- FindStat

Identifier

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001343: Posets ⟶ ℤ

Values

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

$F_{1} = q$

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

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

Description

The dimension of the reduced incidence algebra of a poset.
The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets.
Thus, this statistic returns the number of non-isomorphic intervals of the poset.

Map

to poset

Description

Return the poset corresponding to the lattice.

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.