Identifier
Values
[1,0,1,1,0,0] => [[2,1],[]] => ([(0,1),(0,2)],3) => ([(1,2)],3) => 0
[1,1,0,0,1,0] => [[2,2],[1]] => ([(0,2),(1,2)],3) => ([(1,2)],3) => 0
[1,1,1,0,0,0] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0
[1,0,1,0,1,1,0,0] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => ([(1,3),(2,3)],4) => 0
[1,0,1,1,0,1,0,0] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => ([(1,3),(2,3)],4) => 0
[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => 0
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => 0
[1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 0
[1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 0
[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => ([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(6,4),(6,5)],7) => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3],[2,2,2,2]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[4,4,4,4],[3,3,3]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[5,5,5],[4,4]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
search for individual values
searching the database for the individual values of this statistic
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
Map
incomparability graph
Description
The incomparability graph of a poset.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
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 skew partition definded by the diagram of $\gamma(D)$.