Identifier
Values
[1,0] => [1] => [1] => [-1] => 1
[1,0,1,0] => [1,2] => [1,2] => [2,-1] => 2
[1,1,0,0] => [2,1] => [2,1] => [1,-2] => 3
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [2,3,-1] => 3
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [3,2,-1] => 4
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [1,3,-2] => 4
[1,1,0,1,0,0] => [2,3,1] => [2,3,1] => [1,2,-3] => 5
[1,1,1,0,0,0] => [3,1,2] => [3,1,2] => [3,1,-2] => 5
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 4
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 5
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 5
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 6
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => 6
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 5
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 6
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 6
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 7
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 7
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => 6
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => 7
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 8
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => 7
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => 6
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => 6
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => 7
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => 7
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => 6
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => 7
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => 7
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => 8
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => 8
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => 7
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => 8
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => 9
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => 8
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 6
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 7
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 7
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => 8
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => 8
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 7
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => 8
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 8
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 9
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => 9
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => 8
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => 9
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => 10
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => 9
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => 7
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => 8
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => 8
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => 9
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => 9
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => 9
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => 10
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => 11
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => 10
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [4,1,2,3,5] => [3,4,1,5,-2] => 8
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [4,1,2,5,3] => [3,5,1,4,-2] => 9
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [4,1,5,2,3] => [4,5,1,3,-2] => 10
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => 11
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => 9
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Description
The number of B-inversions of a signed permutation.
The number of B-inversions of a signed permutation $\sigma$ of length $n$ is
$$ \operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|, $$
see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
inverse Kreweras complement
Description
The inverse Kreweras complement of a signed permutation.
This is the signed permutation $c \pi^{-1}$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the inverse Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.