Identifier
-
Mp00253:
Decorated permutations
—permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001768: Signed permutations ⟶ ℤ
Values
[+] => [1] => [1] => [1] => 1
[-] => [1] => [1] => [1] => 1
[+,+] => [1,2] => [2,1] => [2,1] => 1
[-,+] => [1,2] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [2,1] => [2,1] => 1
[-,-] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 1
[+,+,+] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[-,+,+] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[+,-,+] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[+,+,-] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[-,-,+] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[-,+,-] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[+,-,-] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[-,-,-] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[+,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[-,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,+] => [2,1,3] => [2,3,1] => [2,3,1] => 1
[2,1,-] => [2,1,3] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[3,+,1] => [3,2,1] => [1,2,3] => [1,2,3] => 1
[3,-,1] => [3,2,1] => [1,2,3] => [1,2,3] => 1
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,+,+,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,-,+,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,+,-,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,-,+,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,+,-,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,+,+,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,-,-,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,-,+,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,-,-,+] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,-,+,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,+,-,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 16
[+,+,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 5
[-,+,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 5
[+,-,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 5
[-,-,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 5
[+,3,2,+] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 6
[-,3,2,+] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 6
[+,3,2,-] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 6
[-,3,2,-] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 6
[+,3,4,2] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 3
[-,3,4,2] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 3
[+,4,2,3] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 3
[-,4,2,3] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 3
[+,4,+,2] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[-,4,+,2] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[+,4,-,2] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[-,4,-,2] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[2,1,+,+] => [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 5
[2,1,-,+] => [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 5
[2,1,+,-] => [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 5
[2,1,-,-] => [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 5
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[2,3,1,+] => [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 3
[2,3,1,-] => [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 3
[2,3,4,1] => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[2,4,+,1] => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 1
[2,4,-,1] => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 1
[3,1,2,+] => [3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 3
[3,1,2,-] => [3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 3
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[3,+,1,+] => [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 1
[3,-,1,+] => [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 1
[3,+,1,-] => [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 1
[3,-,1,-] => [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 1
[3,+,4,1] => [3,2,4,1] => [2,3,1,4] => [2,3,1,4] => 1
[3,-,4,1] => [3,2,4,1] => [2,3,1,4] => [2,3,1,4] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,+,2] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,1,-,2] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,+,1,3] => [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[4,-,1,3] => [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[4,+,+,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,-,+,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,+,-,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,-,-,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
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Description
The number of reduced words of a signed permutation.
This is the number of ways to write a permutation as a minimal length product of simple reflections.
This is the number of ways to write a permutation as a minimal length product of simple reflections.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
permutation
Description
The underlying permutation of the decorated permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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