- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ

Values

[1,0] => [2,1] => [1,1] => ([(0,1)],2) => -1
[1,0,1,0] => [3,1,2] => [1,2] => ([(1,2)],3) => 0
[1,1,0,0] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[1,0,1,0,1,0] => [4,1,2,3] => [1,3] => ([(2,3)],4) => 0
[1,1,0,0,1,0] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,1,0,1,0,0] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0
[1,1,1,0,0,0] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,4] => ([(3,4)],5) => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,5] => ([(4,5)],6) => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,6] => ([(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0

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$F_{1} = q^{-1}$

$F_{2} = 2$

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

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .

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

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.