- FindStat

edit this statistic or download as text // json

Identifier

St000213: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 2

[2,1] => 1

[1,2,3] => 3

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 2

[3,1,2] => 1

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 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

$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = q + 11\ q^{2} + 11\ q^{3} + q^{4}$

$F_{5} = q + 26\ q^{2} + 66\ q^{3} + 26\ q^{4} + q^{5}$

$F_{6} = q + 57\ q^{2} + 302\ q^{3} + 302\ q^{4} + 57\ q^{5} + q^{6}$

Description

The number of weak exceedances (also weak excedences) of a permutation.
This is defined as

$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$
The number of weak exceedances is given by the number of exceedances (see St000155The number of exceedances (also excedences) of a permutation.) plus the number of fixed points (see St000022The number of fixed points of a permutation.) of

$\sigma$ .

References

Triangle of Eulerian numbers T(n,k) (n>=1, 1 <= k <= n) read by rows. OEIS:A008292

Code

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

Created

Aug 26, 2014 at 14:33 by Martin Rubey

Updated

Mar 28, 2015 at 23:10 by Martin Rubey