- FindStat

Identifier

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001743: Graphs ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

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

Description

The discrepancy of a graph.
For a subset

$C$ of the set of vertices

$V(G)$ , and a vertex

$v$ , let

$d_{C, v} = |\#(N(v)\cap C) - \#(N(v)\cap(V\setminus C))|$ , and let

$d_C$ be the maximal value of

$d_{C, v}$ over all vertices.
Then the discrepancy of the graph is the minimal value of

$d_C$ over all subsets of

$V(G)$ .
Graphs with at most

$8$ vertices have discrepancy at most

$2$ , but there are graphs with arbitrary discrepancy.

Map

pattern poset

Description

The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.

Map

incomparability graph

Description

The incomparability graph of a poset.