Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
[[1]] => [1] => [1] => ([],1) => 0
[[1,2]] => [1,2] => [2] => ([],2) => 0
[[1],[2]] => [2,1] => [1,1] => ([(0,1)],2) => 1
[[1,2,3]] => [1,2,3] => [3] => ([],3) => 0
[[1,3],[2]] => [2,1,3] => [1,2] => ([(1,2)],3) => 1
[[1,2],[3]] => [3,1,2] => [1,2] => ([(1,2)],3) => 1
[[1],[2],[3]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 2
[[1,2,3,4]] => [1,2,3,4] => [4] => ([],4) => 0
[[1,3,4],[2]] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 1
[[1,2,4],[3]] => [3,1,2,4] => [1,3] => ([(2,3)],4) => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,3] => ([(2,3)],4) => 1
[[1,4],[2],[3]] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[[1,3],[2],[4]] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[[1,2],[3],[4]] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[[1,2,3,4,5]] => [1,2,3,4,5] => [5] => ([],5) => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [1,4] => ([(3,4)],5) => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,4] => ([(3,4)],5) => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,4] => ([(3,4)],5) => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,4] => ([(3,4)],5) => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [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) => 4
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6] => ([],6) => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,5] => ([(4,5)],6) => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,5] => ([(4,5)],6) => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,5] => ([(4,5)],6) => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,5] => ([(4,5)],6) => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,5] => ([(4,5)],6) => 1
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [7] => ([],7) => 0
[[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [1,6] => ([(5,6)],7) => 1
[[1,2,4,5,6,7],[3]] => [3,1,2,4,5,6,7] => [1,6] => ([(5,6)],7) => 1
[[1,2,3,5,6,7],[4]] => [4,1,2,3,5,6,7] => [1,6] => ([(5,6)],7) => 1
[[1,2,3,4,6,7],[5]] => [5,1,2,3,4,6,7] => [1,6] => ([(5,6)],7) => 1
[[1,2,3,4,5,7],[6]] => [6,1,2,3,4,5,7] => [1,6] => ([(5,6)],7) => 1
[[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [1,6] => ([(5,6)],7) => 1
[[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,3,5,6,7],[2],[4]] => [4,2,1,3,5,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,5,6,7],[3],[4]] => [4,3,1,2,5,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,3,4,6,7],[2],[5]] => [5,2,1,3,4,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,4,6,7],[3],[5]] => [5,3,1,2,4,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,6,7],[4],[5]] => [5,4,1,2,3,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,3,4,5,7],[2],[6]] => [6,2,1,3,4,5,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,4,5,7],[3],[6]] => [6,3,1,2,4,5,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,5,7],[4],[6]] => [6,4,1,2,3,5,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,4,7],[5],[6]] => [6,5,1,2,3,4,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,3,4,5,6],[2],[7]] => [7,2,1,3,4,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,4,5,6],[3],[7]] => [7,3,1,2,4,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,5,6],[4],[7]] => [7,4,1,2,3,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,4,6],[5],[7]] => [7,5,1,2,3,4,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6,7],[2],[3],[5]] => [5,3,2,1,4,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6,7],[2],[4],[5]] => [5,4,2,1,3,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,6,7],[3],[4],[5]] => [5,4,3,1,2,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5,7],[2],[3],[6]] => [6,3,2,1,4,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,7],[2],[4],[6]] => [6,4,2,1,3,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5,7],[3],[4],[6]] => [6,4,3,1,2,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,7],[2],[5],[6]] => [6,5,2,1,3,4,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,7],[3],[5],[6]] => [6,5,3,1,2,4,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,7],[4],[5],[6]] => [6,5,4,1,2,3,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5,6],[2],[3],[7]] => [7,3,2,1,4,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,6],[2],[4],[7]] => [7,4,2,1,3,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5,6],[3],[4],[7]] => [7,4,3,1,2,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6],[2],[5],[7]] => [7,5,2,1,3,4,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6],[3],[5],[7]] => [7,5,3,1,2,4,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,6],[4],[5],[7]] => [7,5,4,1,2,3,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
>>> Load all 127 entries. <<<
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Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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
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.
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