Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤ (values match St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.)
Values
[1,0] => [2,1] => [1,2] => 1
[1,0,1,0] => [3,1,2] => [1,3,2] => 1
[1,1,0,0] => [2,3,1] => [2,1,3] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => 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] => [1,2,4,3] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [3,2,1,4] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,5,4,1,3] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,1,4,2] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,2,1,4] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [4,1,5,3,2] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [2,3,5,1,4] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,3,1,5,2] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,1,2,5,3] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,3,2,5,4] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [4,3,2,1,5] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,6,5,4,1,3] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [3,6,5,1,4,2] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,5,2,4,3] => 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] => [4,6,1,5,3,2] => 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] => [1,6,3,5,4,2] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,2,5,4,3] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,6,3,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [4,6,3,1,5,2] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,6,1,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,3,2,5,4] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [4,6,3,2,1,5] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,6,4,1,3] => 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] => [5,1,6,2,4,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [5,3,6,2,1,4] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [2,4,6,5,1,3] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,3,6,5,4,2] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [2,3,6,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,4,6,1,5,2] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,4,6,2,5,3] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,3,6,2,5,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,4,6,2,1,5] => 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] => [5,4,2,6,1,3] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [5,1,3,6,4,2] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,1,2,6,4,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,2,3,6,1,4] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,4,3,6,5,2] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,4,2,6,5,3] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [2,4,3,6,1,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,4,1,2,6,3] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,1,3,2,6,4] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,4,3,2,6,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [5,4,3,2,1,6] => 5
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => 1
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => [1,3,4,7,2,6,5] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => 1
[] => [1] => [1] => 1
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Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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