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Identifier

Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations

Images

[1] => [-1]

[-1] => [1]

[1,2] => [2,-1]

[1,-2] => [2,1]

[-1,2] => [-2,-1]

[-1,-2] => [-2,1]

[2,1] => [-1,2]

[2,-1] => [1,2]

[-2,1] => [-1,-2]

[-2,-1] => [1,-2]

[1,2,3] => [2,3,-1]

[1,2,-3] => [2,3,1]

[1,-2,3] => [2,-3,-1]

[1,-2,-3] => [2,-3,1]

[-1,2,3] => [-2,3,-1]

[-1,2,-3] => [-2,3,1]

[-1,-2,3] => [-2,-3,-1]

[-1,-2,-3] => [-2,-3,1]

[1,3,2] => [2,-1,3]

[1,3,-2] => [2,1,3]

[1,-3,2] => [2,-1,-3]

[1,-3,-2] => [2,1,-3]

[-1,3,2] => [-2,-1,3]

[-1,3,-2] => [-2,1,3]

[-1,-3,2] => [-2,-1,-3]

[-1,-3,-2] => [-2,1,-3]

[2,1,3] => [3,2,-1]

[2,1,-3] => [3,2,1]

[2,-1,3] => [-3,2,-1]

[2,-1,-3] => [-3,2,1]

[-2,1,3] => [3,-2,-1]

[-2,1,-3] => [3,-2,1]

[-2,-1,3] => [-3,-2,-1]

[-2,-1,-3] => [-3,-2,1]

[2,3,1] => [-1,2,3]

[2,3,-1] => [1,2,3]

[2,-3,1] => [-1,2,-3]

[2,-3,-1] => [1,2,-3]

[-2,3,1] => [-1,-2,3]

[-2,3,-1] => [1,-2,3]

[-2,-3,1] => [-1,-2,-3]

[-2,-3,-1] => [1,-2,-3]

[3,1,2] => [3,-1,2]

[3,1,-2] => [3,1,2]

[3,-1,2] => [-3,-1,2]

[3,-1,-2] => [-3,1,2]

[-3,1,2] => [3,-1,-2]

[-3,1,-2] => [3,1,-2]

[-3,-1,2] => [-3,-1,-2]

[-3,-1,-2] => [-3,1,-2]

[3,2,1] => [-1,3,2]

[3,2,-1] => [1,3,2]

[3,-2,1] => [-1,-3,2]

[3,-2,-1] => [1,-3,2]

[-3,2,1] => [-1,3,-2]

[-3,2,-1] => [1,3,-2]

[-3,-2,1] => [-1,-3,-2]

[-3,-2,-1] => [1,-3,-2]

[1,2,3,4] => [2,3,4,-1]

[1,2,3,-4] => [2,3,4,1]

[1,2,-3,4] => [2,3,-4,-1]

[1,2,-3,-4] => [2,3,-4,1]

[1,-2,3,4] => [2,-3,4,-1]

[1,-2,3,-4] => [2,-3,4,1]

[1,-2,-3,4] => [2,-3,-4,-1]

[1,-2,-3,-4] => [2,-3,-4,1]

[-1,2,3,4] => [-2,3,4,-1]

[-1,2,3,-4] => [-2,3,4,1]

[-1,2,-3,4] => [-2,3,-4,-1]

[-1,2,-3,-4] => [-2,3,-4,1]

[-1,-2,3,4] => [-2,-3,4,-1]

[-1,-2,3,-4] => [-2,-3,4,1]

[-1,-2,-3,4] => [-2,-3,-4,-1]

[-1,-2,-3,-4] => [-2,-3,-4,1]

[1,2,4,3] => [2,3,-1,4]

[1,2,4,-3] => [2,3,1,4]

[1,2,-4,3] => [2,3,-1,-4]

[1,2,-4,-3] => [2,3,1,-4]

[1,-2,4,3] => [2,-3,-1,4]

[1,-2,4,-3] => [2,-3,1,4]

[1,-2,-4,3] => [2,-3,-1,-4]

[1,-2,-4,-3] => [2,-3,1,-4]

[-1,2,4,3] => [-2,3,-1,4]

[-1,2,4,-3] => [-2,3,1,4]

[-1,2,-4,3] => [-2,3,-1,-4]

[-1,2,-4,-3] => [-2,3,1,-4]

[-1,-2,4,3] => [-2,-3,-1,4]

[-1,-2,4,-3] => [-2,-3,1,4]

[-1,-2,-4,3] => [-2,-3,-1,-4]

[-1,-2,-4,-3] => [-2,-3,1,-4]

[1,3,2,4] => [2,4,3,-1]

[1,3,2,-4] => [2,4,3,1]

[1,3,-2,4] => [2,-4,3,-1]

[1,3,-2,-4] => [2,-4,3,1]

[1,-3,2,4] => [2,4,-3,-1]

[1,-3,2,-4] => [2,4,-3,1]

[1,-3,-2,4] => [2,-4,-3,-1]

[1,-3,-2,-4] => [2,-4,-3,1]

[-1,3,2,4] => [-2,4,3,-1]

[-1,3,2,-4] => [-2,4,3,1]

[-1,3,-2,4] => [-2,-4,3,-1]

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click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

The Kreweras complement of a signed permutation.
This is the signed permutation $\pi^{-1}c$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.

Properties

bijective, graded

Sage code

def mapping(pi):
    n = len(list(pi))
    if n == 0:
        return SignedPermutations(0)([])
    else:
        c = SignedPermutations(n)(list(range(2,n+1))+[-1])
        return pi.inverse()*c

Weight

Created

Sep 11, 2020 at 16:25 by Martin Rubey

Updated

Sep 11, 2020 at 16:25 by Martin Rubey