Identifier
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
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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.