Identifier
Values
[[1]] => [1] => [1] => ([],1) => 0
[[1,2]] => [2] => [2] => ([],2) => 0
[[1],[2]] => [2] => [2] => ([],2) => 0
[[1,2,3]] => [3] => [3] => ([],3) => 0
[[1,3],[2]] => [2,1] => [1,2] => ([(1,2)],3) => 1
[[1,2],[3]] => [3] => [3] => ([],3) => 0
[[1],[2],[3]] => [3] => [3] => ([],3) => 0
[[1,2,3,4]] => [4] => [4] => ([],4) => 0
[[1,2,4],[3]] => [3,1] => [1,3] => ([(2,3)],4) => 1
[[1,2,3],[4]] => [4] => [4] => ([],4) => 0
[[1,2],[3,4]] => [3,1] => [1,3] => ([(2,3)],4) => 1
[[1,4],[2],[3]] => [3,1] => [1,3] => ([(2,3)],4) => 1
[[1,2],[3],[4]] => [4] => [4] => ([],4) => 0
[[1],[2],[3],[4]] => [4] => [4] => ([],4) => 0
[[1,2,3,4,5]] => [5] => [5] => ([],5) => 0
[[1,2,3,5],[4]] => [4,1] => [1,4] => ([(3,4)],5) => 1
[[1,2,3,4],[5]] => [5] => [5] => ([],5) => 0
[[1,2,3],[4,5]] => [4,1] => [1,4] => ([(3,4)],5) => 1
[[1,2,5],[3],[4]] => [4,1] => [1,4] => ([(3,4)],5) => 1
[[1,2,3],[4],[5]] => [5] => [5] => ([],5) => 0
[[1,2],[3,5],[4]] => [4,1] => [1,4] => ([(3,4)],5) => 1
[[1,5],[2],[3],[4]] => [4,1] => [1,4] => ([(3,4)],5) => 1
[[1,2],[3],[4],[5]] => [5] => [5] => ([],5) => 0
[[1],[2],[3],[4],[5]] => [5] => [5] => ([],5) => 0
[[1,2,3,4,5,6]] => [6] => [6] => ([],6) => 0
[[1,3,4,5,6],[2]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,4,6],[5]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,2,3,4,5],[6]] => [6] => [6] => ([],6) => 0
[[1,3,4,6],[2,5]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,4,5],[2,6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,4],[5,6]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,3,4,6],[2],[5]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3,6],[4],[5]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,3,4,5],[2],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,4],[5],[6]] => [6] => [6] => ([],6) => 0
[[1,3,4],[2,5,6]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,6],[2,4],[5]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,4],[2,6],[5]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4,6],[5]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,3,4],[2,5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,3,6],[2],[4],[5]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,6],[3],[4],[5]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,3,4],[2],[5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3],[4],[5],[6]] => [6] => [6] => ([],6) => 0
[[1,3],[2,4],[5,6]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3],[2,6],[4],[5]] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2],[3,6],[4],[5]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,3],[2,4],[5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,6],[2],[3],[4],[5]] => [5,1] => [1,5] => ([(4,5)],6) => 1
[[1,3],[2],[4],[5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2],[3],[4],[5],[6]] => [6] => [6] => ([],6) => 0
[[1],[2],[3],[4],[5],[6]] => [6] => [6] => ([],6) => 0
[[1,2,3,4,5,6,7]] => [7] => [7] => ([],7) => 0
[[1,2,4,5,6,7],[3]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4,5,7],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,2,3,4,5,6],[7]] => [7] => [7] => ([],7) => 0
[[1,2,5,6,7],[3,4]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4,5,7],[3,6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,5,6],[3,7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4,5],[6,7]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,4,5,6,7],[2],[3]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4,5,7],[3],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,4,7],[5],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,2,4,5,6],[3],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4,5],[6],[7]] => [7] => [7] => ([],7) => 0
[[1,2,5,7],[3,4,6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5,6],[3,4,7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4,5],[3,6,7]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5,7],[2,6],[3]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5,7],[3,4],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,7],[3,5],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5,6],[2,7],[3]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4,5],[3,7],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,4],[5,7],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,2,5,6],[3,4],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4,5],[3,6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,4,5,7],[2],[3],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,7],[3],[5],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,7],[4],[5],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,4,5,6],[2],[3],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4,5],[3],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4],[5],[6],[7]] => [7] => [7] => ([],7) => 0
[[1,4,5],[2,6,7],[3]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5],[3,4,7],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4],[3,5,7],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5],[3,4,6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,4,7],[2,5],[3,6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5],[2,6],[3,7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5],[3,4],[6,7]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4],[3,5],[6,7]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,7],[2,5],[3],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,7],[3,4],[5],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5],[2,7],[3],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4],[3,7],[5],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3],[4,7],[5],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,4,5],[2,6],[3],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5],[3,4],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,4],[3,5],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,4,7],[2],[3],[5],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,7],[3],[4],[5],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,4,5],[2],[3],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
>>> Load all 114 entries. <<<
[[1,2,4],[3],[5],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3],[4],[5],[6],[7]] => [7] => [7] => ([],7) => 0
[[1,4],[2,5],[3,7],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2],[3,4],[5,7],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4],[2,5],[3,6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,4],[2,7],[3],[5],[6]] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2],[3,7],[4],[5],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,4],[2,5],[3],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2],[3,4],[5],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,7],[2],[3],[4],[5],[6]] => [6,1] => [1,6] => ([(5,6)],7) => 1
[[1,4],[2],[3],[5],[6],[7]] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2],[3],[4],[5],[6],[7]] => [7] => [7] => ([],7) => 0
[[1],[2],[3],[4],[5],[6],[7]] => [7] => [7] => ([],7) => 0
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 largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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
rotate front to back
Description
The front to back rotation of the entries of an integer composition.
Map
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.