Identifier
Values
[1,1] => [1,0,1,0] => [1,2] => [1,2] => 0
[2] => [1,1,0,0] => [2,1] => [2,1] => 0
[1,1,1] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[1,2] => [1,0,1,1,0,0] => [1,3,2] => [3,1,2] => 1
[2,1] => [1,1,0,0,1,0] => [2,1,3] => [2,3,1] => 1
[3] => [1,1,1,0,0,0] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,1,2,3] => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,3,1,2] => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,3,4,1] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,4,2,1] => 1
[4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,4,2,5] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [3,1,5,2,4] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,4,3,1,2] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,3,4,5,1] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,4,1,5,3] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [5,2,1,4,3] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,5,4,1] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,5,3,2,1] => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,5,2,3,4,6] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [4,1,6,2,3,5] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,5,4,2,3,6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,4,5,2,6] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [3,1,2,6,4,5] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [6,3,1,5,2,4] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,5,3,2,6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [4,1,6,3,2,5] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [2,3,4,5,6,1] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [2,3,1,5,6,4] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [6,2,1,3,5,4] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,4,5,1,6,3] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [2,5,4,1,6,3] => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [6,5,2,1,4,3] => 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [3,4,5,6,2,1] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [3,2,4,6,5,1] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [3,5,2,6,4,1] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [4,5,6,3,2,1] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [4,3,6,5,2,1] => 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => [5,6,4,3,2,1] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => [1,2,5,3,4,6,7] => 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => [1,2,4,5,3,6,7] => 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => [1,4,2,6,3,5,7] => 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => [1,3,4,5,6,2,7] => 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => 0
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => [1,3,5,2,6,4,7] => 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => [1,4,5,6,3,2,7] => 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => [1,4,3,6,5,2,7] => 1
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Description
The number of cycles of length at least 3 of a permutation.
Map
bounce path
Description
The bounce path determined by an integer composition.
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
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.