Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [-2,1] => [1,-2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [-3,1,2] => [2,1,-3] => 2
[2,1,3] => [2,1,3] => [-2,1,3] => [3,1,-2] => 3
[2,3,1] => [2,3,1] => [-3,-2,1] => [1,-2,-3] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[3,2,1] => [3,2,1] => [2,-3,1] => [1,-3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => [3,2,1,-4] => 3
[1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => [4,2,1,-3] => 4
[1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => [2,1,-3,-4] => 2
[1,4,2,3] => [1,4,2,3] => [4,1,2,3] => [3,2,1,4] => 0
[1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => [2,1,-4,3] => 2
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [4,3,1,-2] => 5
[2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => [3,1,-2,-4] => 6
[2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => [4,1,-2,-3] => 3
[2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => [1,-2,-3,-4] => 1
[2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => [3,1,-4,2] => 2
[2,4,3,1] => [2,4,3,1] => [3,-4,-2,1] => [1,-2,-4,3] => 3
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 0
[3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => [2,1,-3,4] => 2
[3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => [4,1,-3,2] => 3
[3,2,4,1] => [3,2,4,1] => [2,-4,-3,1] => [1,-3,-4,2] => 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [2,4,-3,1] => [1,-3,4,2] => 2
[4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => [3,2,-4,1] => 2
[4,1,3,2] => [4,1,3,2] => [-3,-4,1,2] => [2,1,-4,-3] => 2
[4,2,1,3] => [4,2,1,3] => [-2,-4,1,3] => [3,1,-4,-2] => 5
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => [1,-4,3,2] => 1
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => [2,1,3,-4] => 3
[4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => [1,-4,2,-3] => 2
[2,3,4,5,1] => [2,3,4,5,1] => [-5,-4,-3,-2,1] => [1,-2,-3,-4,-5] => 1
[2,3,5,4,1] => [2,3,5,4,1] => [4,-5,-3,-2,1] => [1,-2,-3,-5,4] => 4
[2,4,3,5,1] => [2,4,3,5,1] => [3,-5,-4,-2,1] => [1,-2,-4,-5,3] => 3
[2,4,5,3,1] => [2,4,5,3,1] => [3,5,-4,-2,1] => [1,-2,-4,5,3] => 4
[2,5,3,4,1] => [2,5,3,4,1] => [3,4,-5,-2,1] => [1,-2,-5,4,3] => 3
[2,5,4,3,1] => [2,5,4,3,1] => [-4,3,-5,-2,1] => [1,-2,-5,3,-4] => 4
[3,2,4,5,1] => [3,2,4,5,1] => [2,-5,-4,-3,1] => [1,-3,-4,-5,2] => 1
[3,2,5,4,1] => [3,2,5,4,1] => [2,4,-5,-3,1] => [1,-3,-5,4,2] => 4
[3,4,2,5,1] => [3,4,2,5,1] => [2,5,-4,-3,1] => [1,-3,-4,5,2] => 2
[3,4,5,2,1] => [3,4,5,2,1] => [2,4,5,-3,1] => [1,-3,5,4,2] => 3
[3,5,2,4,1] => [3,5,2,4,1] => [-4,2,-5,-3,1] => [1,-3,-5,2,-4] => 2
[3,5,4,2,1] => [3,5,4,2,1] => [-5,2,4,-3,1] => [1,-3,4,2,-5] => 6
[4,2,3,5,1] => [4,2,3,5,1] => [2,3,-5,-4,1] => [1,-4,-5,3,2] => 1
[4,2,5,3,1] => [4,2,5,3,1] => [2,3,5,-4,1] => [1,-4,5,3,2] => 2
[4,3,2,5,1] => [4,3,2,5,1] => [-3,2,-5,-4,1] => [1,-4,-5,2,-3] => 2
[4,3,5,2,1] => [4,3,5,2,1] => [-3,2,5,-4,1] => [1,-4,5,2,-3] => 4
[4,5,2,3,1] => [4,5,2,3,1] => [-5,2,3,-4,1] => [1,-4,3,2,-5] => 3
[4,5,3,2,1] => [4,5,3,2,1] => [-5,-3,2,-4,1] => [1,-4,2,-3,-5] => 4
[5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,-5,1] => [1,-5,4,3,2] => 1
[5,2,4,3,1] => [5,2,4,3,1] => [-4,2,3,-5,1] => [1,-5,3,2,-4] => 3
[5,3,2,4,1] => [5,3,2,4,1] => [-3,2,4,-5,1] => [1,-5,4,2,-3] => 5
[5,3,4,2,1] => [5,3,4,2,1] => [-4,-3,2,-5,1] => [1,-5,2,-3,-4] => 2
[5,4,2,3,1] => [5,4,2,3,1] => [4,2,3,-5,1] => [1,-5,3,2,4] => 1
[5,4,3,2,1] => [5,4,3,2,1] => [3,-4,2,-5,1] => [1,-5,2,-4,3] => 2
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Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
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.
Map
reverse
Description
The reversal of a signed permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.