Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000007: Permutations ⟶ ℤ
Values
[1,0] => [2,1] => [1,2] => 1
[1,0,1,0] => [3,1,2] => [2,1,3] => 1
[1,1,0,0] => [2,3,1] => [1,3,2] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [3,2,1,4] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [2,1,3,4] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [3,5,2,1,4] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,5,1,3] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,2,4,1,5] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,4,1,3] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [4,3,1,5,2] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,1,4,2] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [4,2,1,3,5] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,2,1,4,5] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [2,5,1,3,4] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,5,3,2] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,1,4,5,2] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,1,4,3,5] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [4,6,3,2,1,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,3,6,2,1,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [4,3,5,2,1,6] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [3,6,5,2,1,4] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [5,4,2,6,1,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [5,3,2,4,1,6] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [4,3,2,5,1,6] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [3,6,2,4,1,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,2,6,4,1,3] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,2,5,6,1,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [3,2,5,4,1,6] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,5,4,1,3] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [5,4,3,1,6,2] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [4,6,3,1,5,2] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [5,3,6,1,4,2] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [4,3,5,1,6,2] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,6,5,1,4,2] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [5,4,2,1,3,6] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [4,6,2,1,3,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [5,3,2,1,4,6] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,6,2,1,4,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,2,6,1,3,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,2,5,1,3,6] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,2,5,1,4,6] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [2,6,5,1,3,4] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,6,1,5,3,2] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [5,3,1,4,6,2] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,3,1,5,6,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [3,6,1,4,5,2] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [5,2,1,4,3,6] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,2,1,5,3,6] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [2,6,1,4,3,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,6,4,3,2] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [4,1,5,6,3,2] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,1,5,4,6,2] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [2,1,5,4,3,6] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,6,5,4,3,2] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [5,4,6,3,2,1,7] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [6,4,3,5,2,1,7] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [5,4,3,6,2,1,7] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [4,3,6,5,2,1,7] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [6,4,7,2,5,1,3] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [6,4,3,2,5,1,7] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [6,3,2,5,4,1,7] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [5,3,2,6,1,4,7] => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [5,2,6,7,1,3,4] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [8,1,2,3,4,7,5,6] => [6,5,7,4,3,2,1,8] => 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [8,1,2,3,7,4,5,6] => [6,5,4,7,3,2,1,8] => 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [8,1,2,5,3,4,6,7] => [7,6,4,3,5,2,1,8] => 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [8,1,2,6,3,4,5,7] => [7,5,4,3,6,2,1,8] => 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [8,1,2,5,6,3,4,7] => [7,4,3,6,5,2,1,8] => 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [8,1,4,2,3,7,5,6] => [6,5,7,3,2,4,1,8] => 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [6,1,8,2,3,4,5,7] => [7,5,4,3,2,8,1,6] => 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [8,1,4,5,6,7,2,3] => [3,2,7,6,5,4,1,8] => 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [7,3,1,2,4,5,8,6] => [6,8,5,4,2,1,3,7] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [5,8,1,2,6,7,3,4] => [4,3,7,6,2,1,8,5] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [8,3,1,5,2,7,4,6] => [6,4,7,2,5,1,3,8] => 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [2,6,7,1,3,4,8,5] => [5,8,4,3,1,7,6,2] => 4
[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [6,3,4,1,2,7,8,5] => [5,8,7,2,1,4,3,6] => 3
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [6,3,8,1,2,7,4,5] => [5,4,7,2,1,8,3,6] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 3
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [7,8,4,1,6,2,3,5] => [5,3,2,6,1,4,8,7] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => [7,1,8,6,5,4,3,2] => 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [9,1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1,9] => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,9,6,8] => [8,6,9,4,7,2,5,1,3] => 4
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0] => [9,8,5,1,7,2,3,4,6] => [6,4,3,2,7,1,5,8,9] => 1
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0] => [9,5,4,1,8,7,2,3,6] => [6,3,2,7,8,1,4,5,9] => 1
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Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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