Identifier
Values
[1,0] => [(1,2)] => [2,1] => [1,2] => 0
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [1,2,3,4] => 0
[1,1,0,0] => [(1,4),(2,3)] => [4,3,2,1] => [1,4,2,3] => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,2,3,6,4,5] => 1
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,4,2,3,5,6] => 1
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,6,2,3,4,5] => 3
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => 4
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Description
The number of occurrences of the pattern 31-2.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $31\!\!-\!\!2$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.