Identifier
  • Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
    St001234: Dyck paths ⟶ ℤ (values match St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.)
Values
[1,0] => [1,0] => 0
[1,0,1,0] => [1,1,0,0] => 0
[1,1,0,0] => [1,0,1,0] => 0
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 0
[1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 0
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 0
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 0
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 0
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 0
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 0
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 0
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 3
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 0
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 4
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 0
>>> Load all 196 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 3
[1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 0
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 0
[1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 0
[1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 0
[1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 0
[1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 1
[1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 0
[1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 0
[1,1,1,0,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
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Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.