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Identifier

Mp00267: Signed permutations —signs⟶ Binary words

Images

[1] => 0

[-1] => 1

[1,2] => 00

[1,-2] => 01

[-1,2] => 10

[-1,-2] => 11

[2,1] => 00

[2,-1] => 01

[-2,1] => 10

[-2,-1] => 11

[1,2,3] => 000

[1,2,-3] => 001

[1,-2,3] => 010

[1,-2,-3] => 011

[-1,2,3] => 100

[-1,2,-3] => 101

[-1,-2,3] => 110

[-1,-2,-3] => 111

[1,3,2] => 000

[1,3,-2] => 001

[1,-3,2] => 010

[1,-3,-2] => 011

[-1,3,2] => 100

[-1,3,-2] => 101

[-1,-3,2] => 110

[-1,-3,-2] => 111

[2,1,3] => 000

[2,1,-3] => 001

[2,-1,3] => 010

[2,-1,-3] => 011

[-2,1,3] => 100

[-2,1,-3] => 101

[-2,-1,3] => 110

[-2,-1,-3] => 111

[2,3,1] => 000

[2,3,-1] => 001

[2,-3,1] => 010

[2,-3,-1] => 011

[-2,3,1] => 100

[-2,3,-1] => 101

[-2,-3,1] => 110

[-2,-3,-1] => 111

[3,1,2] => 000

[3,1,-2] => 001

[3,-1,2] => 010

[3,-1,-2] => 011

[-3,1,2] => 100

[-3,1,-2] => 101

[-3,-1,2] => 110

[-3,-1,-2] => 111

[3,2,1] => 000

[3,2,-1] => 001

[3,-2,1] => 010

[3,-2,-1] => 011

[-3,2,1] => 100

[-3,2,-1] => 101

[-3,-2,1] => 110

[-3,-2,-1] => 111

[1,2,3,4] => 0000

[1,2,3,-4] => 0001

[1,2,-3,4] => 0010

[1,2,-3,-4] => 0011

[1,-2,3,4] => 0100

[1,-2,3,-4] => 0101

[1,-2,-3,4] => 0110

[1,-2,-3,-4] => 0111

[-1,2,3,4] => 1000

[-1,2,3,-4] => 1001

[-1,2,-3,4] => 1010

[-1,2,-3,-4] => 1011

[-1,-2,3,4] => 1100

[-1,-2,3,-4] => 1101

[-1,-2,-3,4] => 1110

[-1,-2,-3,-4] => 1111

[1,2,4,3] => 0000

[1,2,4,-3] => 0001

[1,2,-4,3] => 0010

[1,2,-4,-3] => 0011

[1,-2,4,3] => 0100

[1,-2,4,-3] => 0101

[1,-2,-4,3] => 0110

[1,-2,-4,-3] => 0111

[-1,2,4,3] => 1000

[-1,2,4,-3] => 1001

[-1,2,-4,3] => 1010

[-1,2,-4,-3] => 1011

[-1,-2,4,3] => 1100

[-1,-2,4,-3] => 1101

[-1,-2,-4,3] => 1110

[-1,-2,-4,-3] => 1111

[1,3,2,4] => 0000

[1,3,2,-4] => 0001

[1,3,-2,4] => 0010

[1,3,-2,-4] => 0011

[1,-3,2,4] => 0100

[1,-3,2,-4] => 0101

[1,-3,-2,4] => 0110

[1,-3,-2,-4] => 0111

[-1,3,2,4] => 1000

[-1,3,2,-4] => 1001

[-1,3,-2,4] => 1010

>>> Load all 6397 entries. <<<

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Description

The binary word recording the signs of a signed permutation.
This map sends a signed permutation

$\pi\in\mathfrak H_n$ to the binary word

$w$ of length

$n$ such that

$w_i = 0$ if

$\pi(i) > 0$ .

Properties

surjective, graded

Sage code

def mapping(pi):
    return Words([0, 1], len(pi))([0 if e > 0 else 1 for e in pi])

Weight

Created

Dec 01, 2022 at 19:54 by Martin Rubey

Updated

Dec 01, 2022 at 19:54 by Martin Rubey