Identifier
Values
{{1}} => [1] => [1,0] => [(1,2)] => 0
{{1,2}} => [2] => [1,1,0,0] => [(1,4),(2,3)] => 0
{{1},{2}} => [1,1] => [1,0,1,0] => [(1,2),(3,4)] => 1
{{1,2,3}} => [3] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 0
{{1,2},{3}} => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 2
{{1,3},{2}} => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 2
{{1},{2,3}} => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 2
{{1},{2},{3}} => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 3
{{1,2,3,4}} => [4] => [1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 0
{{1,2,3},{4}} => [3,1] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 3
{{1,2,4},{3}} => [3,1] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 3
{{1,2},{3,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 4
{{1,2},{3},{4}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
{{1,3,4},{2}} => [3,1] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 3
{{1,3},{2,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 4
{{1,3},{2},{4}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
{{1,4},{2,3}} => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 4
{{1},{2,3,4}} => [1,3] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 3
{{1},{2,3},{4}} => [1,2,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
{{1,4},{2},{3}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
{{1},{2,4},{3}} => [1,2,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
{{1},{2},{3,4}} => [1,1,2] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
{{1},{2},{3},{4}} => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 6
{{1,2,3,4,5}} => [5] => [1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => 0
{{1,2,3,4},{5}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 4
{{1,2,3,5},{4}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 4
{{1,2,3},{4,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
{{1,2,3},{4},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 7
{{1,2,4,5},{3}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 4
{{1,2,4},{3,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
{{1,2,4},{3},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 7
{{1,2,5},{3,4}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
{{1,2},{3,4,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 6
{{1,2},{3,4},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1,2,5},{3},{4}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 7
{{1,2},{3,5},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1,2},{3},{4,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => 8
{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => 9
{{1,3,4,5},{2}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 4
{{1,3,4},{2,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
{{1,3,4},{2},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 7
{{1,3,5},{2,4}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
{{1,3},{2,4,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 6
{{1,3},{2,4},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1,3,5},{2},{4}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 7
{{1,3},{2,5},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1,3},{2},{4,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => 8
{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => 9
{{1,4,5},{2,3}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
{{1,4},{2,3,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 6
{{1,4},{2,3},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1,5},{2,3,4}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 6
{{1},{2,3,4,5}} => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 4
{{1},{2,3,4},{5}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 7
{{1,5},{2,3},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1},{2,3,5},{4}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 7
{{1},{2,3},{4,5}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => 8
{{1},{2,3},{4},{5}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => 9
{{1,4,5},{2},{3}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 7
{{1,4},{2,5},{3}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1,4},{2},{3,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => 8
{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => 9
{{1,5},{2,4},{3}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => 8
{{1},{2,4,5},{3}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 7
{{1},{2,4},{3,5}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => 8
{{1},{2,4},{3},{5}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => 9
{{1,5},{2},{3,4}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => 8
{{1},{2,5},{3,4}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => 8
{{1},{2},{3,4,5}} => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => 7
{{1},{2},{3,4},{5}} => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10)] => 9
{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => 9
{{1},{2,5},{3},{4}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => 9
{{1},{2},{3,5},{4}} => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10)] => 9
{{1},{2},{3},{4,5}} => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => 9
{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => 10
{{1},{2},{3,4,5,6}} => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => 9
{{1},{2},{3,4,5},{6}} => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] => 12
{{1},{2},{3,4,6},{5}} => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] => 12
{{1},{2},{3,4},{5,6}} => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => 13
{{1},{2},{3,4},{5},{6}} => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] => 14
{{1},{2},{3,5,6},{4}} => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] => 12
{{1},{2},{3,5},{4,6}} => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => 13
{{1},{2},{3,5},{4},{6}} => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] => 14
{{1},{2},{3,6},{4,5}} => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => 13
{{1},{2},{3},{4,5,6}} => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => 12
{{1},{2},{3},{4,5},{6}} => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)] => 14
{{1},{2},{3,6},{4},{5}} => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] => 14
{{1},{2},{3},{4,6},{5}} => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)] => 14
{{1},{2},{3},{4},{5,6}} => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)] => 14
{{1},{2},{3},{4},{5},{6}} => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => 15
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Description
The number of alignments in a perfect matching.
An alignment is a pair of edges $(i,j)$, $(k,l)$ such that $i < j < k < l$.
Since any two edges in a perfect matching are either nesting (St000041The number of nestings of a perfect matching.), crossing (St000042The number of crossings of a perfect matching.) or form an alignment, the sum of these numbers in a perfect matching with $n$ edges is $\binom{n}{2}$.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
Map
bounce path
Description
The bounce path determined by an integer composition.
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.