Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000672: Permutations ⟶ ℤ
Values
[1,0] => [2,1] => [2,1] => [1,2] => 1
[1,0,1,0] => [3,1,2] => [3,2,1] => [2,1,3] => 1
[1,1,0,0] => [2,3,1] => [3,2,1] => [2,1,3] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [4,2,3,1] => [2,3,1,4] => 2
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,3,1] => [2,3,1,4] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [3,4,1,2] => [4,1,2,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [4,3,2,1] => [3,2,1,4] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,2,3,4,1] => [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,2,3,4,1] => [2,3,4,1,5] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,5,1,3] => [2,5,1,3,4] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,2,4,3,1] => [2,4,3,1,5] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,5,1,4,2] => [5,1,4,2,3] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,1,4,2] => [5,1,4,2,3] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,4,3,2,1] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [5,4,3,2,1] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,3,2,4,1] => [3,2,4,1,5] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,2,5,1,3] => [2,5,1,3,4] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,5,3,1,2] => [5,3,1,2,4] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1,5] => 3
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => 4
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,2,3,6,1,4] => [2,3,6,1,4,5] => 4
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,2,3,5,4,1] => [2,3,5,4,1,6] => 3
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => 4
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [2,6,1,5,3,4] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [2,6,1,5,3,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,2,5,4,3,1] => [2,5,4,3,1,6] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,2,5,4,3,1] => [2,5,4,3,1,6] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,2,4,3,5,1] => [2,4,3,5,1,6] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [2,3,6,1,4,5] => 4
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [2,6,4,1,3,5] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,2,5,4,3,1] => [2,5,4,3,1,6] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,6,1,4,5,2] => [6,1,4,5,2,3] => 3
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => [6,1,4,5,2,3] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,5,1,6,2,4] => [5,1,6,2,4,3] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,6,1,4,5,2] => [6,1,4,5,2,3] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,4,3,2,5,1] => [4,3,2,5,1,6] => 2
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,3,2,5,1] => [4,3,2,5,1,6] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [6,5,3,4,2,1] => [5,3,4,2,1,6] => 2
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,5,3,4,2,1] => [5,3,4,2,1,6] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [6,4,3,2,5,1] => [4,3,2,5,1,6] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => [3,2,6,1,4,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,5,3,4,2,1] => [5,3,4,2,1,6] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [6,5,3,4,2,1] => [5,3,4,2,1,6] => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,3,2,4,5,1] => [3,2,4,5,1,6] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,2,6,1,5,3] => [2,6,1,5,3,4] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => [2,6,1,5,3,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,6,3,1,5,2] => [6,3,1,5,2,4] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,4,3,2,5,1] => [4,3,2,5,1,6] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,2,3,6,1,4] => [2,3,6,1,4,5] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,6,4,1,3] => [2,6,4,1,3,5] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,6,4,3,1,2] => [6,4,3,1,2,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,2,3,4,5,6,1] => [2,3,4,5,6,1,7] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [7,2,3,4,5,6,1] => [2,3,4,5,6,1,7] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [6,2,3,4,7,1,5] => [2,3,4,7,1,5,6] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [7,2,3,4,5,6,1] => [2,3,4,5,6,1,7] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [6,2,3,4,7,1,5] => [2,3,4,7,1,5,6] => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [7,2,3,4,5,6,1] => [2,3,4,5,6,1,7] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [7,2,6,5,4,3,1] => [2,6,5,4,3,1,7] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [7,2,6,5,4,3,1] => [2,6,5,4,3,1,7] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [7,2,6,5,4,3,1] => [2,6,5,4,3,1,7] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [6,2,3,4,7,1,5] => [2,3,4,7,1,5,6] => 5
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [7,2,6,5,4,3,1] => [2,6,5,4,3,1,7] => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [7,2,3,4,5,6,1] => [2,3,4,5,6,1,7] => 5
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [7,6,3,4,5,2,1] => [6,3,4,5,2,1,7] => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => [7,6,3,4,5,2,1] => [6,3,4,5,2,1,7] => 3
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [7,6,3,4,5,2,1] => [6,3,4,5,2,1,7] => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [7,6,3,5,4,2,1] => [6,3,5,4,2,1,7] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [7,6,3,5,4,2,1] => [6,3,5,4,2,1,7] => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [7,6,3,5,4,2,1] => [6,3,5,4,2,1,7] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => [7,6,3,5,4,2,1] => [6,3,5,4,2,1,7] => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [7,6,3,5,4,2,1] => [6,3,5,4,2,1,7] => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => [7,5,4,3,2,6,1] => [5,4,3,2,6,1,7] => 2
[1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => [7,5,4,3,2,6,1] => [5,4,3,2,6,1,7] => 2
[1,1,1,0,1,0,0,1,1,0,0,0] => [6,3,5,1,2,7,4] => [7,5,4,3,2,6,1] => [5,4,3,2,6,1,7] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,5,4,1,2,7,3] => [7,5,4,3,2,6,1] => [5,4,3,2,6,1,7] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => [7,6,4,3,5,2,1] => [6,4,3,5,2,1,7] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [7,5,4,3,2,6,1] => [5,4,3,2,6,1,7] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [6,2,3,4,7,1,5] => [2,3,4,7,1,5,6] => 5
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1,7] => 1
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Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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