Identifier
Values
[1,0] => [(1,2)] => {{1,2}} => 0
[1,0,1,0] => [(1,2),(3,4)] => {{1,2},{3,4}} => 0
[1,1,0,0] => [(1,4),(2,3)] => {{1,4},{2,3}} => 2
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => {{1,2},{3,4},{5,6}} => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => 2
[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] => [(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => 6
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Description
The dimension exponent of a set partition.
This is
$$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$
where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$.
It is thus equal to the difference St000728The dimension of a set partition. - St000211The rank of the set partition..
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block.
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
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.