- FindStat

Identifier

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs

Images

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

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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

graph of inversions

Description

The graph of inversions of a permutation.
For a permutation of

$\{1,\dots,n\}$ , this is the graph with vertices

$\{1,\dots,n\}$ , where

$(i,j)$ is an edge if and only if it is an inversion of the permutation.

Map

core

Description

The core of a graph.
The core of a graph

$G$ is the smallest graph

$C$ such that there is a homomorphism from

$G$ to

$C$ and a homomorphism from

$C$ to

$G$ .
Note that the core of a graph is not necessarily connected, see [2].