Identifier
Values
[1] => 10 => [1,2] => ([(1,2)],3) => 2
[2] => 100 => [1,3] => ([(2,3)],4) => 3
[1,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[3] => 1000 => [1,4] => ([(3,4)],5) => 4
[2,1] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4] => 10000 => [1,5] => ([(4,5)],6) => 5
[3,1] => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[2,2] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
[2,1,1] => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,1,1] => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[5] => 100000 => [1,6] => ([(5,6)],7) => 6
[4,1] => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 5
[3,2] => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,1] => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[2,2,1] => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,1,1,1] => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,1,1,1,1] => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[4,2] => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 5
[3,3] => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 4
[3,2,1] => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[2,2,2] => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,2,1,1] => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[4,3] => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 4
[3,3,1] => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[3,2,2] => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,2,2,1] => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[4,4] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 5
[3,3,2] => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,2,2,2] => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,3,3] => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[] => => [1] => ([],1) => 1
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Description
The number of distinct columns in the nullspace of a graph.
Let $A$ be the adjacency matrix of a graph on $n$ vertices, and $K$ a $n\times d$ matrix whose column vectors form a basis of the nullspace of $A$. Then any other matrix $K'$ whose column vectors also form a basis of the nullspace is related to $K$ by $K' = K T$ for some invertible $d\times d$ matrix $T$. Any two rows of $K$ are equal if and only if they are equal in $K'$.
The nullspace of a graph is usually written as a $d\times n$ matrix, hence the name of this statistic.
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
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.