Identifier
-
Mp00253:
Decorated permutations
—permutation⟶
Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤ
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [1,2] => [1,2] => [1,2] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,-,-] => [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] => [1,3,2] => [1,3,2] => 0
[-,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,+] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[3,+,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[+,+,+,+] => [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
[+,-,+,+] => [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
[+,+,+,-] => [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
[-,+,-,+] => [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
[+,-,-,+] => [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
[+,+,-,-] => [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
[-,-,+,-] => [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
[+,-,-,-] => [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] => [1,2,4,3] => [1,2,4,3] => 0
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,3,4,2] => [1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[-,3,4,2] => [1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[+,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[-,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,+] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[2,3,1,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 2
[2,4,1,3] => [2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,+,1] => [2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 1
[2,4,-,1] => [2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 1
[3,1,2,+] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,2,-] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,4,2] => [3,1,4,2] => [2,3,4,1] => [2,3,4,1] => 2
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,4,1] => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,-,4,1] => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 2
[4,1,+,2] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 1
[4,1,-,2] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 1
[4,+,1,3] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[4,-,1,3] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[4,+,+,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,-,+,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,+,-,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,-,-,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[+,+,+,+,+] => [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
[+,-,+,+,+] => [1,2,3,4,5] => [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
[+,+,+,-,+] => [1,2,3,4,5] => [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
[-,-,+,+,+] => [1,2,3,4,5] => [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
[-,+,+,-,+] => [1,2,3,4,5] => [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
[+,-,-,+,+] => [1,2,3,4,5] => [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
[+,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
- $i < j \leq \pi(i) < \pi(j)$, or
- $-i < j \leq -\pi(i) < \pi(j)$, or
- $i > j > \pi(i) > \pi(j)$.
Map
catalanization
Description
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
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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