Identifier
-
Mp00256:
Decorated permutations
—upper permutation⟶
Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤ
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 2
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[-,3,2] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[2,1,+] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[2,1,-] => [2,1,3] => [2,3,1] => [2,3,1] => 2
[2,3,1] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[3,+,1] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[3,-,1] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 1
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 2
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 3
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[-,+,4,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[+,3,2,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 1
[-,3,2,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 2
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[+,4,2,3] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 1
[-,4,2,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[+,4,+,2] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 1
[-,4,+,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[+,4,-,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,4,-,2] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,1,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[2,1,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 3
[2,1,4,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,1,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[2,3,1,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 2
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,1,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[2,4,+,1] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[2,4,-,1] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[3,1,2,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[3,+,1,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[3,-,1,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[3,+,1,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,-,1,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 2
[3,+,4,1] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[3,-,4,1] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,1,2,3] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,+,2] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,-,2] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,+,1,3] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,-,1,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,+,+,1] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,-,+,1] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,+,-,1] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,-,-,1] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,3,2,1] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,+,+,-] => [1,3,4,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[+,+,-,-,+] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[+,+,-,+,-] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,-,+,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[+,-,-,-,+] => [1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[+,-,-,+,-] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[+,-,+,-,-] => [1,3,2,4,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[+,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,-,-,+] => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 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 $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Map
upper permutation
Description
The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
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