- FindStat

Identifier

Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            
                            
                            

                        [1]=>[1]=>0
[1,2]=>[1,2]=>0
[2,1]=>[2,1]=>0
[1,2,3]=>[1,2,3]=>0
[1,3,2]=>[1,3,2]=>0
[2,1,3]=>[2,1,3]=>0
[2,3,1]=>[2,3,1]=>1
[3,1,2]=>[3,1,2]=>0
[3,2,1]=>[3,2,1]=>0
[1,2,3,4]=>[1,2,3,4]=>0
[1,2,4,3]=>[1,2,4,3]=>0
[1,3,2,4]=>[1,3,2,4]=>0
[1,3,4,2]=>[1,3,4,2]=>1
[1,4,2,3]=>[1,4,2,3]=>0
[1,4,3,2]=>[1,4,3,2]=>0
[2,1,3,4]=>[2,1,3,4]=>0
[2,1,4,3]=>[2,1,4,3]=>0
[2,3,1,4]=>[2,3,1,4]=>1
[2,3,4,1]=>[2,3,4,1]=>2
[2,4,1,3]=>[2,4,1,3]=>1
[2,4,3,1]=>[2,4,3,1]=>1
[3,1,2,4]=>[3,1,2,4]=>0
[3,1,4,2]=>[3,1,4,2]=>1
[3,2,1,4]=>[3,2,1,4]=>0
[3,2,4,1]=>[3,2,4,1]=>1
[3,4,1,2]=>[3,4,1,2]=>2
[3,4,2,1]=>[3,4,2,1]=>1
[4,1,2,3]=>[4,1,2,3]=>0
[4,1,3,2]=>[4,1,3,2]=>0
[4,2,1,3]=>[4,2,1,3]=>0
[4,2,3,1]=>[4,2,3,1]=>0
[4,3,1,2]=>[4,3,1,2]=>1
[4,3,2,1]=>[4,3,2,1]=>0
[1,2,3,4,5]=>[1,2,3,4,5]=>0
[1,2,3,5,4]=>[1,2,3,5,4]=>0
[1,2,4,3,5]=>[1,2,4,3,5]=>0
[1,2,4,5,3]=>[1,2,4,5,3]=>1
[1,2,5,3,4]=>[1,2,5,3,4]=>0
[1,2,5,4,3]=>[1,2,5,4,3]=>0
[1,3,2,4,5]=>[1,3,2,4,5]=>0
[1,3,2,5,4]=>[1,3,2,5,4]=>0
[1,3,4,2,5]=>[1,3,4,2,5]=>1
[1,3,4,5,2]=>[1,3,4,5,2]=>2
[1,3,5,2,4]=>[1,3,5,2,4]=>1
[1,3,5,4,2]=>[1,3,5,4,2]=>1
[1,4,2,3,5]=>[1,4,2,3,5]=>0
[1,4,2,5,3]=>[1,4,2,5,3]=>1
[1,4,3,2,5]=>[1,4,3,2,5]=>0
[1,4,3,5,2]=>[1,4,3,5,2]=>1
[1,4,5,2,3]=>[1,4,5,2,3]=>2
[1,4,5,3,2]=>[1,4,5,3,2]=>1
[1,5,2,3,4]=>[1,5,2,3,4]=>0
[1,5,2,4,3]=>[1,5,2,4,3]=>0
[1,5,3,2,4]=>[1,5,3,2,4]=>0
[1,5,3,4,2]=>[1,5,3,4,2]=>0
[1,5,4,2,3]=>[1,5,4,2,3]=>1
[1,5,4,3,2]=>[1,5,4,3,2]=>0
                    

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 number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that

$i < j \leq \pi(i) < \pi(j)$, or
$-i < j \leq -\pi(i) < \pi(j)$, or
$i > j > \pi(i) > \pi(j)$.

Map

to signed permutation

Description

The signed permutation with all signs positive.