- FindStat

Identifier

Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ

Values

[-,+] => [2,1] => [1,1] => ([(0,1)],2) => -1

[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3) => 0

[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3) => 0

[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3) => 0

[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3) => 0

[-,3,2] => [2,1,3] => [1,2] => ([(1,2)],3) => 0

[2,1,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3) => 0

[3,+,1] => [2,1,3] => [1,2] => ([(1,2)],3) => 0

[3,-,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3) => 0

[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4) => 0

[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4) => 0

[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4) => 0

[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4) => 0

[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4) => 0

[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 0

[-,+,4,3] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[+,-,4,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

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

[+,3,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[-,3,2,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4) => 0

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

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

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

[+,4,+,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[-,4,+,2] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0

[+,4,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[2,1,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[2,1,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4) => 0

[2,1,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[2,1,4,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[2,3,1,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4) => 0

[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

[2,4,+,1] => [3,1,2,4] => [1,3] => ([(2,3)],4) => 0

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

[3,1,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

[3,-,1,+] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0

[3,+,1,-] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 0

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

[3,+,4,1] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 0

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

[3,4,2,1] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 0

[4,1,+,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => 0

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

[4,+,1,3] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 0

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

[4,+,+,1] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0

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

[4,3,1,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[+,+,4,3,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

[+,+,5,+,3] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0

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

>>> Load all 1176 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

lower permutation

Description

The lower bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.

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

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