- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤ

Values

[1] => [1,0,1,0] => [1,1,0,0] => [(1,4),(2,3)] => 2
[2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 2
[1,1] => [1,0,1,1,0,0] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 2
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 4
[2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 6
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => 2
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => 6
[3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => 4
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => 2
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 4
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [(1,10),(2,9),(3,6),(4,5),(7,8)] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => 2
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => 4
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => 2
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => 2
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => 2
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [(1,10),(2,3),(4,9),(5,8),(6,7)] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => 2
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => 2
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [(1,10),(2,5),(3,4),(6,9),(7,8)] => 2
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [(1,10),(2,3),(4,9),(5,6),(7,8)] => 2
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [(1,10),(2,3),(4,7),(5,6),(8,9)] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [(1,10),(2,3),(4,5),(6,9),(7,8)] => 2
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => 2
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => 2
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)] => 2
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => 2
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)] => 2
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)] => 2
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)] => 2
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)] => 2
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)] => 2

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$F_{1} = q^{2}$

$F_{2} = 2\ q^{2}$

$F_{3} = 2\ q^{2} + q^{4}$

$F_{4} = 4\ q^{2} + q^{6}$

Description

The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of

$\{1,\dots,2n\}$ and trees with

$n+1$ leaves is described in Example 5.2.6 of [1].

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 Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

inverse promotion

Description

The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.