Identifier
Values
[1,0] => [1,0] => [(1,2)] => {{1,2}} => 0
[1,0,1,0] => [1,1,0,0] => [(1,4),(2,3)] => {{1,4},{2,3}} => 1
[1,1,0,0] => [1,0,1,0] => [(1,2),(3,4)] => {{1,2},{3,4}} => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => 3
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => 5
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => 5
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => {{1,6},{2,3},{4,5}} => 4
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => {{1,2},{3,4},{5,6}} => 6
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Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
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 set partition
Description
Return the set partition corresponding to the perfect matching.