Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001199: Dyck paths ⟶ ℤ
Values
[1,1,0,0] => [2,3,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 2
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 3
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 1
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 3
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 1
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,1,0,0] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,1,1,1,0,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,1,1,1,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 4
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0,1,0] => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0] => 3
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0] => 4
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,1,0,0] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
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Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
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