Identifier
Values
[[1],[2],[3]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[[1,4],[2],[3]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0
[[1,3],[2],[4]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[[1,2],[3],[4]] => [4,3,1,2] => [1,3] => ([(2,3)],4) => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,4] => ([(3,4)],5) => 0
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5) => 0
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,4] => ([(3,4)],5) => 0
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,4] => ([(3,4)],5) => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5) => 0
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,4] => ([(3,4)],5) => 0
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,4] => ([(3,4)],5) => 0
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,5] => ([(4,5)],6) => 0
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,5] => ([(4,5)],6) => 0
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,5] => ([(4,5)],6) => 0
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,5] => ([(4,5)],6) => 0
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,5] => ([(4,5)],6) => 0
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,5] => ([(4,5)],6) => 0
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,5] => ([(4,5)],6) => 0
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,5] => ([(4,5)],6) => 0
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,5] => ([(4,5)],6) => 0
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,5] => ([(4,5)],6) => 0
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,5] => ([(4,5)],6) => 0
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,5] => ([(4,5)],6) => 0
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,4] => ([(3,5),(4,5)],6) => 0
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,5] => ([(4,5)],6) => 0
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,5] => ([(4,5)],6) => 0
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,5,6,7],[2],[4]] => [4,2,1,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5,6,7],[3],[4]] => [4,3,1,2,5,6,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,6,7],[2],[5]] => [5,2,1,3,4,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,6,7],[3],[5]] => [5,3,1,2,4,6,7] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,6,7],[4],[5]] => [5,4,1,2,3,6,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,5,7],[2],[6]] => [6,2,1,3,4,5,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,5,7],[3],[6]] => [6,3,1,2,4,5,7] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,5,7],[4],[6]] => [6,4,1,2,3,5,7] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,4,7],[5],[6]] => [6,5,1,2,3,4,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,5,6],[2],[7]] => [7,2,1,3,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,5,6],[3],[7]] => [7,3,1,2,4,5,6] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,5,6],[4],[7]] => [7,4,1,2,3,5,6] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,4,6],[5],[7]] => [7,5,1,2,3,4,6] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,6] => ([(5,6)],7) => 0
[[1,4,6,7],[2,5],[3]] => [3,2,5,1,4,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,6,7],[3,5],[4]] => [4,3,5,1,2,6,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,6,7],[2,4],[5]] => [5,2,4,1,3,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,6,7],[3,4],[5]] => [5,3,4,1,2,6,7] => [1,6] => ([(5,6)],7) => 0
[[1,4,5,7],[2,6],[3]] => [3,2,6,1,4,5,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,5,7],[2,6],[4]] => [4,2,6,1,3,5,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5,7],[3,6],[4]] => [4,3,6,1,2,5,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,7],[2,6],[5]] => [5,2,6,1,3,4,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,7],[3,6],[5]] => [5,3,6,1,2,4,7] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,7],[4,6],[5]] => [5,4,6,1,2,3,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,5,7],[2,4],[6]] => [6,2,4,1,3,5,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5,7],[3,4],[6]] => [6,3,4,1,2,5,7] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,7],[2,5],[6]] => [6,2,5,1,3,4,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,7],[3,5],[6]] => [6,3,5,1,2,4,7] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,7],[4,5],[6]] => [6,4,5,1,2,3,7] => [1,6] => ([(5,6)],7) => 0
[[1,4,5,6],[2,7],[3]] => [3,2,7,1,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,5,6],[2,7],[4]] => [4,2,7,1,3,5,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5,6],[3,7],[4]] => [4,3,7,1,2,5,6] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,6],[2,7],[5]] => [5,2,7,1,3,4,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,6],[3,7],[5]] => [5,3,7,1,2,4,6] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,6],[4,7],[5]] => [5,4,7,1,2,3,6] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,5],[2,7],[6]] => [6,2,7,1,3,4,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,5],[3,7],[6]] => [6,3,7,1,2,4,5] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,5],[4,7],[6]] => [6,4,7,1,2,3,5] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,4],[5,7],[6]] => [6,5,7,1,2,3,4] => [1,6] => ([(5,6)],7) => 0
[[1,3,5,6],[2,4],[7]] => [7,2,4,1,3,5,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5,6],[3,4],[7]] => [7,3,4,1,2,5,6] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,6],[2,5],[7]] => [7,2,5,1,3,4,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,6],[3,5],[7]] => [7,3,5,1,2,4,6] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,6],[4,5],[7]] => [7,4,5,1,2,3,6] => [1,6] => ([(5,6)],7) => 0
[[1,3,4,5],[2,6],[7]] => [7,2,6,1,3,4,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4,5],[3,6],[7]] => [7,3,6,1,2,4,5] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,5],[4,6],[7]] => [7,4,6,1,2,3,5] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => [1,6] => ([(5,6)],7) => 0
[[1,2,3,7],[4],[5],[6]] => [6,5,4,1,2,3,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[1,2,3,6],[4],[5],[7]] => [7,5,4,1,2,3,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[1,2,3,5],[4],[6],[7]] => [7,6,4,1,2,3,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
>>> Load all 146 entries. <<<
[[1,4,6],[2,5,7],[3]] => [3,2,5,7,1,4,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,6],[2,5,7],[4]] => [4,2,5,7,1,3,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,6],[3,5,7],[4]] => [4,3,5,7,1,2,6] => [1,6] => ([(5,6)],7) => 0
[[1,3,6],[2,4,7],[5]] => [5,2,4,7,1,3,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,6],[3,4,7],[5]] => [5,3,4,7,1,2,6] => [1,6] => ([(5,6)],7) => 0
[[1,4,5],[2,6,7],[3]] => [3,2,6,7,1,4,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,5],[2,6,7],[4]] => [4,2,6,7,1,3,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5],[3,6,7],[4]] => [4,3,6,7,1,2,5] => [1,6] => ([(5,6)],7) => 0
[[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4],[3,6,7],[5]] => [5,3,6,7,1,2,4] => [1,6] => ([(5,6)],7) => 0
[[1,2,3],[4,6,7],[5]] => [5,4,6,7,1,2,3] => [1,6] => ([(5,6)],7) => 0
[[1,3,5],[2,4,7],[6]] => [6,2,4,7,1,3,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5],[3,4,7],[6]] => [6,3,4,7,1,2,5] => [1,6] => ([(5,6)],7) => 0
[[1,3,4],[2,5,7],[6]] => [6,2,5,7,1,3,4] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4],[3,5,7],[6]] => [6,3,5,7,1,2,4] => [1,6] => ([(5,6)],7) => 0
[[1,2,3],[4,5,7],[6]] => [6,4,5,7,1,2,3] => [1,6] => ([(5,6)],7) => 0
[[1,3,5],[2,4,6],[7]] => [7,2,4,6,1,3,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,5],[3,4,6],[7]] => [7,3,4,6,1,2,5] => [1,6] => ([(5,6)],7) => 0
[[1,3,4],[2,5,6],[7]] => [7,2,5,6,1,3,4] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,4],[3,5,6],[7]] => [7,3,5,6,1,2,4] => [1,6] => ([(5,6)],7) => 0
[[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [1,6] => ([(5,6)],7) => 0
[[1,4,7],[2,5],[3,6]] => [3,6,2,5,1,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,3,7],[2,5],[4,6]] => [4,6,2,5,1,3,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,7],[3,5],[4,6]] => [4,6,3,5,1,2,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,3,7],[2,4],[5,6]] => [5,6,2,4,1,3,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[1,4,6],[2,5],[3,7]] => [3,7,2,5,1,4,6] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,3,6],[2,5],[4,7]] => [4,7,2,5,1,3,6] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,6],[3,5],[4,7]] => [4,7,3,5,1,2,6] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,3,6],[2,4],[5,7]] => [5,7,2,4,1,3,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[1,2,6],[3,4],[5,7]] => [5,7,3,4,1,2,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[1,4,5],[2,6],[3,7]] => [3,7,2,6,1,4,5] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,3,5],[2,6],[4,7]] => [4,7,2,6,1,3,5] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,5],[3,6],[4,7]] => [4,7,3,6,1,2,5] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,3,4],[2,6],[5,7]] => [5,7,2,6,1,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,4],[3,6],[5,7]] => [5,7,3,6,1,2,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,3],[4,6],[5,7]] => [5,7,4,6,1,2,3] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,3,5],[2,4],[6,7]] => [6,7,2,4,1,3,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[1,2,5],[3,4],[6,7]] => [6,7,3,4,1,2,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[1,3,4],[2,5],[6,7]] => [6,7,2,5,1,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,4],[3,5],[6,7]] => [6,7,3,5,1,2,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => [2,5] => ([(4,6),(5,6)],7) => 0
[[1,2,3],[4,7],[5],[6]] => [6,5,4,7,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[1,2,3],[4,6],[5],[7]] => [7,5,4,6,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [1,1,5] => ([(4,5),(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
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
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.