- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ

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. <<<

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

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

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

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.