Identifier
Values
[1,0] => [1,0] => [1] => [1] => 0
[1,0,1,0] => [1,0,1,0] => [1,2] => [1,2] => 0
[1,1,0,0] => [1,1,0,0] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [3,1,2] => [3,1,2] => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,1,4,2] => 2
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,2,5,3] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
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Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
reverse
Description
The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.