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Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000338: 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)] => [3,4,2,1,6,5] => 2
[1,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 0
[2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 0

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$F_{1} = 1$

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

Description

The number of pixed points of a permutation.
For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{pix} \, \sigma = \textrm{length} (p)$ .

Map

non-nesting-exceedence permutation

Description

The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given 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.