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Identifier

St001893: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 2

[-1,2] => 1

[-1,-2] => 3

[2,1] => 2

[2,-1] => 2

[-2,1] => 1

[-2,-1] => 1

[1,2,3] => 0

[1,2,-3] => 2

[1,-2,3] => 2

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

[-1,2,3] => 1

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

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

[-1,-2,-3] => 5

[1,3,2] => 2

[1,3,-2] => 2

[1,-3,2] => 2

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

[-1,3,2] => 3

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

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

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

[2,1,3] => 2

[2,1,-3] => 4

[2,-1,3] => 2

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

[-2,1,3] => 1

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

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

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

[2,3,1] => 2

[2,3,-1] => 2

[2,-3,1] => 2

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

[-2,3,1] => 3

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

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

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

[3,1,2] => 2

[3,1,-2] => 4

[3,-1,2] => 2

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

[-3,1,2] => 1

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

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

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

[3,2,1] => 4

[3,2,-1] => 4

[3,-2,1] => 2

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

[-3,2,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[-1,2,-3,-4] => 5

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

[-1,-2,3,-4] => 5

[-1,-2,-3,4] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

[-1,-2,4,3] => 5

[-1,-2,4,-3] => 5

[-1,-2,-4,3] => 5

[-1,-2,-4,-3] => 5

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

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

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

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

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

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

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

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

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

[-1,3,2,-4] => 5

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

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Description

The flag descent of a signed permutation.
$$ fdes(\sigma) = 2 \lvert \{ i \in [n-1] \mid \sigma(i) > \sigma(i+1) \} \rvert + \chi( \sigma(1) < 0 ) $$
It has the same distribution as the flag excedance statistic.

References

[1] Foata, D., Han, G.-N. Signed words and permutations. V. A sextuple distribution MathSciNet:2501235

Code

def statistic(pi):
    l = list(pi)
    return 2*len([i for i in range(len(l)-1) if l[i]> l[i+1]]) + int(l[0]<0)

Created

Mar 31, 2023 at 11:59 by Jose Bastidas

Updated

Mar 31, 2023 at 11:59 by Jose Bastidas