Identifier
Values
[1] => [1] => ([],1) => ([],1) => 1
[1,1] => [2] => ([],2) => ([],1) => 1
[2] => [1,1] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,1,1] => [3] => ([],3) => ([],1) => 1
[1,2] => [1,2] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,1] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,1,1,1] => [4] => ([],4) => ([],1) => 1
[1,1,2] => [1,3] => ([(2,3)],4) => ([(0,1)],2) => 1
[1,2,1] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[2,1,1] => [3,1] => ([(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) => 3
[1,1,1,1,1] => [5] => ([],5) => ([],1) => 1
[1,1,1,2] => [1,4] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,2,1,1] => [3,2] => ([(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) => 3
[1,1,1,1,1,1] => [6] => ([],6) => ([],1) => 1
[1,1,1,1,2] => [1,5] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,1,2,1,1] => [3,3] => ([(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) => 3
[1,1,1,1,1,1,1] => [7] => ([],7) => ([],1) => 1
[1,1,1,1,1,2] => [1,6] => ([(5,6)],7) => ([(0,1)],2) => 1
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,1,1,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,1,1,2,1,1] => [3,4] => ([(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) => 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
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
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
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.