Identifier
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 2
[1,1,1,0,0,0] => [3,2,1] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 3
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 3
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 3
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => 2
>>> Load all 196 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 3
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,6,4,3,5,2] => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => 3
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => 3
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => 3
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,6,4,5,3] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 4
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 3
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 4
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 4
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 5
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => 4
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => 4
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => 4
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => 4
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,5,3,4,1,6] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,5,3,4,6,1] => 3
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => 3
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => 3
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => 4
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => 3
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,6,4,5,3,1] => 4
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => 3
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => 3
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => 3
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => 3
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => 4
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => 3
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => 3
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,2,3,1,5,6] => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,2,3,1,6,5] => 2
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,2,3,5,1,6] => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => 3
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,2,3,6,5,1] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,3,4,1,6] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => 2
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,2,4,3,1,6] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => 3
[1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 3
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => 3
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => 4
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => 3
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,3,2,4,1,6] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 3
[1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => 3
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,3,4,2,1,6] => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => 4
[1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => 4
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => 3
[1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => 3
[1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 3
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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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..