- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000719: Perfect matchings ⟶ ℤ

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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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q^{2} + q^{3}$

$F_{4} = 1 + 4\ q^{3} + 3\ q^{4} + 6\ q^{5} + q^{6}$

$F_{5} = 1 + 5\ q^{4} + 10\ q^{6} + 10\ q^{7} + 15\ q^{8} + 10\ q^{9} + q^{10}$

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

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.

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.