- FindStat

Identifier

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤ

Values

[1,1,1] => [3] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1

[1,1,2] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,1] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,1,3] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,2] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,3,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,2,1] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[3,1,1] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,1,4] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,3,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,4,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,2,2] => [3] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1

[2,3,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,1,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,2,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,1,1] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,1,5] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,3,3] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,4,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,5,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,2,3] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[2,3,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,4,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,1,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,2,2] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[3,3,1] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[4,1,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,2,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[5,1,1] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,1,6] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,3,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,4,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,5,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,6,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,2,4] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[2,3,3] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[2,4,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,5,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,1,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,2,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,3,2] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[3,4,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,1,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,2,2] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[4,3,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[5,1,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[5,2,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[6,1,1] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,1,7] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,6] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,3,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,4,4] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,5,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,6,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,7,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,6] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,2,5] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[2,3,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,4,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,5,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,6,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,1,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,2,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,3,3] => [3] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1

[3,4,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,5,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,1,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,2,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,3,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[4,4,1] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[5,1,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[5,2,2] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[5,3,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[6,1,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[6,2,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[7,1,1] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[1,1,8] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[1,2,7] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,3,6] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,4,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,5,4] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,6,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,7,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[1,8,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,1,7] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,2,6] => [2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 1

[2,3,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,4,4] => [1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1

[2,5,3] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,6,2] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[2,7,1] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,1,6] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

[3,2,5] => [1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1

>>> Load all 164 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin

$([1,3],[2,4])$ , insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])

$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$

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

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition

$C$ is the compositions composed of the lengths of consecutive runs of the same integer in

$C$ .