Identifier
-
Mp00039:
Integer compositions
—complement⟶
Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001846: Lattices ⟶ ℤ
Values
[1] => [1] => ([],1) => ([],1) => 0
[1,1] => [2] => ([],2) => ([],1) => 0
[2] => [1,1] => ([(0,1)],2) => ([(0,1)],2) => 0
[1,1,1] => [3] => ([],3) => ([],1) => 0
[1,2] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1] => [1,2] => ([(1,2)],3) => ([(0,1)],2) => 0
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,1] => [4] => ([],4) => ([],1) => 0
[1,1,2] => [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) => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1,1] => [1,3] => ([(2,3)],4) => ([(0,1)],2) => 0
[3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,1,1] => [5] => ([],5) => ([],1) => 0
[1,1,2,1] => [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) => 0
[1,2,1,1] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1,1,1] => [1,4] => ([(3,4)],5) => ([(0,1)],2) => 0
[3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,1,1,1] => [6] => ([],6) => ([],1) => 0
[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) => 0
[1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1,1,1,1] => [1,5] => ([(4,5)],6) => ([(0,1)],2) => 0
[3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,1,1,1,1] => [7] => ([],7) => ([],1) => 0
[1,1,2,1,1,1] => [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) => 0
[1,2,1,1,1,1] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1,1,1,1,1] => [1,6] => ([(5,6)],7) => ([(0,1)],2) => 0
[3,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
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Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
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.
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
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.
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
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
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