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1] => 12
[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1] => 4
[1,1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1] => 12
[1,1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1] => 6
[1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1,1] => 12
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1] => 12
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1] => 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1,1] => 4
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1] => 4
[1,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1] => 4
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1] => 4
[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1] => 4
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1] => 4
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0] => [4,4,4] => 1
[1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0] => [5,4,3] => 6
[1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0] => [4,4,3] => 3
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0,0] => [6,3,3] => 3
[1,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0,0] => [4,3,3] => 3
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3] => 1
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0] => [6,4,2] => 6
[1,1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2] => 6
[1,1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0,0] => [4,4,2] => 3
[1,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2] => 6
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2] => 6
[1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2] => 3
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0,0] => [7,2,2] => 3
[1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2] => 3
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2] => 3
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0] => [6,5,1] => 6
[1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0] => [5,4,1] => 6
[1,1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1] => 3
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1] => 6
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0] => [5,3,1] => 6
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1] => 6
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1] => 3
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0] => [7,2,1] => 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0,0] => [6,2,1] => 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1] => 6
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1] => 6
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => 6
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1] => 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1] => 3
[1,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1] => 3
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => 3
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => 3
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0] => [6,6] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0] => [7,5] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0] => [5,5] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0] => [4,4] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0] => [7,3] => 2
[1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0] => [4,3] => 2
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0] => [7,2] => 2
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0] => [4,2] => 2
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => 2
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0] => [7,1] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0] => [6,1] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0] => [5,1] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0] => [4,1] => 2
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => 2
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [7] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [6] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => 3
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => 2
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0] => [8] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0] => [9] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0] => [8,1] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => 1
[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
[1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0] => [7] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0] => [6] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0] => [5] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0] => [9,1] => 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0,0] => [8,2,1] => 6
[1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1,1] => 56
[1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0] => [3,3] => 1
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0] => [3,3,3] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0] => [4,4,4] => 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0] => [] => 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => 3
[1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1,1] => 8
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0] => [8,1,1] => 3
[1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1,1,1] => 8
[1,1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1,1] => 28
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2] => 6
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0] => [6,6] => 1
[1,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => 6
[1,1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1] => 6
[1,1,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [4,1,1,1,1,1] => 6
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [5,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1] => 24
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0,0] => [7,2,1] => 6
[1,1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1] => 42
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0,0,0] => [8,2,1] => 6
[1,1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [3,2,1,1,1,1,1,1] => 56
[1,1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [4,1,1,1,1,1,1,1] => 8
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [5,1,1,1,1,1,1] => 7
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [6,1,1,1,1,1] => 6
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [7,1,1,1,1] => 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0,0] => [8,1,1,1] => 4
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0] => [5,3,2,1] => 24
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0] => [5,4,2,1] => 24
[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0] => [4,2] => 2
[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] => 6
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0,0] => [6,3,2,1] => 24
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1,1,1] => 120
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,2,1] => 60
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1,1] => 60
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0] => [6,5,1] => 6
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0] => [5,4,1] => 6
[1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0] => [4,3,1] => 6
[1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2] => 1
[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
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0] => [7,1] => 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0] => [6,1] => 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0] => [5,1] => 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0] => [4,1] => 2
[1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1] => 4
[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] => 5
[1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 7
[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] => 3
[1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,1] => 12
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1] => 3
[1,1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1] => 6
[1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1] => 12
[1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2] => 1
[1,1,1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,2,1,1,1] => 140
[1,1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0] => [3,3] => 1
[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
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0,0] => [8,1] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0] => [7,1] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0] => [6,1] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0] => [5,1] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0] => [4,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0] => [5,5] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0] => [4,4] => 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0] => [5,3,1] => 6
[1,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,1] => 4
[1,1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0] => [3,3,2] => 3
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Generating function
click to show known generating functions
Search the OEIS for these generating functions
Search the Online Encyclopedia of Integer
Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
4,1 7,3,3,0,0,1 11,6,8,4,0,6,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1
$F_{1} = q$
$F_{2} = 2\ q$
$F_{3} = 4\ q + q^{2}$
$F_{4} = 7\ q + 3\ q^{2} + 3\ q^{3} + q^{6}$
$F_{5} = 11\ q + 6\ q^{2} + 8\ q^{3} + 4\ q^{4} + 6\ q^{6} + 6\ q^{12} + q^{24}$
Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
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 $(n-1,\ldots,2,1)$.
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 $(n-1,\ldots,2,1)$.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!