Identifier
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0] => [2,1,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,1,0,0] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0] => [3,2,1] => [1,3,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [1,4,2,5,3] => [1,4,2,5,3] => 2
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => 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 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.