Identifier
Values
[1,0] => ([],1) => ([],1) => ([],1) => 1
[1,0,1,0] => ([(0,1)],2) => ([],2) => ([],2) => 2
[1,1,0,0] => ([(0,1)],2) => ([],2) => ([],2) => 2
[1,0,1,0,1,0] => ([(0,2),(2,1)],3) => ([],3) => ([],3) => 3
[1,0,1,1,0,0] => ([(0,2),(2,1)],3) => ([],3) => ([],3) => 3
[1,1,0,0,1,0] => ([(0,2),(2,1)],3) => ([],3) => ([],3) => 3
[1,1,0,1,0,0] => ([(0,2),(2,1)],3) => ([],3) => ([],3) => 3
[1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => ([(2,3)],4) => 3
[1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,0,1,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,0,1,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(3,4)],5) => 4
[1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,1,0,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([],4) => 4
[1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(3,4)],5) => 4
[1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => ([(3,4)],5) => 4
[1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => ([(3,4)],5) => 4
[1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 4
[1,1,1,1,0,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 4
[1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,0,1,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([],5) => 5
[1,1,0,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,0,1,1,1,0,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(4,5)],6) => 5
[1,1,1,0,0,1,1,0,0,0] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 5
[1,1,1,0,1,0,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,0,1,0,0,1,0,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,1,0,0,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,1,0,0,0,1,0,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8) => ([(4,7),(5,6)],8) => ([(4,7),(5,6)],8) => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,1,1,1,0,0,0,0,1,0] => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,0,1,1,1,1,0,0,0,1,0,0] => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => ([(4,7),(5,6),(6,7)],8) => ([(4,7),(5,6),(6,7)],8) => 6
[1,1,0,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,1,0,0,1,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,1,0,0,1,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,1,0,0,1,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([],6) => 6
[1,1,0,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => ([(5,6)],7) => 6
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Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Map
connected complement
Description
The componentwise connected complement of a graph.
For a connected graph $G$, this map returns the complement of $G$ if it is connected, otherwise $G$ itself. If $G$ is not connected, the map is applied to each connected component separately.
For a connected graph $G$, this map returns the complement of $G$ if it is connected, otherwise $G$ itself. If $G$ is not connected, the map is applied to each connected component separately.
Map
parallelogram poset
Description
The cell poset of the parallelogram polyomino corresponding to the Dyck path.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.
Map
incomparability graph
Description
The incomparability graph of a poset.
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