- FindStat

Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000558: Set partitions ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

Description

The number of occurrences of the pattern {{1,2}} in a set partition.

Map

intertwining number to dual major index

Description

A bijection sending the intertwining number of a set partition to its dual major index.
More precisely, St000490The intertwining number of a set partition.

$(P) =$ St000493The los statistic of a set partition.

$(\phi(P))$ for all set partitions

$P$ .

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.