Identifier
-
Mp00068:
Permutations
—Simion-Schmidt map⟶
Permutations
Mp00248: Permutations —DEX 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
[2,1] => [2,1] => [2] => ([],2) => 0
[1,2,3] => [1,3,2] => [1,2] => ([(1,2)],3) => 1
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3) => 1
[2,1,3] => [2,1,3] => [3] => ([],3) => 0
[2,3,1] => [2,3,1] => [3] => ([],3) => 0
[3,1,2] => [3,1,2] => [3] => ([],3) => 0
[2,3,1,4] => [2,4,1,3] => [4] => ([],4) => 0
[2,4,1,3] => [2,4,1,3] => [4] => ([],4) => 0
[3,4,1,2] => [3,4,1,2] => [4] => ([],4) => 0
[4,3,1,2] => [4,3,1,2] => [1,3] => ([(2,3)],4) => 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,3,1,5,4] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,4,1,3,5] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,4,1,5,3] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,5,1,3,4] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,5,1,4,3] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,4,1,2,5] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,4,1,5,2] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,4,2,5,1] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,5,1,2,4] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,5,1,4,2] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,5,2,4,1] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,3,5,1,2] => [4,3,5,1,2] => [1,4] => ([(3,4)],5) => 1
[4,5,1,2,3] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,1,3,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,2,3,1] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,3,2,1] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,3,4,1,2] => [5,3,4,1,2] => [1,4] => ([(3,4)],5) => 1
[4,5,3,2,1,6] => [4,6,3,2,1,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,6,3,2,1,5] => [4,6,3,2,1,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,6,2,3,1,4] => [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,6,2,4,1,3] => [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,6,3,2,1,4] => [5,6,3,2,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,6,3,4,1,2] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,5,3,2,6,1,7] => [4,7,3,2,6,1,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,5,3,2,7,1,6] => [4,7,3,2,6,1,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,6,3,2,5,1,7] => [4,7,3,2,6,1,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,6,3,2,7,1,5] => [4,7,3,2,6,1,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,7,3,2,5,1,6] => [4,7,3,2,6,1,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,7,3,2,6,1,5] => [4,7,3,2,6,1,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,6,3,2,4,1,7] => [5,7,3,2,6,1,4] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,6,3,2,7,1,4] => [5,7,3,2,6,1,4] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,7,3,2,4,1,6] => [5,7,3,2,6,1,4] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,7,3,2,6,1,4] => [5,7,3,2,6,1,4] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
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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
DEX composition
Description
The DEX composition of a permutation.
Let π be a permutation in Sn. Let ˉπ be the word in the ordered set ˉ1<⋯<ˉn<1⋯<n obtained from π by replacing every excedance π(i)>i by ¯π(i). Then the DEX set of π is the set of indices 1≤i<n such that ˉπ(i)>ˉπ(i+1). Finally, the DEX composition c1,…,ck of n corresponds to the DEX subset {c1,c1+c2,…,c1+⋯+ck−1}.
The (quasi)symmetric function
∑π∈Sλ,jFDEX(π),
where the sum is over the set of permutations of cycle type λ with j excedances, is the Eulerian quasisymmetric function.
Let π be a permutation in Sn. Let ˉπ be the word in the ordered set ˉ1<⋯<ˉn<1⋯<n obtained from π by replacing every excedance π(i)>i by ¯π(i). Then the DEX set of π is the set of indices 1≤i<n such that ˉπ(i)>ˉπ(i+1). Finally, the DEX composition c1,…,ck of n corresponds to the DEX subset {c1,c1+c2,…,c1+⋯+ck−1}.
The (quasi)symmetric function
∑π∈Sλ,jFDEX(π),
where the sum is over the set of permutations of cycle type λ with j excedances, is the Eulerian quasisymmetric function.
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
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a 123-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between 132-avoiding permutations and 123-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between 132-avoiding permutations and 123-avoiding permutations, see [1, Proposition 19].
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