- FindStat

Identifier

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001719: Lattices ⟶ ℤ

Values

([],1) => ([],1) => ([],1) => 1

([],2) => ([],1) => ([],1) => 1

([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => 1

([],3) => ([],1) => ([],1) => 1

([(1,2)],3) => ([(1,2)],3) => ([(0,1)],2) => 1

([(0,2),(1,2)],3) => ([(0,1)],2) => ([(0,1)],2) => 1

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

([],4) => ([],1) => ([],1) => 1

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

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

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

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

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

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

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

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

([],5) => ([],1) => ([],1) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([],1) => ([],1) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],7) => ([],1) => ([],1) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q$

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

Map

connected vertex partitions

Description

Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph

$G = (V,E)$ is a set partition of

$V$ such that each part induced a connected subgraph of

$G$ . The connected vertex partitions of

$G$ form a lattice under refinement. If

$G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on

$n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.

Map

de-duplicate

Description

The de-duplicate of a graph.
Let

$G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods

$\{N_v | v \in V\}$ of

$G$ , and has an edge

$(N_a, N_b)$ between two vertices if and only if

$(a, b)$ is an edge of

$G$ . This is well-defined, because if

$N_a = N_c$ and

$N_b = N_d$ , then

$(a, b)\in E$ if and only if

$(c, d)\in E$ .
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.