- FindStat

Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000404: Permutations ⟶ ℤ

Values

[1,0] => [(1,2)] => [2,1] => [2,1] => 0
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => [3,4,2,1] => 0
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [3,6,2,5,4,1] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,6,5,3,2,1] => 0
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10)] => [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0] => [(1,2),(3,8),(4,5),(6,7),(9,10)] => [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)] => [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)] => [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)] => [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] => [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)] => [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)] => [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)] => [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] => [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)] => [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)] => [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)] => [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)] => [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)] => [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)] => [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)] => [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)] => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)] => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)] => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)] => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)] => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)] => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)] => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)] => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)] => [2,1,6,7,8,5,4,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)] => [2,1,6,7,8,5,4,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)] => [2,1,6,7,9,5,4,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)] => [2,1,6,7,9,5,4,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)] => [2,1,6,7,10,5,4,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)] => [2,1,6,8,9,5,10,7,4,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)] => [2,1,6,8,9,5,11,7,4,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)] => [2,1,6,8,10,5,11,7,12,9,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)] => [2,1,6,9,10,5,11,12,8,7,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)] => [2,1,7,8,9,10,6,5,4,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,1,0,0] => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)] => [2,1,7,8,9,11,6,5,4,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)] => [2,1,7,8,10,11,6,5,12,9,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)] => [2,1,7,9,10,11,6,12,8,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)] => [2,1,8,9,10,11,12,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0

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$F_{1} = 1$

$F_{2} = 2$

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

Description

The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.
A permutation avoids these two pattern if and only if it is an input-restricted deques, see [1].

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

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

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.