Identifier
-
Mp00159:
Permutations
—Demazure product with inverse⟶
Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ
Values
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3) => 0
[2,3,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[3,1,2] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[3,2,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[1,2,4,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,3,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4) => 0
[2,1,4,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[2,3,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0
[2,3,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[3,1,2,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0
[3,1,4,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[3,2,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0
[3,2,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[4,1,2,3] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[4,1,3,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,2,3,5,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,4] => ([(3,4)],5) => 0
[1,3,5,2,4] => [1,4,5,2,3] => [1,4] => ([(3,4)],5) => 0
[2,1,3,5,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[2,3,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[2,3,1,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[2,3,4,1,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[2,3,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,3,5,1,4] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[2,4,1,5,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,4,5,1,3] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[2,5,1,3,4] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,5,1,4,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,5,3,1,4] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[2,5,4,1,3] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[3,1,2,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[3,1,2,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[3,1,4,2,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[3,1,4,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[3,1,5,2,4] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[3,2,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[3,2,4,1,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[3,2,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[3,2,5,1,4] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[3,5,1,2,4] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[3,5,2,1,4] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[4,1,2,3,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[4,1,2,5,3] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[4,1,3,2,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[4,1,3,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[5,1,2,3,4] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[5,1,2,4,3] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,5] => ([(4,5)],6) => 0
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,5] => ([(4,5)],6) => 0
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [2,4] => ([(3,5),(4,5)],6) => 0
[2,3,1,4,5,6] => [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[2,3,1,5,4,6] => [3,2,1,5,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,3,4,1,5,6] => [4,2,3,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,3,4,5,1,6] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,3,4,5,6,1] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,3,4,6,1,5] => [5,2,3,6,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,3,5,1,4,6] => [4,2,5,1,3,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[2,4,1,3,6,5] => [3,4,1,2,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,4,1,5,6,3] => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,4,5,1,3,6] => [4,5,3,1,2,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,4,5,6,1,3] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,5,1,3,4,6] => [3,5,1,4,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,5,1,4,3,6] => [3,5,1,4,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,5,1,4,6,3] => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,5,3,1,4,6] => [4,5,3,1,2,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,5,3,6,1,4] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,5,4,1,3,6] => [4,5,3,1,2,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[2,5,4,6,1,3] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,6,1,3,4,5] => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,6,1,3,5,4] => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,6,3,4,1,5] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,6,3,5,1,4] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,1,2,4,5,6] => [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[3,1,2,5,4,6] => [3,2,1,5,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[3,1,4,2,5,6] => [4,2,3,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[3,1,4,5,2,6] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,1,4,5,6,2] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,1,4,6,2,5] => [5,2,3,6,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[3,1,5,2,4,6] => [4,2,5,1,3,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[3,2,1,4,5,6] => [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[3,2,1,5,4,6] => [3,2,1,5,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[3,2,4,1,5,6] => [4,2,3,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[3,2,4,5,1,6] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,2,4,5,6,1] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,2,4,6,1,5] => [5,2,3,6,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 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
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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
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.
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