Identifier
Values
{{1,2}} => [2] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1
{{1},{2}} => [1,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
{{1,2},{3}} => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
{{1,3},{2}} => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
{{1},{2,3}} => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
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searching the database for statistics with the same generating function
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Map
shape
Description
Sends a set partition to the integer partition obtained by the sizes of the blocks.
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 Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.