- FindStat

edit this statistic or download as text // json

Identifier

St001892: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 1

[-1,2] => 1

[-1,-2] => 2

[2,1] => 2

[2,-1] => 3

[-2,1] => 1

[-2,-1] => 2

[1,2,3] => 0

[1,2,-3] => 1

[1,-2,3] => 1

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

[-1,2,3] => 1

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

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

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

[1,3,2] => 2

[1,3,-2] => 3

[1,-3,2] => 1

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

[-1,3,2] => 3

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

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

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

[2,1,3] => 2

[2,1,-3] => 3

[2,-1,3] => 3

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

[-2,1,3] => 1

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

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

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

[2,3,1] => 4

[2,3,-1] => 5

[2,-3,1] => 3

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

[-2,3,1] => 3

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

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

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

[3,1,2] => 2

[3,1,-2] => 3

[3,-1,2] => 3

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

[-3,1,2] => 1

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

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

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

[3,2,1] => 2

[3,2,-1] => 3

[3,-2,1] => 3

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

[-3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The flag excedance statistic of a signed permutation.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.

References

[1] Bastidas, J. The polytope algebra of generalized permutahedra MathSciNet:4339357
[2] Foata, D., Han, G.-N. Signed words and permutations. V. A sextuple distribution MathSciNet:2501235

Code

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

Created

Mar 31, 2023 at 09:34 by Jose Bastidas

Updated

Jul 05, 2023 at 17:38 by Jose Bastidas