Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ
Values
[1,0,1,1,0,0] => [1,3,2] => [1,2] => ([(1,2)],3) => 0
[1,1,1,0,0,0] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3] => ([(2,3)],4) => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3] => ([(2,3)],4) => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,4] => ([(3,4)],5) => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4] => ([(3,4)],5) => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,4] => ([(3,4)],5) => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,5] => ([(4,5)],6) => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,5,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,5,3,4,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,2,3,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [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
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,2,3,5,1,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,3,5,6,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,4,5,3,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,4,5,6,3,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => [1,6] => ([(5,6)],7) => 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,4,2,5,6,7] => [1,6] => ([(5,6)],7) => 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,3,4,5,2,6,7] => [1,6] => ([(5,6)],7) => 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,3,4,5,6,2,7] => [1,6] => ([(5,6)],7) => 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => [1,6] => ([(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,7,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,1,3,4,6,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,1,3,4,6,7,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [2,1,3,5,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,6,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [2,1,3,5,6,7,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [2,1,4,5,3,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [2,1,4,5,6,3,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [2,1,4,5,6,7,3] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [2,3,1,4,5,7,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
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searching the database for the individual values of this statistic
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
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
to non-crossing permutation
Description
Sends a Dyck path D with valley at positions {(i1,j1),…,(ik,jk)} to the unique non-crossing permutation π having descents {i1,…,ik} and whose inverse has descents {j1,…,jk}.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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