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Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000492: Set partitions ⟶ ℤ

Values

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

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$F_{1} = 1$

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

$F_{3} = q^{3} + q^{4} + 2\ q^{5} + q^{6}$

Description

The rob statistic of a set partition.
Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ .
According to [1, Definition 3], a rob (right-opener-bigger) of

$S$ is given by a pair

$i < j$ such that

$j = \operatorname{min} B_b$ and

$i \in B_a$ for

$a < b$ .
This is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

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.