Identifier
Values
[1,0] => [(1,2)] => 0
[1,0,1,0] => [(1,2),(3,4)] => 0
[1,1,0,0] => [(1,4),(2,3)] => 2
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 2
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 6
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 4
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 6
[1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => 6
[1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => 8
[1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => 8
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => 10
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 12
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Description
The number of crossings plus two-nestings of a perfect matching.

This is $C+2N$ where $C$ is the number of crossings (St000042The number of crossings of a perfect matching.) and $N$ is the number of nestings (St000041The number of nestings of a perfect matching.).
The generating series $\sum_{m} q^{\textrm{cn}(m)}$, where the sum is over the perfect matchings of $2n$ and $\textrm{cn}(m)$ is this statistic is $[2n-1]_q[2n-3]_q\cdots [3]_q[1]_q$ where $[m]_q = 1+q+q^2+\cdots + q^{m-1}$ [1, Equation (5,4)].
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.