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Identifier

Mp00245: Signed permutations —standardize⟶ Permutations

Images

[1] => [1]

[-1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 entries. <<<

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

Description

Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let $\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order $1 < \dots < n < -n < \dots < -1$, this map returns the permutation in $\mathfrak S_n$ which is order isomorphic to $\pi(1),\dots,\pi(n)$.

Properties

surjective, graded

Sage code

def mapping(pi):
    from sage.combinat.permutation import to_standard
    n = len(list(pi))
    return Permutation(to_standard([e if e > 0 else 2*n+1+e for e in pi]))

Weight

Created

Feb 09, 2022 at 10:04 by Martin Rubey

Updated

Feb 09, 2022 at 10:04 by Martin Rubey