Identifier
Values
[1,0] => [(1,2)] => [2,1] => 0
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 0
[1,1,0,0] => [(1,4),(2,3)] => [4,3,2,1] => 2
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => 6
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Description
The number of even inversions of a permutation.
An inversion $i < j$ of a permutation is even if $i \equiv j~(\operatorname{mod} 2)$. See St000539The number of odd inversions of a permutation. for odd inversions.
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.