- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000372: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

Description

The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) < \pi(j) < \pi(k)$ .
The generating function is given by [1].

Map

to permutation

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.

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.