Identifier
Values
[1] => [1,0] => [(1,2)] => [2,1] => 2
[2] => [1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 2
[1,1] => [1,1,0,0] => [(1,4),(2,3)] => [4,3,2,1] => 2
[3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 2
[2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => 4
[1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => 2
[2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => 2
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Description
The size of the symmetry class of a permutation.
The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations inverse (Mp00066inverse), reverse (Mp00064reverse), and complement (Mp00069complement).
Two elements in the same symmetry class are also in the same Wilf-equivalence class, see for example [2].
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.