Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [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] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [-3,1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [-2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [2,-3,1] => 2
[3,1,2] => [3,2,1] => [3,2,1] => [2,-3,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [2,-3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 2
[2,4,1,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[4,1,2,3] => [4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 2
[4,1,3,2] => [4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 2
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The flag descent of a signed permutation.
$$ fdes(\sigma) = 2 \lvert \{ i \in [n-1] \mid \sigma(i) > \sigma(i+1) \} \rvert + \chi( \sigma(1) < 0 ) $$
It has the same distribution as the flag excedance statistic.
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 $\pi = \pi^{-1}$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Foata-Han inverse
Description
The inverse of the Foata-Han bijection for signed permutations.
This map sends the the flag inversion number St001428The number of B-inversions of a signed permutation. to the flag major index St001433The flag major index of a signed permutation.