Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤ
Values
[1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,6,3,4] => [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] => [4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,6,1,4,2,5] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,5,1,2,6,4] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,1,6,3,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,6,1,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,5,6,2,4] => [1,1,1,0,0,1,1,0,1,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,1,5,6,3,2] => [1,1,1,1,0,0,1,0,1,0,0,0] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,1,5,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,4,6,2,5,1] => [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] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,5,6,3,1] => [1,1,1,1,0,0,1,0,1,0,0,0] => 1
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,1,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,2,1,6,4,3] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,7,4,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,2,7,3,4,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,2,7,3,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [4,1,7,2,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [6,1,5,7,2,3,4] => [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] => [4,1,7,2,5,3,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [4,1,6,2,3,7,5] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,7,2,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [3,7,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [3,6,1,2,7,4,5] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [3,7,1,6,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [6,4,2,1,7,3,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [6,7,2,3,4,1,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [6,5,2,7,1,3,4] => [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] => [5,4,7,1,2,3,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [5,4,7,1,2,6,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [3,7,1,4,2,5,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [3,6,1,4,7,2,5] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [3,5,1,2,7,4,6] => [1,1,1,0,1,1,0,0,0,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [3,6,1,2,4,7,5] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [3,5,1,6,7,4,2] => [1,1,1,0,1,1,0,0,1,0,1,0,0,0] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [6,5,7,1,4,2,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [3,7,1,4,5,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [3,6,1,4,2,7,5] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [3,6,1,2,7,5,4] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,7,1,4,5,6,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [6,2,1,3,7,4,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [5,2,1,7,3,4,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [5,2,1,7,3,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [4,2,7,1,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [6,2,5,7,1,3,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [4,2,7,1,5,3,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [4,2,6,1,3,7,5] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [3,1,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [3,1,6,2,7,4,5] => [1,1,1,0,0,1,1,1,0,0,1,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [3,1,5,7,2,4,6] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => [3,1,7,6,4,2,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [3,1,5,7,2,6,4] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [4,1,2,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [4,1,2,6,7,3,5] => [1,1,1,1,0,0,0,1,1,0,1,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [6,1,2,7,5,3,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => [4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => [4,1,7,6,2,3,5] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => [4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [3,4,7,2,1,5,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => [3,4,6,2,7,1,5] => [1,1,1,0,1,0,1,1,0,0,1,0,0,0] => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [3,7,5,1,2,4,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,0,0,1,0,1,0,0] => [7,3,1,6,2,4,5] => [3,7,6,1,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => [3,6,5,7,2,1,4] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [4,7,1,5,2,3,6] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [4,7,1,6,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => [4,6,5,7,2,3,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0] => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => [3,4,7,2,5,1,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 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
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
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