Identifier
-
Mp00223:
Permutations
—runsort⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤ
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,5,2,4,3] => [1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,5,3,2,4] => [1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,1,3,5,4] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,3,1,4,5] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[2,4,1,3,5] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,4,3,1,5] => [1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[2,5,1,4,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,5,3,1,4] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,1,2,5,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,1,4,2,5] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,1,4,5,2] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,1,5,2,4] => [1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,1,5,4,2] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,2,1,4,5] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,2,1,5,4] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,2,4,1,5] => [1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,2,5,1,4] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,2,5,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,4,1,2,5] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,4,1,5,2] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,4,2,1,5] => [1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[3,5,1,4,2] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
>>> Load all 153 entries. <<<
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Description
The order of a signed permutation.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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