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Identifier

St001848: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 1

[-1,2] => 6

[-1,-2] => 7

[2,1] => 1

[2,-1] => 3

[-2,1] => 4

[-2,-1] => 6

[1,2,3] => 0

[1,2,-3] => 1

[1,-2,3] => 6

[1,-2,-3] => 7

[-1,2,3] => 15

[-1,2,-3] => 16

[-1,-2,3] => 21

[-1,-2,-3] => 22

[1,3,2] => 1

[1,3,-2] => 3

[1,-3,2] => 4

[1,-3,-2] => 6

[-1,3,2] => 16

[-1,3,-2] => 18

[-1,-3,2] => 19

[-1,-3,-2] => 21

[2,1,3] => 1

[2,1,-3] => 2

[2,-1,3] => 10

[2,-1,-3] => 11

[-2,1,3] => 11

[-2,1,-3] => 12

[-2,-1,3] => 20

[-2,-1,-3] => 21

[2,3,1] => 3

[2,3,-1] => 6

[2,-3,1] => 6

[2,-3,-1] => 9

[-2,3,1] => 13

[-2,3,-1] => 16

[-2,-3,1] => 16

[-2,-3,-1] => 19

[3,1,2] => 3

[3,1,-2] => 5

[3,-1,2] => 12

[3,-1,-2] => 14

[-3,1,2] => 8

[-3,1,-2] => 10

[-3,-1,2] => 17

[-3,-1,-2] => 19

[3,2,1] => 4

[3,2,-1] => 7

[3,-2,1] => 10

[3,-2,-1] => 13

[-3,2,1] => 9

[-3,2,-1] => 12

[-3,-2,1] => 15

[-3,-2,-1] => 18

[1,2,3,4] => 0

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

[1,2,-3,4] => 6

[1,2,-3,-4] => 7

[1,-2,3,4] => 15

[1,-2,3,-4] => 16

[1,-2,-3,4] => 21

[1,-2,-3,-4] => 22

[-1,2,3,4] => 28

[-1,2,3,-4] => 29

[-1,2,-3,4] => 34

[-1,2,-3,-4] => 35

[-1,-2,3,4] => 43

[-1,-2,3,-4] => 44

[-1,-2,-3,4] => 49

[-1,-2,-3,-4] => 50

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

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

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

[1,2,-4,-3] => 6

[1,-2,4,3] => 16

[1,-2,4,-3] => 18

[1,-2,-4,3] => 19

[1,-2,-4,-3] => 21

[-1,2,4,3] => 29

[-1,2,4,-3] => 31

[-1,2,-4,3] => 32

[-1,2,-4,-3] => 34

[-1,-2,4,3] => 44

[-1,-2,4,-3] => 46

[-1,-2,-4,3] => 47

[-1,-2,-4,-3] => 49

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

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

[1,3,-2,4] => 10

[1,3,-2,-4] => 11

[1,-3,2,4] => 11

[1,-3,2,-4] => 12

[1,-3,-2,4] => 20

[1,-3,-2,-4] => 21

[-1,3,2,4] => 29

[-1,3,2,-4] => 30

[-1,3,-2,4] => 38

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Description

The atomic length of a signed permutation.
The atomic length of an element $w$ of a Weyl group is the sum of the heights of the inversions of $w$.

References

[1] not yet indexed arXiv 2211.12359

Code

def atomic_length(pi):
    """
    EXAMPLES::

        sage: l = [atomic_length(SignedPermutations(n).long_element()) for n in range(1,8)]
        sage: l
        sage: fricas.guess(l)[0].sage().factor()
        1/6*(4*n + 3)*(n + 2)*(n + 1)

    """
    W = WeylGroup(pi.parent().coxeter_type())
    w = W.from_reduced_word(pi.reduced_word())
    return sum(a.height() for a in w.inversions(inversion_type="roots"))

Created

Nov 23, 2022 at 10:16 by Martin Rubey

Updated

Nov 23, 2022 at 10:16 by Martin Rubey