- FindStat

Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000373: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

Description

The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ .
Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j \geq j$ and there exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ .
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a 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.

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

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.