Identifier
-
Mp00075:
Semistandard tableaux
—reading word permutation⟶
Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤ
Values
[[1],[2],[3]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[2],[4]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[3],[4]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[3],[4]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[2],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[3],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[4],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[3],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[4],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[3],[4],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[3],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[4],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[3],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[4],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[3],[4],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[3],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[4],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[2],[3],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[4],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[3],[4],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[3],[4],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[3],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[4],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[3],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[4],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[2],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[3],[4],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[3],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[3],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[4],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[4],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[5],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
[[1],[2],[3],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[4],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[3],[4],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[3],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[4],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[3],[4],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[3],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[4],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[3],[4],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[1]] => [1] => [1] => ([],1) => 1
[[2]] => [1] => [1] => ([],1) => 1
[[3]] => [1] => [1] => ([],1) => 1
[[4]] => [1] => [1] => ([],1) => 1
[[5]] => [1] => [1] => ([],1) => 1
[[6]] => [1] => [1] => ([],1) => 1
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,1,2,1,1] => ([(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) => 4
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Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).
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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