Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000622: Permutations ⟶ ℤ
Values
[1,0] => [2,1] => [2,1] => [2,1] => 0
[1,0,1,0] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[1,1,0,0] => [2,3,1] => [2,3,1] => [1,3,2] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,3,2] => [3,2,4,1] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => [1,2,3,4] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => [4,3,2,1] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [4,3,1,2] => [3,2,1,4] => 0
[1,1,1,0,0,0] => [2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,3,2] => [1,3,4,2,5] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,4,2] => [1,2,4,3,5] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,1,4,3,2] => [4,3,1,2,5] => 0
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,5,4,2] => [1,2,4,3,5] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,5,1,4,3] => [5,4,3,1,2] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,5,1,4,3] => [5,4,3,1,2] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => 0
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [5,4,1,3,2] => [4,3,2,5,1] => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,1,5,2] => [1,3,2,4,5] => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,4,1,3] => [5,4,3,2,1] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,5,4,1,3] => [5,4,3,2,1] => 0
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,3,4,1,2] => [4,2,3,1,5] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,5,4,3,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,3,5,4,2,6] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,4,3,2] => [1,4,5,3,2,6] => 0
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,1,5,4,3,2] => [5,4,1,3,2,6] => 0
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,2,5,4,3,6] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,6,1,5,4,3] => [6,5,4,1,3,2] => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,3,1,5,4,2] => [1,5,2,4,3,6] => 0
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,6,4,2] => [1,4,2,5,3,6] => 0
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [6,4,1,5,3,2] => [1,5,3,4,2,6] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,6,1,4,3,2] => [4,5,3,1,2,6] => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,6,3,2] => [1,4,3,5,2,6] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,5,2] => [1,3,2,5,4,6] => 0
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,3,1,5,4,2] => [1,5,2,4,3,6] => 0
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [6,4,1,5,3,2] => [1,5,3,4,2,6] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,1,6,5,2] => [1,3,2,5,4,6] => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,6,5,1,4,3] => [6,5,4,3,1,2] => 0
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,3,5,1,4,2] => [5,2,4,3,6,1] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [6,3,5,1,4,2] => [5,2,4,3,6,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [6,5,4,1,3,2] => [5,4,3,2,6,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,3,6,1,4,2] => [4,2,5,3,6,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,6,5,4,1,3] => [6,5,4,3,2,1] => 0
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,3,5,4,1,2] => [5,2,4,3,1,6] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => [1,3,6,5,4,2,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => [1,4,6,5,3,2,7] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => [1,2,6,5,4,3,7] => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => [1,4,2,6,5,3,7] => 0
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => [5,4,1,7,6,3,2] => [1,4,3,6,5,2,7] => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => [5,4,1,7,6,3,2] => [1,4,3,6,5,2,7] => 0
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [4,3,1,7,6,5,2] => [1,3,2,6,5,4,7] => 0
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Description
The number of occurrences of the patterns 2143 or 4231 in a permutation.
It is a necessary and sufficient condition that a permutation $\pi$ avoids these two patterns for the Schubert variety associated to $\pi$ to be smooth [1].
It is a necessary and sufficient condition that a permutation $\pi$ avoids these two patterns for the Schubert variety associated to $\pi$ to be smooth [1].
Map
toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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