Identifier
Values
([],1) => ([],1) => 1
([],2) => ([],1) => 1
([(0,1)],2) => ([(0,1)],2) => 2
([],3) => ([],1) => 1
([(1,2)],3) => ([(0,1)],2) => 2
([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5
([],4) => ([],1) => 1
([(2,3)],4) => ([(0,1)],2) => 2
([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5
([],5) => ([],1) => 1
([(3,4)],5) => ([(0,1)],2) => 2
([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5
([],6) => ([],1) => 1
([(4,5)],6) => ([(0,1)],2) => 2
([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5
([],7) => ([],1) => 1
([(5,6)],7) => ([(0,1)],2) => 2
([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5
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Description
The number of left modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
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.