Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000155: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,3,1] => [3,2,1] => 1
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,2,1,3] => 1
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,1,3,2] => 1
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,3,2,1] => 2
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [5,2,1,4,3] => 1
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [5,2,4,3,1] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,1,3,4,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,1,3,2,5] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [5,3,2,4,1] => 2
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,4,3,2,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [5,1,3,2,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [5,3,2,1,4] => 2
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,2,4,3] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,5,4,3,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [5,1,2,4,3] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [5,1,4,3,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [5,4,2,3,1] => 2
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,2,3,1,6,5] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,2,3,1,5,6] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [6,2,3,1,5,4] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [6,2,3,5,4,1] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,2,3,1,4,6] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [3,2,1,6,4,5] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [2,1,6,4,3,5] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [6,2,1,4,5,3] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [5,2,1,4,3,6] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [6,2,4,3,5,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [6,2,5,4,3,1] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [5,2,4,3,1,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [6,2,1,4,3,5] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [6,2,4,3,1,5] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [4,2,1,3,5,6] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,6,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [6,2,1,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [6,2,1,5,4,3] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [6,2,5,3,4,1] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [5,2,1,3,4,6] => 1
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Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $den$ is the Denert index of a permutation, see St000156The Denert index of a permutation..
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $den$ is the Denert index of a permutation, see St000156The Denert index of a permutation..
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
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