- FindStat

edit this statistic or download as text // json

Identifier

St001864: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 0

[1,2] => 0

[1,-2] => 0

[-1,2] => 0

[-1,-2] => 0

[2,1] => 1

[2,-1] => 1

[-2,1] => 0

[-2,-1] => 0

[1,2,3] => 0

[1,2,-3] => 0

[1,-2,3] => 0

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

[-1,2,3] => 0

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

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

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

[1,3,2] => 1

[1,3,-2] => 1

[1,-3,2] => 0

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

[-1,3,2] => 1

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

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

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

[2,1,3] => 1

[2,1,-3] => 1

[2,-1,3] => 1

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

[-2,1,3] => 0

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

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

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

[2,3,1] => 2

[2,3,-1] => 2

[2,-3,1] => 1

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

[-2,3,1] => 1

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

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

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

[3,1,2] => 1

[3,1,-2] => 1

[3,-1,2] => 1

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

[-3,1,2] => 0

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

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

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

[3,2,1] => 1

[3,2,-1] => 1

[3,-2,1] => 1

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

[-3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> 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

$F_{1} = 2$

$F_{2} = 6 + 2\ q$

$F_{3} = 26 + 20\ q + 2\ q^{2}$

$F_{4} = 150 + 182\ q + 50\ q^{2} + 2\ q^{3}$

Description

The number of excedances of a signed permutation.
For a signed permutation

$\pi\in\mathfrak H_n$ , this is

$\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$ .

References

[1] Corteel, S., Josuat-Vergès, M., Kim, J. S. Crossings of signed permutations and $q$ -Eulerian numbers of type $B$ MathSciNet:3096133

Code

def statistic(pi):
    return sum(1 for i in range(len(pi)) if pi[i] > i + 1)

Created

Nov 30, 2022 at 23:53 by Martin Rubey

Updated

Nov 30, 2022 at 23:53 by Martin Rubey