- FindStat

Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001394: Permutations ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 8\ q + 5\ q^{2}$

Description

The genus of a permutation.
The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$
where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

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 permutation

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.

Map

Kasraoui-Zeng

Description

The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.