Identifier
Values
[1,0] => [1,0] => 1
[1,0,1,0] => [1,0,1,0] => 2
[1,1,0,0] => [1,1,0,0] => 1
[1,0,1,0,1,0] => [1,1,0,1,0,0] => 3
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 3
[1,1,0,1,0,0] => [1,0,1,0,1,0] => 3
[1,1,1,0,0,0] => [1,1,1,0,0,0] => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => 4
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 4
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => 4
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 4
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 4
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 4
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => 4
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 4
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => 4
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 4
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 3
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => 5
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 5
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 5
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 5
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 5
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 5
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 5
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 5
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 5
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => 5
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 5
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 5
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 5
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 5
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => 5
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => 5
[1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 5
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 5
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => 5
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 5
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 5
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 5
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 5
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 5
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => 4
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 6
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 6
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
>>> Load all 196 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 6
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 6
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 6
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 6
[1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 6
[1,1,0,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 6
[1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 6
[1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 6
[1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 6
[1,1,0,0,1,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 6
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 6
[1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 6
[1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 6
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 6
[1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 6
[1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 6
[1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 6
[1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 6
[1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 5
[1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 6
[1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 6
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 6
[1,1,0,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 5
[1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 6
[1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 6
[1,1,0,1,0,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 6
[1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 6
[1,1,0,1,1,0,0,0,1,0,1,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 6
[1,1,0,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 6
[1,1,0,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 6
[1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 6
[1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 6
[1,1,0,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 6
[1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 6
[1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 6
[1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => 6
[1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 6
[1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 6
[1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 6
[1,1,0,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 6
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 6
[1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 6
[1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5
[1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 6
[1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => 6
[1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 6
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 6
[1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 5
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 6
[1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 5
[1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 6
[1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 6
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 6
[1,1,1,0,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 6
[1,1,1,0,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 6
[1,1,1,0,1,0,0,1,0,0,1,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => 6
[1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 6
[1,1,1,0,1,0,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 6
[1,1,1,0,1,0,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 6
[1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 6
[1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 6
[1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 6
[1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 6
[1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 6
[1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 6
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 6
[1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 6
[1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5
[1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 6
[1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 6
[1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 5
[1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 6
[1,1,1,1,0,0,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 6
[1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 5
[1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 6
[1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 6
[1,1,1,1,0,1,0,0,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 6
[1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 5
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 6
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 6
[1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 5
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 5
[1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
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Description
Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra.
Map
Barnabei-Castronuovo involution
Description
The Barnabei-Castronuovo Schützenberger involution on Dyck paths.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let $D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is $\dots, 1, D_2, 1, D_1$.