- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000670: Permutations ⟶ ℤ

Values

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

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$F_{1} = q$

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

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

$F_{4} = q + 3\ q^{2} + 6\ q^{3} + 4\ q^{4}$

Description

The reversal length of a permutation.
A reversal in a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form

$\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for

$1 \leq i < j \leq n$ .
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .

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.