Identifier
-
Mp00207:
Standard tableaux
—horizontal strip sizes⟶
Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001742: Graphs ⟶ ℤ
Values
[[1]] => [1] => ([],1) => 0
[[1,2]] => [2] => ([],2) => 0
[[1],[2]] => [1,1] => ([(0,1)],2) => 0
[[1,2,3]] => [3] => ([],3) => 0
[[1,3],[2]] => [1,2] => ([(1,2)],3) => 1
[[1,2],[3]] => [2,1] => ([(0,2),(1,2)],3) => 1
[[1],[2],[3]] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
[[1,2,3,4]] => [4] => ([],4) => 0
[[1,3,4],[2]] => [1,3] => ([(2,3)],4) => 1
[[1,2,4],[3]] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,2,3],[4]] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[[1,3],[2,4]] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1,2],[3,4]] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[3]] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[[1,3],[2],[4]] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1,2],[3],[4]] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[2],[3],[4]] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[[1,2,3,4,5]] => [5] => ([],5) => 0
[[1,3,4,5],[2]] => [1,4] => ([(3,4)],5) => 1
[[1,2,4,5],[3]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,2,3,5],[4]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[[1,3,5],[2,4]] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,5],[3,4]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,3,4],[2,5]] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,4],[3,5]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,3],[4,5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,4,5],[2],[3]] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,3,5],[2],[4]] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,5],[3],[4]] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,3,4],[2],[5]] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,4],[3],[5]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,3],[4],[5]] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2,5],[3]] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,3],[2,5],[4]] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2],[3,5],[4]] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,3],[2,4],[5]] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[3,4],[5]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,5],[2],[3],[4]] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,4],[2],[3],[5]] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,3],[2],[4],[5]] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[3],[4],[5]] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[[1],[2],[3],[4],[5]] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[[1,2,3,4,5,6]] => [6] => ([],6) => 0
[[1,3,4,5,6],[2]] => [1,5] => ([(4,5)],6) => 1
[[1,2,4,5,6],[3]] => [2,4] => ([(3,5),(4,5)],6) => 2
[[1,2,3,5,6],[4]] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 3
[[1,2,3,4,6],[5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,4,5],[6]] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,5,6],[2,4]] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,5,6],[3,4]] => [2,4] => ([(3,5),(4,5)],6) => 2
[[1,3,4,6],[2,5]] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,4,6],[3,5]] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3,6],[4,5]] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 3
[[1,3,4,5],[2,6]] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,4,5],[3,6]] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3,5],[4,6]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3,4],[5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,4,5,6],[2],[3]] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,3,5,6],[2],[4]] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,5,6],[3],[4]] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,4,6],[2],[5]] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,4,6],[3],[5]] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3,6],[4],[5]] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,4,5],[2],[6]] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,4,5],[3],[6]] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3,5],[4],[6]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3,4],[5],[6]] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5],[2,4,6]] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,5],[3,4,6]] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,4],[2,5,6]] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,4],[3,5,6]] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3],[4,5,6]] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 3
[[1,4,6],[2,5],[3]] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,6],[2,5],[4]] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,6],[3,5],[4]] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,6],[2,4],[5]] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,6],[3,4],[5]] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,4,5],[2,6],[3]] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,5],[2,6],[4]] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,5],[3,6],[4]] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,4],[2,6],[5]] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,4],[3,6],[5]] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,3],[4,6],[5]] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,5],[2,4],[6]] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,5],[3,4],[6]] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,4],[2,5],[6]] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,4],[3,5],[6]] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4,5],[6]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,5,6],[2],[3],[4]] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,4,6],[2],[3],[5]] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,6],[2],[4],[5]] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,6],[3],[4],[5]] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,4,5],[2],[3],[6]] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,5],[2],[4],[6]] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2,5],[3],[4],[6]] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3,4],[2],[5],[6]] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,4],[3],[5],[6]] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4],[5],[6]] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,4],[2,5],[3,6]] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3],[2,5],[4,6]] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
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Description
The difference of the maximal and the minimal degree in a graph.
The graph is regular if and only if this statistic is zero.
The graph is regular if and only if this statistic is zero.
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
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.
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