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Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000665: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

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

Description

The number of rafts of a permutation.
Let

$\pi$ be a permutation of length

$n$ . A small ascent of

$\pi$ is an index

$i$ such that

$\pi(i+1)= \pi(i)+1$ , see St000441The number of successions of a permutation., and a raft of

$\pi$ is a non-empty maximal sequence of consecutive small ascents.

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

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

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.