Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤ
Values
[1,0,1,0] => [1,2] => [1] => [1] => 0
[1,1,0,0] => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0] => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0] => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0] => [3,2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1] => [2,3,1] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,1] => [2,3,1] => 2
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,3,1] => [2,3,1] => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [2,4,3,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [2,4,3,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,5,4,3,2] => [1,5,4,3,2] => 2
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Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
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..
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
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
This map is undefined for the empty permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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