Identifier
Values
([],1) => ([],1) => 0
([],2) => ([],1) => 0
([(0,1)],2) => ([(0,1)],2) => 0
([],3) => ([],1) => 0
([(1,2)],3) => ([(0,1)],2) => 0
([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([],4) => ([],1) => 0
([(2,3)],4) => ([(0,1)],2) => 0
([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(0,3),(1,3),(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) => 0
([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(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) => 0
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([],5) => ([],1) => 0
([(3,4)],5) => ([(0,1)],2) => 0
([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(1,4),(2,4),(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) => 0
([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(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) => 0
([(0,1),(2,4),(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) => 0
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([],6) => ([],1) => 0
([(4,5)],6) => ([(0,1)],2) => 0
([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(2,5),(3,5),(4,5)],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) => 0
([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(2,5),(3,4),(4,5)],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) => 0
([(1,2),(3,5),(4,5)],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) => 0
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([(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) => 0
([],7) => ([],1) => 0
([(5,6)],7) => ([(0,1)],2) => 0
([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(3,6),(4,6),(5,6)],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) => 0
([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
([(3,6),(4,5),(5,6)],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) => 0
([(2,3),(4,6),(5,6)],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) => 0
([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([(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) => 0
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
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
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.