Identifier
Values
[.,.] => [1] => [1] => [-1] => 0
[.,[.,.]] => [2,1] => [2,1] => [1,-2] => 1
[[.,.],.] => [1,2] => [1,2] => [-2,1] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => [1,2,-3] => 2
[.,[[.,.],.]] => [2,3,1] => [2,3,1] => [2,1,-3] => 1
[[.,.],[.,.]] => [3,1,2] => [3,1,2] => [2,-3,1] => 1
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => [1,-3,2] => 1
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [-3,1,2] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 3
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => [2,1,3,-4] => 2
[.,[[.,.],[.,.]]] => [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 2
[.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => [1,3,2,-4] => 2
[.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,4,1] => [3,1,2,-4] => 1
[[.,.],[.,[.,.]]] => [4,3,1,2] => [4,3,1,2] => [2,3,-4,1] => 2
[[.,.],[[.,.],.]] => [3,4,1,2] => [3,4,1,2] => [3,2,-4,1] => 2
[[.,[.,.]],[.,.]] => [4,2,1,3] => [4,2,1,3] => [1,3,-4,2] => 2
[[[.,.],.],[.,.]] => [4,1,2,3] => [4,1,2,3] => [3,-4,1,2] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1,4] => [2,1,-4,3] => 1
[[[.,.],[.,.]],.] => [3,1,2,4] => [3,1,2,4] => [2,-4,1,3] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => [1,-4,2,3] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,-5] => 4
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [4,3,5,2,1] => [1,3,2,4,-5] => 3
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,-5] => 3
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [4,3,2,5,1] => [1,2,4,3,-5] => 3
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [3,2,4,5,1] => [1,4,2,3,-5] => 2
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 3
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => 3
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [5,2,1,3,4] => [1,4,-5,2,3] => 2
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 3
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,2,4,1,5] => [1,3,2,-5,4] => 2
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => [1,-5,2,3,4] => 1
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Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
Map
rowmotion
Description
The rowmotion of a signed permutation with respect to the sorting order.
The sorting order on signed permutations (with respect to the Coxeter element $-n, 1, 2,\dots, n-1$) is defined in [1].