Identifier
-
Mp00327:
Dyck paths
—inverse Kreweras complement⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => 1
[1,0,1,0] => [1,1,0,0] => [1,2] => 2
[1,1,0,0] => [1,0,1,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 3
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [3,1,2] => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [2,1,3] => 3
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [1,3,2] => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [2,3,1] => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 4
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 2
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => 3
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 4
[1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 3
[1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 2
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 4
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 4
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 5
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 5
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 3
[1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 5
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 4
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 5
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 3
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 4
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 5
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 5
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 2
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 5
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 3
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 5
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => 3
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => 5
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,6,2,3,4] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [5,1,6,2,3,4] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4,6] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,6,2,4,5] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,5,6,1,2,4] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,6,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [3,4,6,1,2,5] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,1,5,2,3,6] => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4,6] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [3,1,2,6,4,5] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [3,1,5,6,2,4] => 4
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Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
This statistic is undefined for the empty permutation.
Map
inverse Kreweras complement
Description
Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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