Identifier
-
Mp00038:
Integer compositions
—reverse⟶
Integer compositions
St000382: Integer compositions ⟶ ℤ
Values
[1] => [1] => 1
[1,1] => [1,1] => 1
[2] => [2] => 2
[1,1,1] => [1,1,1] => 1
[1,2] => [2,1] => 2
[2,1] => [1,2] => 1
[3] => [3] => 3
[1,1,1,1] => [1,1,1,1] => 1
[1,1,2] => [2,1,1] => 2
[1,2,1] => [1,2,1] => 1
[1,3] => [3,1] => 3
[2,1,1] => [1,1,2] => 1
[2,2] => [2,2] => 2
[3,1] => [1,3] => 1
[4] => [4] => 4
[1,1,1,1,1] => [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1] => 2
[1,1,2,1] => [1,2,1,1] => 1
[1,1,3] => [3,1,1] => 3
[1,2,1,1] => [1,1,2,1] => 1
[1,2,2] => [2,2,1] => 2
[1,3,1] => [1,3,1] => 1
[1,4] => [4,1] => 4
[2,1,1,1] => [1,1,1,2] => 1
[2,1,2] => [2,1,2] => 2
[2,2,1] => [1,2,2] => 1
[2,3] => [3,2] => 3
[3,1,1] => [1,1,3] => 1
[3,2] => [2,3] => 2
[4,1] => [1,4] => 1
[5] => [5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1] => 2
[1,1,1,2,1] => [1,2,1,1,1] => 1
[1,1,1,3] => [3,1,1,1] => 3
[1,1,2,1,1] => [1,1,2,1,1] => 1
[1,1,2,2] => [2,2,1,1] => 2
[1,1,3,1] => [1,3,1,1] => 1
[1,1,4] => [4,1,1] => 4
[1,2,1,1,1] => [1,1,1,2,1] => 1
[1,2,1,2] => [2,1,2,1] => 2
[1,2,2,1] => [1,2,2,1] => 1
[1,2,3] => [3,2,1] => 3
[1,3,1,1] => [1,1,3,1] => 1
[1,3,2] => [2,3,1] => 2
[1,4,1] => [1,4,1] => 1
[1,5] => [5,1] => 5
[2,1,1,1,1] => [1,1,1,1,2] => 1
[2,1,1,2] => [2,1,1,2] => 2
[2,1,2,1] => [1,2,1,2] => 1
[2,1,3] => [3,1,2] => 3
[2,2,1,1] => [1,1,2,2] => 1
[2,2,2] => [2,2,2] => 2
[2,3,1] => [1,3,2] => 1
[2,4] => [4,2] => 4
[3,1,1,1] => [1,1,1,3] => 1
[3,1,2] => [2,1,3] => 2
[3,2,1] => [1,2,3] => 1
[3,3] => [3,3] => 3
[4,1,1] => [1,1,4] => 1
[4,2] => [2,4] => 2
[5,1] => [1,5] => 1
[6] => [6] => 6
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 2
[1,1,1,1,2,1] => [1,2,1,1,1,1] => 1
[1,1,1,1,3] => [3,1,1,1,1] => 3
[1,1,1,2,1,1] => [1,1,2,1,1,1] => 1
[1,1,1,2,2] => [2,2,1,1,1] => 2
[1,1,1,3,1] => [1,3,1,1,1] => 1
[1,1,1,4] => [4,1,1,1] => 4
[1,1,2,1,1,1] => [1,1,1,2,1,1] => 1
[1,1,2,1,2] => [2,1,2,1,1] => 2
[1,1,2,2,1] => [1,2,2,1,1] => 1
[1,1,2,3] => [3,2,1,1] => 3
[1,1,3,1,1] => [1,1,3,1,1] => 1
[1,1,3,2] => [2,3,1,1] => 2
[1,1,4,1] => [1,4,1,1] => 1
[1,1,5] => [5,1,1] => 5
[1,2,1,1,1,1] => [1,1,1,1,2,1] => 1
[1,2,1,1,2] => [2,1,1,2,1] => 2
[1,2,1,2,1] => [1,2,1,2,1] => 1
[1,2,1,3] => [3,1,2,1] => 3
[1,2,2,1,1] => [1,1,2,2,1] => 1
[1,2,2,2] => [2,2,2,1] => 2
[1,2,3,1] => [1,3,2,1] => 1
[1,2,4] => [4,2,1] => 4
[1,3,1,1,1] => [1,1,1,3,1] => 1
[1,3,1,2] => [2,1,3,1] => 2
[1,3,2,1] => [1,2,3,1] => 1
[1,3,3] => [3,3,1] => 3
[1,4,1,1] => [1,1,4,1] => 1
[1,4,2] => [2,4,1] => 2
[1,5,1] => [1,5,1] => 1
[1,6] => [6,1] => 6
[2,1,1,1,1,1] => [1,1,1,1,1,2] => 1
[2,1,1,1,2] => [2,1,1,1,2] => 2
[2,1,1,2,1] => [1,2,1,1,2] => 1
[2,1,1,3] => [3,1,1,2] => 3
[2,1,2,1,1] => [1,1,2,1,2] => 1
[2,1,2,2] => [2,2,1,2] => 2
>>> Load all 1018 entries. <<<[2,1,3,1] => [1,3,1,2] => 1
[2,1,4] => [4,1,2] => 4
[2,2,1,1,1] => [1,1,1,2,2] => 1
[2,2,1,2] => [2,1,2,2] => 2
[2,2,2,1] => [1,2,2,2] => 1
[2,2,3] => [3,2,2] => 3
[2,3,1,1] => [1,1,3,2] => 1
[2,3,2] => [2,3,2] => 2
[2,4,1] => [1,4,2] => 1
[2,5] => [5,2] => 5
[3,1,1,1,1] => [1,1,1,1,3] => 1
[3,1,1,2] => [2,1,1,3] => 2
[3,1,2,1] => [1,2,1,3] => 1
[3,1,3] => [3,1,3] => 3
[3,2,1,1] => [1,1,2,3] => 1
[3,2,2] => [2,2,3] => 2
[3,3,1] => [1,3,3] => 1
[3,4] => [4,3] => 4
[4,1,1,1] => [1,1,1,4] => 1
[4,1,2] => [2,1,4] => 2
[4,2,1] => [1,2,4] => 1
[4,3] => [3,4] => 3
[5,1,1] => [1,1,5] => 1
[5,2] => [2,5] => 2
[6,1] => [1,6] => 1
[7] => [7] => 7
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 2
[1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => 1
[1,1,1,1,1,3] => [3,1,1,1,1,1] => 3
[1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => 1
[1,1,1,1,2,2] => [2,2,1,1,1,1] => 2
[1,1,1,1,3,1] => [1,3,1,1,1,1] => 1
[1,1,1,1,4] => [4,1,1,1,1] => 4
[1,1,1,2,1,1,1] => [1,1,1,2,1,1,1] => 1
[1,1,1,2,1,2] => [2,1,2,1,1,1] => 2
[1,1,1,2,2,1] => [1,2,2,1,1,1] => 1
[1,1,1,2,3] => [3,2,1,1,1] => 3
[1,1,1,3,1,1] => [1,1,3,1,1,1] => 1
[1,1,1,3,2] => [2,3,1,1,1] => 2
[1,1,1,4,1] => [1,4,1,1,1] => 1
[1,1,1,5] => [5,1,1,1] => 5
[1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => 1
[1,1,2,1,1,2] => [2,1,1,2,1,1] => 2
[1,1,2,1,2,1] => [1,2,1,2,1,1] => 1
[1,1,2,1,3] => [3,1,2,1,1] => 3
[1,1,2,2,1,1] => [1,1,2,2,1,1] => 1
[1,1,2,2,2] => [2,2,2,1,1] => 2
[1,1,2,3,1] => [1,3,2,1,1] => 1
[1,1,2,4] => [4,2,1,1] => 4
[1,1,3,1,1,1] => [1,1,1,3,1,1] => 1
[1,1,3,1,2] => [2,1,3,1,1] => 2
[1,1,3,2,1] => [1,2,3,1,1] => 1
[1,1,3,3] => [3,3,1,1] => 3
[1,1,4,1,1] => [1,1,4,1,1] => 1
[1,1,4,2] => [2,4,1,1] => 2
[1,1,5,1] => [1,5,1,1] => 1
[1,1,6] => [6,1,1] => 6
[1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => 1
[1,2,1,1,1,2] => [2,1,1,1,2,1] => 2
[1,2,1,1,2,1] => [1,2,1,1,2,1] => 1
[1,2,1,1,3] => [3,1,1,2,1] => 3
[1,2,1,2,1,1] => [1,1,2,1,2,1] => 1
[1,2,1,2,2] => [2,2,1,2,1] => 2
[1,2,1,3,1] => [1,3,1,2,1] => 1
[1,2,1,4] => [4,1,2,1] => 4
[1,2,2,1,1,1] => [1,1,1,2,2,1] => 1
[1,2,2,1,2] => [2,1,2,2,1] => 2
[1,2,2,2,1] => [1,2,2,2,1] => 1
[1,2,2,3] => [3,2,2,1] => 3
[1,2,3,1,1] => [1,1,3,2,1] => 1
[1,2,3,2] => [2,3,2,1] => 2
[1,2,4,1] => [1,4,2,1] => 1
[1,2,5] => [5,2,1] => 5
[1,3,1,1,1,1] => [1,1,1,1,3,1] => 1
[1,3,1,1,2] => [2,1,1,3,1] => 2
[1,3,1,2,1] => [1,2,1,3,1] => 1
[1,3,1,3] => [3,1,3,1] => 3
[1,3,2,1,1] => [1,1,2,3,1] => 1
[1,3,2,2] => [2,2,3,1] => 2
[1,3,3,1] => [1,3,3,1] => 1
[1,3,4] => [4,3,1] => 4
[1,4,1,1,1] => [1,1,1,4,1] => 1
[1,4,1,2] => [2,1,4,1] => 2
[1,4,2,1] => [1,2,4,1] => 1
[1,4,3] => [3,4,1] => 3
[1,5,1,1] => [1,1,5,1] => 1
[1,5,2] => [2,5,1] => 2
[1,6,1] => [1,6,1] => 1
[1,7] => [7,1] => 7
[2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => 1
[2,1,1,1,1,2] => [2,1,1,1,1,2] => 2
[2,1,1,1,2,1] => [1,2,1,1,1,2] => 1
[2,1,1,1,3] => [3,1,1,1,2] => 3
[2,1,1,2,1,1] => [1,1,2,1,1,2] => 1
[2,1,1,2,2] => [2,2,1,1,2] => 2
[2,1,1,3,1] => [1,3,1,1,2] => 1
[2,1,1,4] => [4,1,1,2] => 4
[2,1,2,1,1,1] => [1,1,1,2,1,2] => 1
[2,1,2,1,2] => [2,1,2,1,2] => 2
[2,1,2,2,1] => [1,2,2,1,2] => 1
[2,1,2,3] => [3,2,1,2] => 3
[2,1,3,1,1] => [1,1,3,1,2] => 1
[2,1,3,2] => [2,3,1,2] => 2
[2,1,4,1] => [1,4,1,2] => 1
[2,1,5] => [5,1,2] => 5
[2,2,1,1,1,1] => [1,1,1,1,2,2] => 1
[2,2,1,1,2] => [2,1,1,2,2] => 2
[2,2,1,2,1] => [1,2,1,2,2] => 1
[2,2,1,3] => [3,1,2,2] => 3
[2,2,2,1,1] => [1,1,2,2,2] => 1
[2,2,2,2] => [2,2,2,2] => 2
[2,2,3,1] => [1,3,2,2] => 1
[2,2,4] => [4,2,2] => 4
[2,3,1,1,1] => [1,1,1,3,2] => 1
[2,3,1,2] => [2,1,3,2] => 2
[2,3,2,1] => [1,2,3,2] => 1
[2,3,3] => [3,3,2] => 3
[2,4,1,1] => [1,1,4,2] => 1
[2,4,2] => [2,4,2] => 2
[2,5,1] => [1,5,2] => 1
[2,6] => [6,2] => 6
[3,1,1,1,1,1] => [1,1,1,1,1,3] => 1
[3,1,1,1,2] => [2,1,1,1,3] => 2
[3,1,1,2,1] => [1,2,1,1,3] => 1
[3,1,1,3] => [3,1,1,3] => 3
[3,1,2,1,1] => [1,1,2,1,3] => 1
[3,1,2,2] => [2,2,1,3] => 2
[3,1,3,1] => [1,3,1,3] => 1
[3,1,4] => [4,1,3] => 4
[3,2,1,1,1] => [1,1,1,2,3] => 1
[3,2,1,2] => [2,1,2,3] => 2
[3,2,2,1] => [1,2,2,3] => 1
[3,2,3] => [3,2,3] => 3
[3,3,1,1] => [1,1,3,3] => 1
[3,3,2] => [2,3,3] => 2
[3,4,1] => [1,4,3] => 1
[3,5] => [5,3] => 5
[4,1,1,1,1] => [1,1,1,1,4] => 1
[4,1,1,2] => [2,1,1,4] => 2
[4,1,2,1] => [1,2,1,4] => 1
[4,1,3] => [3,1,4] => 3
[4,2,1,1] => [1,1,2,4] => 1
[4,2,2] => [2,2,4] => 2
[4,3,1] => [1,3,4] => 1
[4,4] => [4,4] => 4
[5,1,1,1] => [1,1,1,5] => 1
[5,1,2] => [2,1,5] => 2
[5,2,1] => [1,2,5] => 1
[5,3] => [3,5] => 3
[6,1,1] => [1,1,6] => 1
[6,2] => [2,6] => 2
[7,1] => [1,7] => 1
[8] => [8] => 8
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => 3
[1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1] => 1
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => 2
[1,1,1,1,1,3,1] => [1,3,1,1,1,1,1] => 1
[1,1,1,1,1,4] => [4,1,1,1,1,1] => 4
[1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => 1
[1,1,1,1,2,1,2] => [2,1,2,1,1,1,1] => 2
[1,1,1,1,2,2,1] => [1,2,2,1,1,1,1] => 1
[1,1,1,1,2,3] => [3,2,1,1,1,1] => 3
[1,1,1,1,3,1,1] => [1,1,3,1,1,1,1] => 1
[1,1,1,1,3,2] => [2,3,1,1,1,1] => 2
[1,1,1,1,4,1] => [1,4,1,1,1,1] => 1
[1,1,1,1,5] => [5,1,1,1,1] => 5
[1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1] => 1
[1,1,1,2,1,1,2] => [2,1,1,2,1,1,1] => 2
[1,1,1,2,1,2,1] => [1,2,1,2,1,1,1] => 1
[1,1,1,2,1,3] => [3,1,2,1,1,1] => 3
[1,1,1,2,2,1,1] => [1,1,2,2,1,1,1] => 1
[1,1,1,2,2,2] => [2,2,2,1,1,1] => 2
[1,1,1,2,3,1] => [1,3,2,1,1,1] => 1
[1,1,1,2,4] => [4,2,1,1,1] => 4
[1,1,1,3,1,1,1] => [1,1,1,3,1,1,1] => 1
[1,1,1,3,1,2] => [2,1,3,1,1,1] => 2
[1,1,1,3,2,1] => [1,2,3,1,1,1] => 1
[1,1,1,3,3] => [3,3,1,1,1] => 3
[1,1,1,4,1,1] => [1,1,4,1,1,1] => 1
[1,1,1,4,2] => [2,4,1,1,1] => 2
[1,1,1,5,1] => [1,5,1,1,1] => 1
[1,1,1,6] => [6,1,1,1] => 6
[1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1] => 1
[1,1,2,1,1,1,2] => [2,1,1,1,2,1,1] => 2
[1,1,2,1,1,2,1] => [1,2,1,1,2,1,1] => 1
[1,1,2,1,1,3] => [3,1,1,2,1,1] => 3
[1,1,2,1,2,1,1] => [1,1,2,1,2,1,1] => 1
[1,1,2,1,2,2] => [2,2,1,2,1,1] => 2
[1,1,2,1,3,1] => [1,3,1,2,1,1] => 1
[1,1,2,1,4] => [4,1,2,1,1] => 4
[1,1,2,2,1,1,1] => [1,1,1,2,2,1,1] => 1
[1,1,2,2,1,2] => [2,1,2,2,1,1] => 2
[1,1,2,2,2,1] => [1,2,2,2,1,1] => 1
[1,1,2,2,3] => [3,2,2,1,1] => 3
[1,1,2,3,1,1] => [1,1,3,2,1,1] => 1
[1,1,2,3,2] => [2,3,2,1,1] => 2
[1,1,2,4,1] => [1,4,2,1,1] => 1
[1,1,2,5] => [5,2,1,1] => 5
[1,1,3,1,1,1,1] => [1,1,1,1,3,1,1] => 1
[1,1,3,1,1,2] => [2,1,1,3,1,1] => 2
[1,1,3,1,2,1] => [1,2,1,3,1,1] => 1
[1,1,3,1,3] => [3,1,3,1,1] => 3
[1,1,3,2,1,1] => [1,1,2,3,1,1] => 1
[1,1,3,2,2] => [2,2,3,1,1] => 2
[1,1,3,3,1] => [1,3,3,1,1] => 1
[1,1,3,4] => [4,3,1,1] => 4
[1,1,4,1,1,1] => [1,1,1,4,1,1] => 1
[1,1,4,1,2] => [2,1,4,1,1] => 2
[1,1,4,2,1] => [1,2,4,1,1] => 1
[1,1,4,3] => [3,4,1,1] => 3
[1,1,5,1,1] => [1,1,5,1,1] => 1
[1,1,5,2] => [2,5,1,1] => 2
[1,1,6,1] => [1,6,1,1] => 1
[1,1,7] => [7,1,1] => 7
[1,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1] => 1
[1,2,1,1,1,1,2] => [2,1,1,1,1,2,1] => 2
[1,2,1,1,1,2,1] => [1,2,1,1,1,2,1] => 1
[1,2,1,1,1,3] => [3,1,1,1,2,1] => 3
[1,2,1,1,2,1,1] => [1,1,2,1,1,2,1] => 1
[1,2,1,1,2,2] => [2,2,1,1,2,1] => 2
[1,2,1,1,3,1] => [1,3,1,1,2,1] => 1
[1,2,1,1,4] => [4,1,1,2,1] => 4
[1,2,1,2,1,1,1] => [1,1,1,2,1,2,1] => 1
[1,2,1,2,1,2] => [2,1,2,1,2,1] => 2
[1,2,1,2,2,1] => [1,2,2,1,2,1] => 1
[1,2,1,2,3] => [3,2,1,2,1] => 3
[1,2,1,3,1,1] => [1,1,3,1,2,1] => 1
[1,2,1,3,2] => [2,3,1,2,1] => 2
[1,2,1,4,1] => [1,4,1,2,1] => 1
[1,2,1,5] => [5,1,2,1] => 5
[1,2,2,1,1,1,1] => [1,1,1,1,2,2,1] => 1
[1,2,2,1,1,2] => [2,1,1,2,2,1] => 2
[1,2,2,1,2,1] => [1,2,1,2,2,1] => 1
[1,2,2,1,3] => [3,1,2,2,1] => 3
[1,2,2,2,1,1] => [1,1,2,2,2,1] => 1
[1,2,2,2,2] => [2,2,2,2,1] => 2
[1,2,2,3,1] => [1,3,2,2,1] => 1
[1,2,2,4] => [4,2,2,1] => 4
[1,2,3,1,1,1] => [1,1,1,3,2,1] => 1
[1,2,3,1,2] => [2,1,3,2,1] => 2
[1,2,3,2,1] => [1,2,3,2,1] => 1
[1,2,3,3] => [3,3,2,1] => 3
[1,2,4,1,1] => [1,1,4,2,1] => 1
[1,2,4,2] => [2,4,2,1] => 2
[1,2,5,1] => [1,5,2,1] => 1
[1,2,6] => [6,2,1] => 6
[1,3,1,1,1,1,1] => [1,1,1,1,1,3,1] => 1
[1,3,1,1,1,2] => [2,1,1,1,3,1] => 2
[1,3,1,1,2,1] => [1,2,1,1,3,1] => 1
[1,3,1,1,3] => [3,1,1,3,1] => 3
[1,3,1,2,1,1] => [1,1,2,1,3,1] => 1
[1,3,1,2,2] => [2,2,1,3,1] => 2
[1,3,1,3,1] => [1,3,1,3,1] => 1
[1,3,1,4] => [4,1,3,1] => 4
[1,3,2,1,1,1] => [1,1,1,2,3,1] => 1
[1,3,2,1,2] => [2,1,2,3,1] => 2
[1,3,2,2,1] => [1,2,2,3,1] => 1
[1,3,2,3] => [3,2,3,1] => 3
[1,3,3,1,1] => [1,1,3,3,1] => 1
[1,3,3,2] => [2,3,3,1] => 2
[1,3,4,1] => [1,4,3,1] => 1
[1,3,5] => [5,3,1] => 5
[1,4,1,1,1,1] => [1,1,1,1,4,1] => 1
[1,4,1,1,2] => [2,1,1,4,1] => 2
[1,4,1,2,1] => [1,2,1,4,1] => 1
[1,4,1,3] => [3,1,4,1] => 3
[1,4,2,1,1] => [1,1,2,4,1] => 1
[1,4,2,2] => [2,2,4,1] => 2
[1,4,3,1] => [1,3,4,1] => 1
[1,4,4] => [4,4,1] => 4
[1,5,1,1,1] => [1,1,1,5,1] => 1
[1,5,1,2] => [2,1,5,1] => 2
[1,5,2,1] => [1,2,5,1] => 1
[1,5,3] => [3,5,1] => 3
[1,6,1,1] => [1,1,6,1] => 1
[1,6,2] => [2,6,1] => 2
[1,7,1] => [1,7,1] => 1
[1,8] => [8,1] => 8
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => 1
[2,1,1,1,1,1,2] => [2,1,1,1,1,1,2] => 2
[2,1,1,1,1,2,1] => [1,2,1,1,1,1,2] => 1
[2,1,1,1,1,3] => [3,1,1,1,1,2] => 3
[2,1,1,1,2,1,1] => [1,1,2,1,1,1,2] => 1
[2,1,1,1,2,2] => [2,2,1,1,1,2] => 2
[2,1,1,1,3,1] => [1,3,1,1,1,2] => 1
[2,1,1,1,4] => [4,1,1,1,2] => 4
[2,1,1,2,1,1,1] => [1,1,1,2,1,1,2] => 1
[2,1,1,2,1,2] => [2,1,2,1,1,2] => 2
[2,1,1,2,2,1] => [1,2,2,1,1,2] => 1
[2,1,1,2,3] => [3,2,1,1,2] => 3
[2,1,1,3,1,1] => [1,1,3,1,1,2] => 1
[2,1,1,3,2] => [2,3,1,1,2] => 2
[2,1,1,4,1] => [1,4,1,1,2] => 1
[2,1,1,5] => [5,1,1,2] => 5
[2,1,2,1,1,1,1] => [1,1,1,1,2,1,2] => 1
[2,1,2,1,1,2] => [2,1,1,2,1,2] => 2
[2,1,2,1,2,1] => [1,2,1,2,1,2] => 1
[2,1,2,1,3] => [3,1,2,1,2] => 3
[2,1,2,2,1,1] => [1,1,2,2,1,2] => 1
[2,1,2,2,2] => [2,2,2,1,2] => 2
[2,1,2,3,1] => [1,3,2,1,2] => 1
[2,1,2,4] => [4,2,1,2] => 4
[2,1,3,1,1,1] => [1,1,1,3,1,2] => 1
[2,1,3,1,2] => [2,1,3,1,2] => 2
[2,1,3,2,1] => [1,2,3,1,2] => 1
[2,1,3,3] => [3,3,1,2] => 3
[2,1,4,1,1] => [1,1,4,1,2] => 1
[2,1,4,2] => [2,4,1,2] => 2
[2,1,5,1] => [1,5,1,2] => 1
[2,1,6] => [6,1,2] => 6
[2,2,1,1,1,1,1] => [1,1,1,1,1,2,2] => 1
[2,2,1,1,1,2] => [2,1,1,1,2,2] => 2
[2,2,1,1,2,1] => [1,2,1,1,2,2] => 1
[2,2,1,1,3] => [3,1,1,2,2] => 3
[2,2,1,2,1,1] => [1,1,2,1,2,2] => 1
[2,2,1,2,2] => [2,2,1,2,2] => 2
[2,2,1,3,1] => [1,3,1,2,2] => 1
[2,2,1,4] => [4,1,2,2] => 4
[2,2,2,1,1,1] => [1,1,1,2,2,2] => 1
[2,2,2,1,2] => [2,1,2,2,2] => 2
[2,2,2,2,1] => [1,2,2,2,2] => 1
[2,2,2,3] => [3,2,2,2] => 3
[2,2,3,1,1] => [1,1,3,2,2] => 1
[2,2,3,2] => [2,3,2,2] => 2
[2,2,4,1] => [1,4,2,2] => 1
[2,2,5] => [5,2,2] => 5
[2,3,1,1,1,1] => [1,1,1,1,3,2] => 1
[2,3,1,1,2] => [2,1,1,3,2] => 2
[2,3,1,2,1] => [1,2,1,3,2] => 1
[2,3,1,3] => [3,1,3,2] => 3
[2,3,2,1,1] => [1,1,2,3,2] => 1
[2,3,2,2] => [2,2,3,2] => 2
[2,3,3,1] => [1,3,3,2] => 1
[2,3,4] => [4,3,2] => 4
[2,4,1,1,1] => [1,1,1,4,2] => 1
[2,4,1,2] => [2,1,4,2] => 2
[2,4,2,1] => [1,2,4,2] => 1
[2,4,3] => [3,4,2] => 3
[2,5,1,1] => [1,1,5,2] => 1
[2,5,2] => [2,5,2] => 2
[2,6,1] => [1,6,2] => 1
[2,7] => [7,2] => 7
[3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => 1
[3,1,1,1,1,2] => [2,1,1,1,1,3] => 2
[3,1,1,1,2,1] => [1,2,1,1,1,3] => 1
[3,1,1,1,3] => [3,1,1,1,3] => 3
[3,1,1,2,1,1] => [1,1,2,1,1,3] => 1
[3,1,1,2,2] => [2,2,1,1,3] => 2
[3,1,1,3,1] => [1,3,1,1,3] => 1
[3,1,1,4] => [4,1,1,3] => 4
[3,1,2,1,1,1] => [1,1,1,2,1,3] => 1
[3,1,2,1,2] => [2,1,2,1,3] => 2
[3,1,2,2,1] => [1,2,2,1,3] => 1
[3,1,2,3] => [3,2,1,3] => 3
[3,1,3,1,1] => [1,1,3,1,3] => 1
[3,1,3,2] => [2,3,1,3] => 2
[3,1,4,1] => [1,4,1,3] => 1
[3,1,5] => [5,1,3] => 5
[3,2,1,1,1,1] => [1,1,1,1,2,3] => 1
[3,2,1,1,2] => [2,1,1,2,3] => 2
[3,2,1,2,1] => [1,2,1,2,3] => 1
[3,2,1,3] => [3,1,2,3] => 3
[3,2,2,1,1] => [1,1,2,2,3] => 1
[3,2,2,2] => [2,2,2,3] => 2
[3,2,3,1] => [1,3,2,3] => 1
[3,2,4] => [4,2,3] => 4
[3,3,1,1,1] => [1,1,1,3,3] => 1
[3,3,1,2] => [2,1,3,3] => 2
[3,3,2,1] => [1,2,3,3] => 1
[3,3,3] => [3,3,3] => 3
[3,4,1,1] => [1,1,4,3] => 1
[3,4,2] => [2,4,3] => 2
[3,5,1] => [1,5,3] => 1
[3,6] => [6,3] => 6
[4,1,1,1,1,1] => [1,1,1,1,1,4] => 1
[4,1,1,1,2] => [2,1,1,1,4] => 2
[4,1,1,2,1] => [1,2,1,1,4] => 1
[4,1,1,3] => [3,1,1,4] => 3
[4,1,2,1,1] => [1,1,2,1,4] => 1
[4,1,2,2] => [2,2,1,4] => 2
[4,1,3,1] => [1,3,1,4] => 1
[4,1,4] => [4,1,4] => 4
[4,2,1,1,1] => [1,1,1,2,4] => 1
[4,2,1,2] => [2,1,2,4] => 2
[4,2,2,1] => [1,2,2,4] => 1
[4,2,3] => [3,2,4] => 3
[4,3,1,1] => [1,1,3,4] => 1
[4,3,2] => [2,3,4] => 2
[4,4,1] => [1,4,4] => 1
[4,5] => [5,4] => 5
[5,1,1,1,1] => [1,1,1,1,5] => 1
[5,1,1,2] => [2,1,1,5] => 2
[5,1,2,1] => [1,2,1,5] => 1
[5,1,3] => [3,1,5] => 3
[5,2,1,1] => [1,1,2,5] => 1
[5,2,2] => [2,2,5] => 2
[5,3,1] => [1,3,5] => 1
[5,4] => [4,5] => 4
[6,1,1,1] => [1,1,1,6] => 1
[6,1,2] => [2,1,6] => 2
[6,2,1] => [1,2,6] => 1
[6,3] => [3,6] => 3
[7,1,1] => [1,1,7] => 1
[7,2] => [2,7] => 2
[8,1] => [1,8] => 1
[9] => [9] => 9
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => 4
[1,1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1,1] => 1
[1,1,1,1,1,2,3] => [3,2,1,1,1,1,1] => 3
[1,1,1,1,1,5] => [5,1,1,1,1,1] => 5
[1,1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1,1] => 1
[1,1,1,1,2,1,1,2] => [2,1,1,2,1,1,1,1] => 2
[1,1,1,1,2,2,1,1] => [1,1,2,2,1,1,1,1] => 1
[1,1,1,1,2,2,2] => [2,2,2,1,1,1,1] => 2
[1,1,1,1,2,4] => [4,2,1,1,1,1] => 4
[1,1,1,1,3,3] => [3,3,1,1,1,1] => 3
[1,1,1,1,5,1] => [1,5,1,1,1,1] => 1
[1,1,1,1,6] => [6,1,1,1,1] => 6
[1,1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1,1] => 1
[1,1,1,2,1,2,2] => [2,2,1,2,1,1,1] => 2
[1,1,1,2,1,3,1] => [1,3,1,2,1,1,1] => 1
[1,1,1,2,2,1,2] => [2,1,2,2,1,1,1] => 2
[1,1,1,2,2,3] => [3,2,2,1,1,1] => 3
[1,1,1,2,3,1,1] => [1,1,3,2,1,1,1] => 1
[1,1,1,2,5] => [5,2,1,1,1] => 5
[1,1,1,3,1,2,1] => [1,2,1,3,1,1,1] => 1
[1,1,1,3,1,3] => [3,1,3,1,1,1] => 3
[1,1,1,3,2,1,1] => [1,1,2,3,1,1,1] => 1
[1,1,1,3,4] => [4,3,1,1,1] => 4
[1,1,1,7] => [7,1,1,1] => 7
[1,1,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1,1] => 1
[1,1,2,1,1,1,1,2] => [2,1,1,1,1,2,1,1] => 2
[1,1,2,1,1,2,1,1] => [1,1,2,1,1,2,1,1] => 1
[1,1,2,1,1,2,2] => [2,2,1,1,2,1,1] => 2
[1,1,2,1,1,3,1] => [1,3,1,1,2,1,1] => 1
[1,1,2,1,2,2,1] => [1,2,2,1,2,1,1] => 1
[1,1,2,1,2,3] => [3,2,1,2,1,1] => 3
[1,1,2,2,1,1,1,1] => [1,1,1,1,2,2,1,1] => 1
[1,1,2,2,1,1,2] => [2,1,1,2,2,1,1] => 2
[1,1,2,2,2,2] => [2,2,2,2,1,1] => 2
[1,1,2,2,4] => [4,2,2,1,1] => 4
[1,1,2,3,1,1,1] => [1,1,1,3,2,1,1] => 1
[1,1,2,3,3] => [3,3,2,1,1] => 3
[1,1,2,6] => [6,2,1,1] => 6
[1,1,3,1,1,2,1] => [1,2,1,1,3,1,1] => 1
[1,1,3,1,1,3] => [3,1,1,3,1,1] => 3
[1,1,3,2,1,1,1] => [1,1,1,2,3,1,1] => 1
[1,1,3,2,1,2] => [2,1,2,3,1,1] => 2
[1,1,3,3,1,1] => [1,1,3,3,1,1] => 1
[1,1,3,5] => [5,3,1,1] => 5
[1,1,4,4] => [4,4,1,1] => 4
[1,1,7,1] => [1,7,1,1] => 1
[1,1,8] => [8,1,1] => 8
[1,2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2,1] => 1
[1,2,1,1,2,1,2] => [2,1,2,1,1,2,1] => 2
[1,2,1,1,3,1,1] => [1,1,3,1,1,2,1] => 1
[1,2,1,1,4,1] => [1,4,1,1,2,1] => 1
[1,2,1,2,1,1,2] => [2,1,1,2,1,2,1] => 2
[1,2,1,2,2,2] => [2,2,2,1,2,1] => 2
[1,2,1,3,1,1,1] => [1,1,1,3,1,2,1] => 1
[1,2,2,2,2,1] => [1,2,2,2,2,1] => 1
[1,2,2,2,3] => [3,2,2,2,1] => 3
[1,2,2,3,2] => [2,3,2,2,1] => 2
[1,2,2,5] => [5,2,2,1] => 5
[1,2,3,2,2] => [2,2,3,2,1] => 2
[1,2,3,3,1] => [1,3,3,2,1] => 1
[1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [3,4,2,1] => 3
[1,2,6,1] => [1,6,2,1] => 1
[1,2,7] => [7,2,1] => 7
[1,3,1,1,2,1,1] => [1,1,2,1,1,3,1] => 1
[1,3,1,1,3,1] => [1,3,1,1,3,1] => 1
[1,3,1,2,1,1,1] => [1,1,1,2,1,3,1] => 1
[1,3,2,2,2] => [2,2,2,3,1] => 2
[1,3,2,3,1] => [1,3,2,3,1] => 1
[1,3,2,4] => [4,2,3,1] => 4
[1,3,3,2,1] => [1,2,3,3,1] => 1
[1,3,3,3] => [3,3,3,1] => 3
[1,3,4,2] => [2,4,3,1] => 2
[1,3,5,1] => [1,5,3,1] => 1
[1,3,6] => [6,3,1] => 6
[1,4,1,1,2,1] => [1,2,1,1,4,1] => 1
[1,4,2,3] => [3,2,4,1] => 3
[1,4,3,2] => [2,3,4,1] => 2
[1,4,4,1] => [1,4,4,1] => 1
[1,4,5] => [5,4,1] => 5
[1,5,1,1,1,1] => [1,1,1,1,5,1] => 1
[1,5,3,1] => [1,3,5,1] => 1
[1,5,4] => [4,5,1] => 4
[1,6,1,2] => [2,1,6,1] => 2
[1,6,2,1] => [1,2,6,1] => 1
[1,7,1,1] => [1,1,7,1] => 1
[1,8,1] => [1,8,1] => 1
[1,9] => [9,1] => 9
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2] => 1
[2,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,2] => 2
[2,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,2] => 1
[2,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,2] => 1
[2,1,1,1,1,3,1] => [1,3,1,1,1,1,2] => 1
[2,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,2] => 1
[2,1,1,1,2,2,1] => [1,2,2,1,1,1,2] => 1
[2,1,1,1,2,3] => [3,2,1,1,1,2] => 3
[2,1,1,1,3,1,1] => [1,1,3,1,1,1,2] => 1
[2,1,1,1,4,1] => [1,4,1,1,1,2] => 1
[2,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,2] => 1
[2,1,1,2,1,2,1] => [1,2,1,2,1,1,2] => 1
[2,1,1,2,2,1,1] => [1,1,2,2,1,1,2] => 1
[2,1,1,2,2,2] => [2,2,2,1,1,2] => 2
[2,1,1,2,3,1] => [1,3,2,1,1,2] => 1
[2,1,1,3,2,1] => [1,2,3,1,1,2] => 1
[2,1,1,5,1] => [1,5,1,1,2] => 1
[2,1,2,1,1,2,1] => [1,2,1,1,2,1,2] => 1
[2,1,2,1,1,3] => [3,1,1,2,1,2] => 3
[2,1,2,1,3,1] => [1,3,1,2,1,2] => 1
[2,1,2,2,1,1,1] => [1,1,1,2,2,1,2] => 1
[2,1,2,2,1,2] => [2,1,2,2,1,2] => 2
[2,1,2,3,1,1] => [1,1,3,2,1,2] => 1
[2,1,2,4,1] => [1,4,2,1,2] => 1
[2,1,3,3,1] => [1,3,3,1,2] => 1
[2,1,3,4] => [4,3,1,2] => 4
[2,1,4,3] => [3,4,1,2] => 3
[2,1,6,1] => [1,6,1,2] => 1
[2,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,2] => 1
[2,2,1,1,2,1,1] => [1,1,2,1,1,2,2] => 1
[2,2,1,1,2,2] => [2,2,1,1,2,2] => 2
[2,2,1,2,1,1,1] => [1,1,1,2,1,2,2] => 1
[2,2,1,4,1] => [1,4,1,2,2] => 1
[2,2,2,1,1,1,1] => [1,1,1,1,2,2,2] => 1
[2,2,2,1,1,2] => [2,1,1,2,2,2] => 2
[2,2,2,2,1,1] => [1,1,2,2,2,2] => 1
[2,2,2,2,2] => [2,2,2,2,2] => 2
[2,2,2,3,1] => [1,3,2,2,2] => 1
[2,2,2,4] => [4,2,2,2] => 4
[2,2,3,2,1] => [1,2,3,2,2] => 1
[2,2,3,3] => [3,3,2,2] => 3
[2,2,4,2] => [2,4,2,2] => 2
[2,2,5,1] => [1,5,2,2] => 1
[2,2,6] => [6,2,2] => 6
[2,3,1,4] => [4,1,3,2] => 4
[2,3,2,2,1] => [1,2,2,3,2] => 1
[2,3,2,3] => [3,2,3,2] => 3
[2,3,3,2] => [2,3,3,2] => 2
[2,3,4,1] => [1,4,3,2] => 1
[2,3,5] => [5,3,2] => 5
[2,4,1,3] => [3,1,4,2] => 3
[2,4,2,2] => [2,2,4,2] => 2
[2,4,3,1] => [1,3,4,2] => 1
[2,4,4] => [4,4,2] => 4
[2,5,3] => [3,5,2] => 3
[2,7,1] => [1,7,2] => 1
[2,8] => [8,2] => 8
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3] => 1
[3,1,1,1,1,3] => [3,1,1,1,1,3] => 3
[3,1,1,2,1,2] => [2,1,2,1,1,3] => 2
[3,1,1,3,1,1] => [1,1,3,1,1,3] => 1
[3,1,2,4] => [4,2,1,3] => 4
[3,1,4,2] => [2,4,1,3] => 2
[3,1,5,1] => [1,5,1,3] => 1
[3,2,1,1,1,1,1] => [1,1,1,1,1,2,3] => 1
[3,2,1,1,1,2] => [2,1,1,1,2,3] => 2
[3,2,1,2,1,1] => [1,1,2,1,2,3] => 1
[3,2,1,4] => [4,1,2,3] => 4
[3,2,2,1,1,1] => [1,1,1,2,2,3] => 1
[3,2,2,2,1] => [1,2,2,2,3] => 1
[3,2,2,3] => [3,2,2,3] => 3
[3,2,3,2] => [2,3,2,3] => 2
[3,2,4,1] => [1,4,2,3] => 1
[3,2,5] => [5,2,3] => 5
[3,3,1,1,1,1] => [1,1,1,1,3,3] => 1
[3,3,2,1,1] => [1,1,2,3,3] => 1
[3,3,2,2] => [2,2,3,3] => 2
[3,3,3,1] => [1,3,3,3] => 1
[3,3,4] => [4,3,3] => 4
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => 1
[3,4,3] => [3,4,3] => 3
[3,5,2] => [2,5,3] => 2
[3,6,1] => [1,6,3] => 1
[3,7] => [7,3] => 7
[4,1,1,1,1,1,1] => [1,1,1,1,1,1,4] => 1
[4,1,1,4] => [4,1,1,4] => 4
[4,1,2,3] => [3,2,1,4] => 3
[4,1,3,2] => [2,3,1,4] => 2
[4,2,1,1,1,1] => [1,1,1,1,2,4] => 1
[4,2,1,3] => [3,1,2,4] => 3
[4,2,2,1,1] => [1,1,2,2,4] => 1
[4,2,2,2] => [2,2,2,4] => 2
[4,2,3,1] => [1,3,2,4] => 1
[4,2,4] => [4,2,4] => 4
[4,3,1,1,1] => [1,1,1,3,4] => 1
[4,3,1,2] => [2,1,3,4] => 2
[4,3,2,1] => [1,2,3,4] => 1
[4,3,3] => [3,3,4] => 3
[4,4,1,1] => [1,1,4,4] => 1
[4,4,2] => [2,4,4] => 2
[4,5,1] => [1,5,4] => 1
[4,6] => [6,4] => 6
[5,1,1,1,1,1] => [1,1,1,1,1,5] => 1
[5,2,1,1,1] => [1,1,1,2,5] => 1
[5,2,2,1] => [1,2,2,5] => 1
[5,2,3] => [3,2,5] => 3
[5,3,1,1] => [1,1,3,5] => 1
[5,3,2] => [2,3,5] => 2
[5,4,1] => [1,4,5] => 1
[5,5] => [5,5] => 5
[6,1,1,1,1] => [1,1,1,1,6] => 1
[6,2,1,1] => [1,1,2,6] => 1
[6,2,2] => [2,2,6] => 2
[6,3,1] => [1,3,6] => 1
[6,4] => [4,6] => 4
[7,1,1,1] => [1,1,1,7] => 1
[7,2,1] => [1,2,7] => 1
[7,3] => [3,7] => 3
[8,1,1] => [1,1,8] => 1
[8,2] => [2,8] => 2
[9,1] => [1,9] => 1
[10] => [10] => 10
[1,10] => [10,1] => 10
[1,8,1,1] => [1,1,8,1] => 1
[1,7,2,1] => [1,2,7,1] => 1
[1,6,3,1] => [1,3,6,1] => 1
[1,5,4,1] => [1,4,5,1] => 1
[1,4,5,1] => [1,5,4,1] => 1
[1,3,6,1] => [1,6,3,1] => 1
[1,2,7,1] => [1,7,2,1] => 1
[1,1,8,1] => [1,8,1,1] => 1
[10,1] => [1,10] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,2,2,1,1] => [1,1,2,2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,2,1,1,2] => [2,1,1,2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,3,3] => [3,3,1,1,1,1,1,1] => 3
[1,1,1,1,2,2,1,1,1,1] => [1,1,1,1,2,2,1,1,1,1] => 1
[1,1,1,1,2,2,2,2] => [2,2,2,2,1,1,1,1] => 2
[1,1,1,1,2,1,1,2,1,1] => [1,1,2,1,1,2,1,1,1,1] => 1
[1,1,1,1,2,1,1,1,1,2] => [2,1,1,1,1,2,1,1,1,1] => 2
[1,1,1,1,2,1,2,3] => [3,2,1,2,1,1,1,1] => 3
[1,1,1,1,3,3,1,1] => [1,1,3,3,1,1,1,1] => 1
[1,1,1,1,3,2,1,2] => [2,1,2,3,1,1,1,1] => 2
[1,1,1,1,3,1,1,3] => [3,1,1,3,1,1,1,1] => 3
[1,1,1,1,4,4] => [4,4,1,1,1,1] => 4
[1,1,2,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,2,1,1] => 1
[1,1,2,2,1,1,2,2] => [2,2,1,1,2,2,1,1] => 2
[1,1,2,2,2,2,1,1] => [1,1,2,2,2,2,1,1] => 1
[1,1,2,2,2,1,1,2] => [2,1,1,2,2,2,1,1] => 2
[1,1,2,2,3,3] => [3,3,2,2,1,1] => 3
[1,1,2,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,2,1,1] => 1
[1,1,2,1,1,2,2,2] => [2,2,2,1,1,2,1,1] => 2
[1,1,2,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,2,1,1] => 1
[1,1,2,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,2,1,1] => 2
[1,1,2,1,1,1,2,3] => [3,2,1,1,1,2,1,1] => 3
[1,1,2,1,2,3,1,1] => [1,1,3,2,1,2,1,1] => 1
[1,1,2,1,2,2,1,2] => [2,1,2,2,1,2,1,1] => 2
[1,1,2,1,2,1,1,3] => [3,1,1,2,1,2,1,1] => 3
[1,1,2,1,3,4] => [4,3,1,2,1,1] => 4
[1,1,3,3,1,1,1,1] => [1,1,1,1,3,3,1,1] => 1
[1,1,3,3,2,2] => [2,2,3,3,1,1] => 2
[1,1,3,2,1,2,1,1] => [1,1,2,1,2,3,1,1] => 1
[1,1,3,2,1,1,1,2] => [2,1,1,1,2,3,1,1] => 2
[1,1,3,2,2,3] => [3,2,2,3,1,1] => 3
[1,1,3,1,1,3,1,1] => [1,1,3,1,1,3,1,1] => 1
[1,1,3,1,1,2,1,2] => [2,1,2,1,1,3,1,1] => 2
[1,1,3,1,1,1,1,3] => [3,1,1,1,1,3,1,1] => 3
[1,1,3,1,2,4] => [4,2,1,3,1,1] => 4
[1,1,4,4,1,1] => [1,1,4,4,1,1] => 1
[1,1,4,3,1,2] => [2,1,3,4,1,1] => 2
[1,1,4,2,1,3] => [3,1,2,4,1,1] => 3
[1,1,4,1,1,4] => [4,1,1,4,1,1] => 4
[1,1,5,5] => [5,5,1,1] => 5
[2,2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2,2] => 1
[2,2,1,1,1,1,2,2] => [2,2,1,1,1,1,2,2] => 2
[2,2,1,1,2,2,1,1] => [1,1,2,2,1,1,2,2] => 1
[2,2,1,1,2,1,1,2] => [2,1,1,2,1,1,2,2] => 2
[2,2,1,1,3,3] => [3,3,1,1,2,2] => 3
[2,2,2,2,1,1,1,1] => [1,1,1,1,2,2,2,2] => 1
[2,2,2,2,2,2] => [2,2,2,2,2,2] => 2
[2,2,2,1,1,2,1,1] => [1,1,2,1,1,2,2,2] => 1
[2,2,2,1,1,1,1,2] => [2,1,1,1,1,2,2,2] => 2
[2,2,2,1,2,3] => [3,2,1,2,2,2] => 3
[2,2,3,3,1,1] => [1,1,3,3,2,2] => 1
[2,2,3,2,1,2] => [2,1,2,3,2,2] => 2
[2,2,3,1,1,3] => [3,1,1,3,2,2] => 3
[2,2,4,4] => [4,4,2,2] => 4
[2,1,1,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1,1,2] => 1
[2,1,1,2,1,1,2,2] => [2,2,1,1,2,1,1,2] => 2
[2,1,1,2,2,2,1,1] => [1,1,2,2,2,1,1,2] => 1
[2,1,1,2,2,1,1,2] => [2,1,1,2,2,1,1,2] => 2
[2,1,1,2,3,3] => [3,3,2,1,1,2] => 3
[2,1,1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1,1,2] => 1
[2,1,1,1,1,2,2,2] => [2,2,2,1,1,1,1,2] => 2
[2,1,1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1,1,2] => 1
[2,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,2] => 2
[2,1,1,1,1,1,2,3] => [3,2,1,1,1,1,1,2] => 3
[2,1,1,1,2,3,1,1] => [1,1,3,2,1,1,1,2] => 1
[2,1,1,1,2,2,1,2] => [2,1,2,2,1,1,1,2] => 2
[2,1,1,1,2,1,1,3] => [3,1,1,2,1,1,1,2] => 3
[2,1,1,1,3,4] => [4,3,1,1,1,2] => 4
[2,1,2,3,1,1,1,1] => [1,1,1,1,3,2,1,2] => 1
[2,1,2,3,2,2] => [2,2,3,2,1,2] => 2
[2,1,2,2,1,2,1,1] => [1,1,2,1,2,2,1,2] => 1
[2,1,2,2,1,1,1,2] => [2,1,1,1,2,2,1,2] => 2
[2,1,2,2,2,3] => [3,2,2,2,1,2] => 3
[2,1,2,1,1,3,1,1] => [1,1,3,1,1,2,1,2] => 1
[2,1,2,1,1,2,1,2] => [2,1,2,1,1,2,1,2] => 2
[2,1,2,1,1,1,1,3] => [3,1,1,1,1,2,1,2] => 3
[2,1,2,1,2,4] => [4,2,1,2,1,2] => 4
[2,1,3,4,1,1] => [1,1,4,3,1,2] => 1
[2,1,3,3,1,2] => [2,1,3,3,1,2] => 2
[2,1,3,2,1,3] => [3,1,2,3,1,2] => 3
[2,1,3,1,1,4] => [4,1,1,3,1,2] => 4
[2,1,4,5] => [5,4,1,2] => 5
[3,3,1,1,1,1,1,1] => [1,1,1,1,1,1,3,3] => 1
[3,3,1,1,2,2] => [2,2,1,1,3,3] => 2
[3,3,2,2,1,1] => [1,1,2,2,3,3] => 1
[3,3,2,1,1,2] => [2,1,1,2,3,3] => 2
[3,3,3,3] => [3,3,3,3] => 3
[3,2,1,2,1,1,1,1] => [1,1,1,1,2,1,2,3] => 1
[3,2,1,2,2,2] => [2,2,2,1,2,3] => 2
[3,2,1,1,1,2,1,1] => [1,1,2,1,1,1,2,3] => 1
[3,2,1,1,1,1,1,2] => [2,1,1,1,1,1,2,3] => 2
[3,2,1,1,2,3] => [3,2,1,1,2,3] => 3
[3,2,2,3,1,1] => [1,1,3,2,2,3] => 1
[3,2,2,2,1,2] => [2,1,2,2,2,3] => 2
[3,2,2,1,1,3] => [3,1,1,2,2,3] => 3
[3,2,3,4] => [4,3,2,3] => 4
[3,1,1,3,1,1,1,1] => [1,1,1,1,3,1,1,3] => 1
[3,1,1,3,2,2] => [2,2,3,1,1,3] => 2
[3,1,1,2,1,2,1,1] => [1,1,2,1,2,1,1,3] => 1
[3,1,1,2,1,1,1,2] => [2,1,1,1,2,1,1,3] => 2
[3,1,1,2,2,3] => [3,2,2,1,1,3] => 3
[3,1,1,1,1,3,1,1] => [1,1,3,1,1,1,1,3] => 1
[3,1,1,1,1,2,1,2] => [2,1,2,1,1,1,1,3] => 2
[3,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,3] => 3
[3,1,1,1,2,4] => [4,2,1,1,1,3] => 4
[3,1,2,4,1,1] => [1,1,4,2,1,3] => 1
[3,1,2,3,1,2] => [2,1,3,2,1,3] => 2
[3,1,2,2,1,3] => [3,1,2,2,1,3] => 3
[3,1,2,1,1,4] => [4,1,1,2,1,3] => 4
[3,1,3,5] => [5,3,1,3] => 5
[4,4,1,1,1,1] => [1,1,1,1,4,4] => 1
[4,4,2,2] => [2,2,4,4] => 2
[4,3,1,2,1,1] => [1,1,2,1,3,4] => 1
[4,3,1,1,1,2] => [2,1,1,1,3,4] => 2
[4,3,2,3] => [3,2,3,4] => 3
[4,2,1,3,1,1] => [1,1,3,1,2,4] => 1
[4,2,1,2,1,2] => [2,1,2,1,2,4] => 2
[4,2,1,1,1,3] => [3,1,1,1,2,4] => 3
[4,2,2,4] => [4,2,2,4] => 4
[4,1,1,4,1,1] => [1,1,4,1,1,4] => 1
[4,1,1,3,1,2] => [2,1,3,1,1,4] => 2
[4,1,1,2,1,3] => [3,1,2,1,1,4] => 3
[4,1,1,1,1,4] => [4,1,1,1,1,4] => 4
[4,1,2,5] => [5,2,1,4] => 5
[5,5,1,1] => [1,1,5,5] => 1
[5,4,1,2] => [2,1,4,5] => 2
[5,3,1,3] => [3,1,3,5] => 3
[5,2,1,4] => [4,1,2,5] => 4
[5,1,1,5] => [5,1,1,5] => 5
[6,6] => [6,6] => 6
[1,2,2,2,2,2,1] => [1,2,2,2,2,2,1] => 1
[1,2,2,2,3,2] => [2,3,2,2,2,1] => 2
[1,2,2,3,3,1] => [1,3,3,2,2,1] => 1
[1,2,2,3,2,2] => [2,2,3,2,2,1] => 2
[1,2,2,4,3] => [3,4,2,2,1] => 3
[1,2,3,3,2,1] => [1,2,3,3,2,1] => 1
[1,2,3,4,2] => [2,4,3,2,1] => 2
[1,2,3,2,3,1] => [1,3,2,3,2,1] => 1
[1,2,3,2,2,2] => [2,2,2,3,2,1] => 2
[1,2,3,3,3] => [3,3,3,2,1] => 3
[1,2,4,4,1] => [1,4,4,2,1] => 1
[1,2,4,3,2] => [2,3,4,2,1] => 2
[1,2,4,2,3] => [3,2,4,2,1] => 3
[1,2,5,4] => [4,5,2,1] => 4
[1,3,3,2,2,1] => [1,2,2,3,3,1] => 1
[1,3,3,3,2] => [2,3,3,3,1] => 2
[1,3,4,3,1] => [1,3,4,3,1] => 1
[1,3,4,2,2] => [2,2,4,3,1] => 2
[1,3,5,3] => [3,5,3,1] => 3
[1,3,2,3,2,1] => [1,2,3,2,3,1] => 1
[1,3,2,4,2] => [2,4,2,3,1] => 2
[1,3,2,2,3,1] => [1,3,2,2,3,1] => 1
[1,3,2,2,2,2] => [2,2,2,2,3,1] => 2
[1,3,2,3,3] => [3,3,2,3,1] => 3
[1,3,3,4,1] => [1,4,3,3,1] => 1
[1,3,3,2,3] => [3,2,3,3,1] => 3
[1,3,4,4] => [4,4,3,1] => 4
[1,4,4,2,1] => [1,2,4,4,1] => 1
[1,4,5,2] => [2,5,4,1] => 2
[1,4,3,3,1] => [1,3,3,4,1] => 1
[1,4,3,2,2] => [2,2,3,4,1] => 2
[1,4,4,3] => [3,4,4,1] => 3
[1,4,2,4,1] => [1,4,2,4,1] => 1
[1,4,2,3,2] => [2,3,2,4,1] => 2
[1,4,2,2,3] => [3,2,2,4,1] => 3
[1,4,3,4] => [4,3,4,1] => 4
[1,5,5,1] => [1,5,5,1] => 1
[1,5,4,2] => [2,4,5,1] => 2
[1,5,3,3] => [3,3,5,1] => 3
[1,5,2,4] => [4,2,5,1] => 4
[1,6,5] => [5,6,1] => 5
[2,3,2,2,2,1] => [1,2,2,2,3,2] => 1
[2,3,2,3,2] => [2,3,2,3,2] => 2
[2,3,3,3,1] => [1,3,3,3,2] => 1
[2,3,3,2,2] => [2,2,3,3,2] => 2
[2,3,4,3] => [3,4,3,2] => 3
[2,4,3,2,1] => [1,2,3,4,2] => 1
[2,4,4,2] => [2,4,4,2] => 2
[2,4,2,3,1] => [1,3,2,4,2] => 1
[2,4,2,2,2] => [2,2,2,4,2] => 2
[2,4,3,3] => [3,3,4,2] => 3
[2,5,4,1] => [1,4,5,2] => 1
[2,5,3,2] => [2,3,5,2] => 2
[2,5,2,3] => [3,2,5,2] => 3
[2,6,4] => [4,6,2] => 4
[2,2,3,2,2,1] => [1,2,2,3,2,2] => 1
[2,2,3,3,2] => [2,3,3,2,2] => 2
[2,2,4,3,1] => [1,3,4,2,2] => 1
[2,2,4,2,2] => [2,2,4,2,2] => 2
[2,2,5,3] => [3,5,2,2] => 3
[2,2,2,3,2,1] => [1,2,3,2,2,2] => 1
[2,2,2,4,2] => [2,4,2,2,2] => 2
[2,2,2,2,3,1] => [1,3,2,2,2,2] => 1
[2,2,2,3,3] => [3,3,2,2,2] => 3
[2,2,3,4,1] => [1,4,3,2,2] => 1
[2,2,3,2,3] => [3,2,3,2,2] => 3
[2,3,4,2,1] => [1,2,4,3,2] => 1
[2,3,5,2] => [2,5,3,2] => 2
[2,3,2,4,1] => [1,4,2,3,2] => 1
[2,3,2,2,3] => [3,2,2,3,2] => 3
[2,3,3,4] => [4,3,3,2] => 4
[2,4,5,1] => [1,5,4,2] => 1
[2,4,2,4] => [4,2,4,2] => 4
[2,5,5] => [5,5,2] => 5
[3,4,2,2,1] => [1,2,2,4,3] => 1
[3,4,3,2] => [2,3,4,3] => 2
[3,5,3,1] => [1,3,5,3] => 1
[3,5,2,2] => [2,2,5,3] => 2
[3,6,3] => [3,6,3] => 3
[3,3,3,2,1] => [1,2,3,3,3] => 1
[3,3,4,2] => [2,4,3,3] => 2
[3,3,2,3,1] => [1,3,2,3,3] => 1
[3,3,2,2,2] => [2,2,2,3,3] => 2
[3,4,4,1] => [1,4,4,3] => 1
[3,4,2,3] => [3,2,4,3] => 3
[3,5,4] => [4,5,3] => 4
[3,2,4,2,1] => [1,2,4,2,3] => 1
[3,2,5,2] => [2,5,2,3] => 2
[3,2,3,3,1] => [1,3,3,2,3] => 1
[3,2,3,2,2] => [2,2,3,2,3] => 2
[3,2,4,3] => [3,4,2,3] => 3
[3,2,2,4,1] => [1,4,2,2,3] => 1
[3,2,2,3,2] => [2,3,2,2,3] => 2
[3,2,2,2,3] => [3,2,2,2,3] => 3
[3,3,5,1] => [1,5,3,3] => 1
[3,3,2,4] => [4,2,3,3] => 4
[3,4,5] => [5,4,3] => 5
[4,5,2,1] => [1,2,5,4] => 1
[4,6,2] => [2,6,4] => 2
[4,4,3,1] => [1,3,4,4] => 1
[4,5,3] => [3,5,4] => 3
[4,3,4,1] => [1,4,3,4] => 1
[4,3,3,2] => [2,3,3,4] => 2
[4,4,4] => [4,4,4] => 4
[4,2,5,1] => [1,5,2,4] => 1
[4,2,4,2] => [2,4,2,4] => 2
[4,2,3,3] => [3,3,2,4] => 3
[4,3,5] => [5,3,4] => 5
[5,6,1] => [1,6,5] => 1
[5,5,2] => [2,5,5] => 2
[5,4,3] => [3,4,5] => 3
[5,3,4] => [4,3,5] => 4
[5,2,5] => [5,2,5] => 5
[3,3,3,2] => [2,3,3,3] => 2
[3,4,4] => [4,4,3] => 4
[2,3,3,3] => [3,3,3,2] => 3
[4,4,3] => [3,4,4] => 3
[2,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,2] => 1
[2,8,1] => [1,8,2] => 1
[2,1,7,1] => [1,7,1,2] => 1
[2,1,1,1,1,1,3,1] => [1,3,1,1,1,1,1,2] => 1
[2,1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1,2] => 1
[11,1] => [1,11] => 1
[3,7,1] => [1,7,3] => 1
[2,1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1,2] => 1
[2,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,2] => 1
[3,2,1,2,2,1] => [1,2,2,1,2,3] => 1
[4,1,1,2,2,1] => [1,2,2,1,1,4] => 1
[3,1,3,1,2,1] => [1,2,1,3,1,3] => 1
[4,1,1,3,1,1] => [1,1,3,1,1,4] => 1
[4,1,2,2,1,1] => [1,1,2,2,1,4] => 1
[5,1,1,2,1,1] => [1,1,2,1,1,5] => 1
[3,1,4,1,1,1] => [1,1,1,4,1,3] => 1
[6,1,1,1,1,1] => [1,1,1,1,1,6] => 1
[3,1,2,2,2,2,1] => [1,2,2,2,2,1,3] => 1
[3,1,2,3,2,1,1] => [1,1,2,3,2,1,3] => 1
[4,2,6] => [6,2,4] => 6
[3,2,2,5] => [5,2,2,3] => 5
[2,2,2,2,4] => [4,2,2,2,2] => 4
[1,2,2,3,4] => [4,3,2,2,1] => 4
[2,2,3,5] => [5,3,2,2] => 5
[3,3,6] => [6,3,3] => 6
[2,3,2,5] => [5,2,3,2] => 5
[1,3,2,2,4] => [4,2,2,3,1] => 4
[2,4,6] => [6,4,2] => 6
[1,5,6] => [6,5,1] => 6
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[5,7] => [7,5] => 7
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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:
1,1 2,1,1 4,2,1,1 8,4,2,1,1 16,8,4,2,1,1 32,16,8,4,2,1,1 64,32,16,8,4,2,1,1 128,64,32,16,8,4,2,1,1
$F_{1} = q$
$F_{2} = q + q^{2}$
$F_{3} = 2\ q + q^{2} + q^{3}$
$F_{4} = 4\ q + 2\ q^{2} + q^{3} + q^{4}$
$F_{5} = 8\ q + 4\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$
$F_{6} = 16\ q + 8\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$
$F_{7} = 32\ q + 16\ q^{2} + 8\ q^{3} + 4\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$
$F_{8} = 64\ q + 32\ q^{2} + 16\ q^{3} + 8\ q^{4} + 4\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$
$F_{9} = 128\ q + 64\ q^{2} + 32\ q^{3} + 16\ q^{4} + 8\ q^{5} + 4\ q^{6} + 2\ q^{7} + q^{8} + q^{9}$
Description
The first part of an integer composition.
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
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