Identifier
Values
[1,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[2,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[2,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[3,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[3,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[3,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[2,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[4,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[4,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[4,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[3,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[5,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[5,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[5,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[4,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[6,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[6,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[6,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[5,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[7,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[7,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[7,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[6,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[8,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[8,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[8,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[7,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[9,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[9,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[9,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[8,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[10,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[10,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[10,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[9,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[11,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[11,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[11,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[10,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[12,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[12,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[12,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[11,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[13,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[13,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[13,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[12,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[14,3] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[14,2,1] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[14,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[13,2,2] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
search for individual values
searching the database for the individual values of this statistic
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
first row removal
Description
Removes the first entry of an integer partition
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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.