Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ
Values
[2,1] => [[1,2],[3]] => [2,1] => ([(0,2),(1,2)],3) => 0
[3,1] => [[1,2,3],[4]] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[2,2] => [[1,2],[3,4]] => [2,2] => ([(1,3),(2,3)],4) => 0
[2,1,1] => [[1,2],[3],[4]] => [2,2] => ([(1,3),(2,3)],4) => 0
[4,1] => [[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[3,2] => [[1,2,3],[4,5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[3,1,1] => [[1,2,3],[4],[5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[2,1,1,1] => [[1,2],[3],[4],[5]] => [2,3] => ([(2,4),(3,4)],5) => 0
[5,1] => [[1,2,3,4,5],[6]] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[4,2] => [[1,2,3,4],[5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[4,1,1] => [[1,2,3,4],[5],[6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,4] => ([(3,5),(4,5)],6) => 0
[6,1] => [[1,2,3,4,5,6],[7]] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,5] => ([(4,6),(5,6)],7) => 0
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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.
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.
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
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
peak composition
Description
The composition corresponding to the peak 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 peak, if $i-1$ is an ascent and $i$ is a descent.
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 peak set of $T$.
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 peak, if $i-1$ is an ascent and $i$ is a descent.
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 peak set of $T$.
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