Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
St000913: 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] => 1
[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] => 2
[1,0,1,0,1,1,0,0] => [2,2,1] => 1
[1,0,1,1,0,0,1,0] => [3,1,1] => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => 1
[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] => 1
[1,1,0,1,0,1,0,0] => [2,1] => 1
[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] => 28
[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 5
[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 9
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 3
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 1
[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 10
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 5
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 1
[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] => 28
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 5
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 9
[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] => 10
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 5
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 1
[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] => 10
[1,1,1,0,0,0,1,1,0,0] => [3,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [4,2] => 5
[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] => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => 2
[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
[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 3516
[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 518
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 502
[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => 126
[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 14
[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 855
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 137
[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => 203
[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 56
[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 9
[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 44
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 14
[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 982
[1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 159
[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 145
[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 42
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => 349
[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 61
[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 89
[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 28
[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 25
[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 9
[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 99
[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 19
[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 27
[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 10
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 12
[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => 1
>>> Load all 940 entries. <<<[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 4
[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 3516
[1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 518
[1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 502
[1,1,0,0,1,0,1,1,0,1,0,0] => [4,3,3,2] => 126
[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 14
[1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 855
[1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 137
[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => 203
[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => 56
[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => 9
[1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 44
[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => 14
[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
[1,1,0,1,0,0,1,0,1,0,1,0] => [5,4,3,1] => 982
[1,1,0,1,0,0,1,0,1,1,0,0] => [4,4,3,1] => 159
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,3,1] => 145
[1,1,0,1,0,0,1,1,0,1,0,0] => [4,3,3,1] => 42
[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => 5
[1,1,0,1,0,1,0,0,1,0,1,0] => [5,4,2,1] => 349
[1,1,0,1,0,1,0,0,1,1,0,0] => [4,4,2,1] => 61
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,3,2,1] => 89
[1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => 28
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,3,2,1] => 5
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,2,2,1] => 25
[1,1,0,1,0,1,1,0,0,1,0,0] => [4,2,2,1] => 9
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [5,4,1,1] => 99
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,4,1,1] => 19
[1,1,0,1,1,0,0,1,0,0,1,0] => [5,3,1,1] => 27
[1,1,0,1,1,0,0,1,0,1,0,0] => [4,3,1,1] => 10
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => 2
[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] => 5
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => 1
[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] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => 1
[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] => 982
[1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 159
[1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 145
[1,1,1,0,0,0,1,1,0,1,0,0] => [4,3,3] => 42
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 5
[1,1,1,0,0,1,0,0,1,0,1,0] => [5,4,2] => 349
[1,1,1,0,0,1,0,0,1,1,0,0] => [4,4,2] => 61
[1,1,1,0,0,1,0,1,0,0,1,0] => [5,3,2] => 89
[1,1,1,0,0,1,0,1,0,1,0,0] => [4,3,2] => 28
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => 5
[1,1,1,0,0,1,1,0,0,0,1,0] => [5,2,2] => 25
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => 9
[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] => 99
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,4,1] => 19
[1,1,1,0,1,0,0,1,0,0,1,0] => [5,3,1] => 27
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => 10
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => 12
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 5
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => 2
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 4
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => 1
[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] => 99
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 19
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => 27
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => 10
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => 12
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => 5
[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] => 4
[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] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => 1
[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] => 4
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => 2
[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] => 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1,1] => 9
[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2,1,1] => 14
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1,1] => 4
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1,1] => 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1,1] => 5
[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1,1] => 28
[1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1,1] => 25
[1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1,1,1] => 9
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => 3
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1,1,1] => 19
[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1,1] => 27
[1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1,1,1] => 10
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1,1] => 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1,1,1] => 37
[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1,1] => 12
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1,1,1] => 5
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1,1] => 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1,1] => 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1,1] => 11
[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1,1,1] => 4
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1,1] => 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1,1] => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => 1
[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] => 14
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [4,2,2,2,2] => 20
[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
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1] => 3516
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [4,4,3,2,1] => 518
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [5,3,3,2,1] => 502
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [4,3,3,2,1] => 126
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2,1] => 14
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,2,1] => 855
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [4,4,2,2,1] => 137
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2,1] => 203
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2,1] => 56
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1] => 9
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [5,2,2,2,1] => 44
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2,1] => 14
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1] => 4
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1] => 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,1] => 982
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [4,4,3,1,1] => 159
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [5,3,3,1,1] => 145
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [4,3,3,1,1] => 42
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1] => 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,1] => 349
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [4,4,2,1,1] => 61
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,1] => 89
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1] => 28
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1] => 5
[1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1,1] => 85
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1,1] => 25
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1,1] => 9
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1] => 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1] => 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1,1] => 99
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1,1] => 19
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1,1] => 97
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1,1] => 27
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1,1] => 10
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1] => 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1,1] => 37
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1,1] => 12
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1,1] => 5
[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1] => 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1] => 11
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1,1] => 4
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1] => 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => 1
[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] => 14
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2] => 3516
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [4,4,3,2] => 518
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [5,3,3,2] => 502
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [4,3,3,2] => 126
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2] => 14
[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [5,4,2,2] => 855
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [4,4,2,2] => 137
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2] => 203
[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2] => 56
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2] => 9
[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [6,2,2,2] => 163
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [5,2,2,2] => 44
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2] => 14
[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
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1] => 982
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [4,4,3,1] => 159
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [5,3,3,1] => 145
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [4,3,3,1] => 42
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1] => 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1] => 349
[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [4,4,2,1] => 61
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1] => 341
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1] => 89
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1] => 28
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1] => 5
[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [6,2,2,1] => 85
[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1] => 25
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1] => 9
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1] => 3
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [5,5,1,1] => 188
[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [6,4,1,1] => 384
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1] => 99
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1] => 19
[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [6,3,1,1] => 97
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1] => 27
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1] => 10
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1] => 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [6,2,1,1] => 37
[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] => 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1] => 11
[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] => 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => 1
[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] => 296
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => 982
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [4,4,3] => 159
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [6,3,3] => 626
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [5,3,3] => 145
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,3] => 42
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [3,3,3] => 5
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [5,5,2] => 740
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [6,4,2] => 1500
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [5,4,2] => 349
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,4,2] => 61
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [6,3,2] => 341
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [5,3,2] => 89
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [4,3,2] => 28
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [3,3,2] => 5
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [6,2,2] => 85
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [5,2,2] => 25
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,2,2] => 9
[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] => 1407
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [5,5,1] => 188
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [6,4,1] => 384
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [5,4,1] => 99
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [4,4,1] => 19
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [6,3,1] => 97
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [5,3,1] => 27
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [4,3,1] => 10
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [6,2,1] => 37
[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [5,2,1] => 12
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,1,1] => 11
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 4
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => 1
[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] => 1407
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 188
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [6,4] => 384
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,4] => 99
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [4,4] => 19
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [6,3] => 97
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,3] => 27
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [4,3] => 10
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [6,2] => 37
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,2] => 12
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [4,2] => 5
[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] => 11
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => 4
[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] => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => 1
[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] => 11
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => 4
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => 2
[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] => 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1,1] => 4
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1,1] => 1
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1,1,1] => 5
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1,1,1] => 9
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1,1] => 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => 1
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1,1,1] => 10
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1,1,1] => 2
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1,1,1,1] => 12
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1,1,1] => 5
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1,1] => 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1,1] => 11
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1,1,1] => 4
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1,1,1] => 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1,1] => 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 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] => 5
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,1] => 1
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1,1] => 9
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,1,1] => 14
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1] => 4
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1] => 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1,1] => 5
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1,1] => 28
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1,1] => 5
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1,1] => 25
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1,1] => 9
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1] => 3
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1] => 1
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1,1,1] => 19
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1,1,1] => 27
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1,1] => 10
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1,1] => 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1,1,1] => 37
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1,1,1] => 12
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1,1] => 5
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1] => 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1] => 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,1,1,1,1] => 33
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1,1] => 11
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1,1] => 4
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1,1] => 2
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1] => 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1] => 1
[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] => 14
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,2] => 20
[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
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2,1] => 3516
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [4,4,3,2,1] => 518
[1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0] => [5,3,3,2,1] => 502
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2,1] => 126
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,1] => 14
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0] => [5,4,2,2,1] => 855
[1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2,1] => 137
[1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2,1] => 203
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2,1] => 56
[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1] => 9
[1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2,1] => 44
[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,1] => 14
[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => 4
[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1] => 1
[1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0] => [5,4,3,1,1] => 982
[1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0] => [4,4,3,1,1] => 159
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0] => [5,3,3,1,1] => 145
[1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0] => [4,3,3,1,1] => 42
[1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1] => 5
[1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0] => [5,4,2,1,1] => 349
[1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0] => [4,4,2,1,1] => 61
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0] => [5,3,2,1,1] => 89
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1] => 28
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1] => 5
[1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1,1] => 85
[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1,1] => 25
[1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1] => 9
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1] => 3
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1] => 1
[1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0] => [5,4,1,1,1] => 99
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1,1] => 19
[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1,1] => 97
[1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1,1] => 27
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1] => 10
[1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1] => 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,1,1] => 123
[1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1,1] => 37
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1,1] => 12
[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1] => 5
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1] => 2
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,1,1,1] => 33
[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1,1] => 11
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1,1] => 4
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1] => 2
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1] => 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => 1
[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] => 14
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2] => 3516
[1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0] => [4,4,3,2] => 518
[1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0] => [5,3,3,2] => 502
[1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2] => 126
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2] => 14
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0] => [5,4,2,2] => 855
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2] => 137
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2] => 203
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2] => 56
[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2] => 9
[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0] => [6,2,2,2] => 163
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2] => 44
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2] => 14
[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
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0] => [5,4,3,1] => 982
[1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0] => [4,4,3,1] => 159
[1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0] => [5,3,3,1] => 145
[1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => [4,3,3,1] => 42
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1] => 5
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0] => [5,4,2,1] => 349
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0] => [4,4,2,1] => 61
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0] => [6,3,2,1] => 341
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0] => [5,3,2,1] => 89
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1] => 28
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1] => 5
[1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0] => [7,2,2,1] => 308
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0] => [6,2,2,1] => 85
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1] => 25
[1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1] => 9
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1] => 3
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1] => 1
[1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0] => [5,5,1,1] => 188
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0] => [6,4,1,1] => 384
[1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0] => [5,4,1,1] => 99
[1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1] => 19
[1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0] => [7,3,1,1] => 351
[1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,1] => 97
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1] => 27
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1] => 10
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1] => 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,1] => 123
[1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,1] => 37
[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] => 5
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1] => 2
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1] => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [7,1,1,1] => 33
[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0] => [6,1,1,1] => 11
[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] => 2
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => 1
[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] => 296
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0] => [5,4,3] => 982
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0] => [4,4,3] => 159
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [6,3,3] => 626
[1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0] => [5,3,3] => 145
[1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0] => [4,3,3] => 42
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0] => [3,3,3] => 5
[1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0] => [5,5,2] => 740
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0] => [6,4,2] => 1500
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0] => [5,4,2] => 349
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0] => [4,4,2] => 61
[1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0] => [7,3,2] => 1329
[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [6,3,2] => 341
[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [5,3,2] => 89
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [4,3,2] => 28
[1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0] => [3,3,2] => 5
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => [7,2,2] => 308
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0] => [6,2,2] => 85
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0] => [5,2,2] => 25
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0] => [4,2,2] => 9
[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] => 1407
[1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0] => [5,5,1] => 188
[1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0] => [7,4,1] => 1551
[1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0] => [6,4,1] => 384
[1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0] => [5,4,1] => 99
[1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0] => [4,4,1] => 19
[1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0] => [7,3,1] => 351
[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [6,3,1] => 97
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [5,3,1] => 27
[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [4,3,1] => 10
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [7,2,1] => 123
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0] => [6,2,1] => 37
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [5,2,1] => 12
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => 5
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => 2
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [7,1,1] => 33
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [6,1,1] => 11
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0] => [5,1,1] => 4
[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => 1
[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] => 3533
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => [7,5] => 6009
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [6,5] => 1407
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0] => [5,5] => 188
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [7,4] => 1551
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0] => [6,4] => 384
[1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0] => [5,4] => 99
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0] => [4,4] => 19
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [7,3] => 351
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [6,3] => 97
[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0] => [5,3] => 27
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [4,3] => 10
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [7,2] => 123
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [6,2] => 37
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [5,2] => 12
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [4,2] => 5
[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] => 33
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [6,1] => 11
[1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [5,1] => 4
[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] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 33
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [6] => 11
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => 4
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => 2
[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] => 116
[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] => 435
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [8,1] => 116
[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] => 1
[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] => 1832
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [9,1] => 435
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [8,2] => 474
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [8,1,1] => 116
[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] => 1
[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] => 1
[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] => 1
[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] => 8167
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1] => 1832
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0] => [9,2] => 1907
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,1] => 435
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0] => [8,3] => 1470
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0] => [8,2,1] => 474
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,1,1] => 116
[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] => 2
[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] => 2
[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] => 1
[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] => 1
[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] => 1
[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] => 1
[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] => 8167
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0] => [10,2] => 8593
[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] => 1832
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0] => [9,3] => 6314
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0] => [9,2,1] => 1907
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,1,1] => 435
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0] => [8,4] => 7084
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0] => [8,3,1] => 1470
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0] => [8,2,2] => 1285
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0] => [8,2,1,1] => 474
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,1,1,1] => 116
[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] => 4
[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] => 5
[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] => 2
[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] => 2
[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] => 3
[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] => 2
[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] => 1
[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] => 1
[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] => 1
[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] => 1
[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] => 1
[] => [] => 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] => 1
[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] => 3
[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] => 1
[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] => 10
[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] => 2
[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] => 12
[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] => 2
[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] => 1
[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] => 11
[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] => 2
[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] => 1
[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] => 1
[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] => 1
[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
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1,1,1] => 28
[1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2,1,1,1] => 5
[1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [5,2,2,1,1,1] => 25
[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
[1,1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1,1,1,1] => 19
[1,1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1,1,1] => 10
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [6,2,1,1,1,1] => 37
[1,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1] => 2
[1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1] => 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1,1,1,1] => 33
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1,1] => 11
[1,1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1,1] => 4
[1,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1,1,1] => 2
[1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1] => 1
[1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => 1
[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] => 20
[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
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [5,4,3,2,1] => 3516
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0] => [4,4,3,2,1] => 518
[1,1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0,0] => [5,3,3,2,1] => 502
[1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0] => [4,3,3,2,1] => 126
[1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0] => [3,3,3,2,1] => 14
[1,1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0,0] => [5,4,2,2,1] => 855
[1,1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0,0] => [5,3,2,2,1] => 203
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0] => [4,3,2,2,1] => 56
[1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,1] => 4
[1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1] => 1
[1,1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0] => [4,4,3,1,1] => 159
[1,1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2,1,1] => 349
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1,1] => 28
[1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1] => 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
[1,1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0,0] => [5,4,1,1,1] => 99
[1,1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1,1,1] => 19
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1,1,1] => 97
[1,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1,1] => 10
[1,1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1,1] => 2
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0,0] => [7,2,1,1,1] => 123
[1,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1,1,1] => 12
[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1] => 2
[1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1] => 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1,1,1] => 33
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1,1] => 11
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1,1] => 4
[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1] => 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => 1
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0] => [5,4,3,2] => 3516
[1,1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0,0] => [4,4,3,2] => 518
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0] => [4,4,2,2] => 137
[1,1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2] => 9
[1,1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0,0] => [5,2,2,2] => 44
[1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0] => [5,4,3,1] => 982
[1,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0] => [4,3,3,1] => 42
[1,1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3,1] => 5
[1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2,1] => 349
[1,1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0,0] => [4,4,2,1] => 61
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0] => [6,3,2,1] => 341
[1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2,1] => 89
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1] => 28
[1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2,1] => 5
[1,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2,1] => 9
[1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1] => 3
[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1] => 1
[1,1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1] => 10
[1,1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1] => 2
[1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1,1] => 5
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1] => 2
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1] => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1,1] => 33
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1,1] => 11
[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] => 2
[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1] => 1
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1] => 1
[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] => 296
[1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0] => [5,4,3] => 982
[1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0] => [4,4,3] => 159
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0,0] => [6,3,3] => 626
[1,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0,0] => [4,3,3] => 42
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3] => 5
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0] => [6,4,2] => 1500
[1,1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2] => 349
[1,1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0,0] => [4,4,2] => 61
[1,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2] => 89
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2] => 28
[1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2] => 5
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0,0] => [7,2,2] => 308
[1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2] => 9
[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] => 1407
[1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0] => [5,4,1] => 99
[1,1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1] => 19
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1] => 97
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0] => [5,3,1] => 27
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1] => 10
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1] => 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0] => [7,2,1] => 123
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0,0] => [6,2,1] => 37
[1,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1] => 12
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1] => 5
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => 2
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0] => [7,1,1] => 33
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1] => 11
[1,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1] => 2
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => 1
[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] => 3533
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0] => [7,5] => 6009
[1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0] => [5,5] => 188
[1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0] => [4,4] => 19
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0] => [7,3] => 351
[1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0] => [4,3] => 10
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0] => [7,2] => 123
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0] => [4,2] => 5
[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] => 33
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0] => [6,1] => 11
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0] => [5,1] => 4
[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] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [7] => 33
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [6] => 11
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => 4
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => 2
[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
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [4,4,3,2,1] => 518
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0] => [5,4,3,2] => 3516
[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] => 1
[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] => 116
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0] => [9] => 435
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0] => [8,1] => 116
[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] => 33
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0] => [6] => 11
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0] => [5] => 4
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => 2
[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] => 435
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0,0] => [8,2,1] => 474
[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] => 2
[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] => 1
[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] => 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0] => [3,3,3] => 5
[1,1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0] => [4,4,4] => 296
[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] => 1
[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] => 1
[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] => 116
[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] => 1
[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] => 1
[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
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [5,4,3,2,1] => 3516
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2] => 28
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0] => [6,6] => 3533
[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] => 1
[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] => 1
[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] => 2
[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] => 4
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1] => 28
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => 2
[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] => 123
[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] => 2
[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] => 474
[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] => 2
[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] => 2
[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] => 4
[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] => 11
[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] => 33
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0,0] => [8,1,1,1] => 116
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0] => [5,3,2,1] => 89
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0] => [5,4,2,1] => 349
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,0] => [5,4,3,1] => 982
[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
[1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0] => [4,2] => 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
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0,0] => [6,3,2,1] => 341
[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] => 28
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0] => [4,3,3,2,1] => 126
[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] => 56
[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] => 28
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0] => [6,5,1] => 1407
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0] => [5,4,1] => 99
[1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0] => [4,3,1] => 10
[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] => 33
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0] => [6,1] => 11
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0] => [5,1] => 4
[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] => 1
[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
[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] => 1
[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
[1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,1] => 3
[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] => 1
[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
[1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1] => 2
[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
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [5,4,3,2,1] => 3516
[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] => 4
[1,1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0] => [3,3] => 2
[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] => 116
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0] => [7,1] => 33
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0] => [6,1] => 11
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0] => [5,1] => 4
[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] => 188
[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0] => [4,4] => 19
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0] => [5,3,1] => 27
[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
[1,1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0] => [3,3,2] => 5
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
12,2 18,10,2,0,5,0,0,0,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2
$F_{1} = q$
$F_{2} = 2\ q$
$F_{3} = 5\ q$
$F_{4} = 12\ q + 2\ q^{2}$
$F_{5} = 18\ q + 10\ q^{2} + 2\ q^{3} + 5\ q^{5} + 2\ q^{9} + 3\ q^{10} + 2\ q^{28}$
$F_{6} = 25\ q + 13\ q^{2} + 3\ q^{3} + 7\ q^{4} + 10\ q^{5} + 5\ q^{9} + 4\ q^{10} + 4\ q^{12} + 4\ q^{14} + 4\ q^{19} + 3\ q^{25} + 4\ q^{27} + 3\ q^{28} + 3\ q^{42} + 2\ q^{44} + 2\ q^{56} + 3\ q^{61} + 3\ q^{89} + 4\ q^{99} + 2\ q^{126} + 2\ q^{137} + 3\ q^{145} + 3\ q^{159} + 2\ q^{203} + 3\ q^{349} + 2\ q^{502} + 2\ q^{518} + 2\ q^{855} + 3\ q^{982} + 2\ q^{3516}$
Description
The number of ways to refine the partition into singletons.
For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$.
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.
For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$.
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.
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 $(n-1,\ldots,2,1)$.
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