Identifier
Values
[1,0] => [1,1,0,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,0,0] => [1,1,1,0,0,0] => [3,2,1] => [2,3,1] => 2
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,2,4,1] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => 2
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,3,4,1] => 3
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,4,1,2] => 2
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [3,2,5,4,1] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [3,2,4,5,1] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,2,5,1,3] => 2
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,5,3,4,1] => 2
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [2,4,3,5,1] => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [2,3,5,4,1] => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [2,3,4,5,1] => 4
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [2,4,5,1,3] => 3
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,5,1,4,2] => 2
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [3,4,1,5,2] => 3
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [3,5,4,1,2] => 3
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [3,4,5,1,2] => 3
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [5,2,3,4,6,1] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [4,2,3,6,5,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [4,2,3,5,6,1] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [5,2,3,6,1,4] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [3,2,6,4,5,1] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => [3,2,5,4,6,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,5,3,4,6,1] => [3,2,4,6,5,1] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [3,2,4,5,6,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => [3,2,5,6,1,4] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [4,2,6,1,5,3] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => [4,2,5,1,6,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,6,4,5,3,1] => [4,2,6,5,1,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [4,2,5,6,1,3] => 3
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [2,6,3,4,5,1] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [2,5,3,4,6,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [2,4,3,6,5,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => [2,4,3,5,6,1] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [2,5,3,6,1,4] => 3
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [2,3,6,4,5,1] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,2,3,6,5,1] => [2,3,5,4,6,1] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [2,3,4,6,5,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [2,3,4,5,6,1] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [2,3,5,6,1,4] => 4
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => [2,4,6,1,5,3] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => [2,4,5,1,6,3] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [2,4,6,5,1,3] => 4
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [2,4,5,6,1,3] => 4
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,6,1,4,5,2] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [3,5,1,4,6,2] => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,4,1,6,5,2] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => [3,4,1,5,6,2] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => [3,5,1,6,2,4] => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => [3,6,4,1,5,2] => 3
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => [3,5,4,1,6,2] => 4
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [3,6,4,5,1,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => [3,5,4,6,1,2] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [3,4,6,1,5,2] => 3
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => [3,4,5,1,6,2] => 4
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [3,4,6,5,1,2] => 4
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [3,4,5,6,1,2] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => 3
[] => [1,0] => [1] => [1] => 0
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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..
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
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.