Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [2,3,1] => 1
[1,3,2] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 2
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,3,1,4] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 1
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[2,4,1,3] => [3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 0
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[2,4,5,3,1] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[2,5,3,4,1] => [1,4,3,5,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[2,5,4,3,1] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[3,2,4,5,1] => [1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[3,4,2,5,1] => [1,5,2,4,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[4,2,3,5,1] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 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.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.