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Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001644: Graphs ⟶ ℤ

Values

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

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$F_{1} = q$

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

Description

The dimension of a graph.
The dimension of a graph is the least integer

$n$ such that there exists a representation of the graph in the Euclidean space of dimension

$n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

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

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

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.