Identifier
Values
[[1]] => [(1,2)] => 0
[[1,0],[0,1]] => [(1,4),(2,3)] => 2
[[0,1],[1,0]] => [(1,2),(3,4)] => 0
[[1,0,0],[0,1,0],[0,0,1]] => [(1,6),(2,5),(3,4)] => 6
[[0,1,0],[1,0,0],[0,0,1]] => [(1,2),(3,4),(5,6)] => 0
[[1,0,0],[0,0,1],[0,1,0]] => [(1,6),(2,3),(4,5)] => 4
[[0,1,0],[1,-1,1],[0,1,0]] => [(1,2),(3,6),(4,5)] => 2
[[0,0,1],[1,0,0],[0,1,0]] => [(1,6),(2,3),(4,5)] => 4
[[0,1,0],[0,0,1],[1,0,0]] => [(1,2),(3,4),(5,6)] => 0
[[0,0,1],[0,1,0],[1,0,0]] => [(1,4),(2,3),(5,6)] => 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [(1,8),(2,7),(3,6),(4,5)] => 12
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [(1,8),(2,3),(4,5),(6,7)] => 6
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [(1,2),(3,8),(4,5),(6,7)] => 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [(1,8),(2,3),(4,5),(6,7)] => 6
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [(1,8),(2,7),(3,4),(5,6)] => 10
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [(1,8),(2,3),(4,7),(5,6)] => 8
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [(1,2),(3,8),(4,7),(5,6)] => 6
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [(1,8),(2,3),(4,7),(5,6)] => 8
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [(1,8),(2,7),(3,4),(5,6)] => 10
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [(1,8),(2,7),(3,4),(5,6)] => 10
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [(1,8),(2,3),(4,7),(5,6)] => 8
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [(1,8),(2,3),(4,5),(6,7)] => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [(1,2),(3,8),(4,5),(6,7)] => 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [(1,8),(2,3),(4,5),(6,7)] => 6
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [(1,8),(2,5),(3,4),(6,7)] => 8
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [(1,8),(2,5),(3,4),(6,7)] => 8
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [(1,8),(2,3),(4,5),(6,7)] => 6
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [(1,2),(3,8),(4,5),(6,7)] => 4
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [(1,8),(2,3),(4,5),(6,7)] => 6
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [(1,8),(2,5),(3,4),(6,7)] => 8
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [(1,8),(2,3),(4,5),(6,7)] => 6
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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
link pattern
Description
Sends an alternating sign matrix to the link pattern of the corresponding fully packed loop configuration.