Processing math: 100%

Identifier
Values
[1,0] => [1] => ([],1) => ([],1) => 0
[1,0,1,0] => [2,1] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,1,0,0] => [1,2] => ([],2) => ([],2) => 0
[1,0,1,1,0,0] => [2,1,3] => ([(1,2)],3) => ([(1,2)],3) => 1
[1,1,0,0,1,0] => [1,3,2] => ([(1,2)],3) => ([(1,2)],3) => 1
[1,1,1,0,0,0] => [1,2,3] => ([],3) => ([],3) => 0
[1,0,1,1,0,0,1,0] => [2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => 1
[1,0,1,1,1,0,0,0] => [2,1,3,4] => ([(2,3)],4) => ([(2,3)],4) => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => ([(2,3)],4) => ([(2,3)],4) => 1
[1,1,0,1,0,1,0,0] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => ([(2,3)],4) => ([(2,3)],4) => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => ([],4) => ([],4) => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => ([(3,4)],5) => ([(3,4)],5) => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => 1
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => ([(3,4)],5) => ([(3,4)],5) => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => ([(3,4)],5) => ([(3,4)],5) => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => ([(3,4)],5) => ([(3,4)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([],5) => ([],5) => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => ([(4,5)],6) => ([(4,5)],6) => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => ([(4,5)],6) => ([(4,5)],6) => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => ([(4,5)],6) => ([(4,5)],6) => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 3
[1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => ([(4,5)],6) => ([(4,5)],6) => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6) => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => ([(4,5)],6) => ([(4,5)],6) => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => ([],6) => ([],6) => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [2,1,5,6,3,4,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [2,1,3,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => ([(5,6)],7) => ([(5,6)],7) => 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,3,4,5,6,2,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [1,3,2,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,3,2,4,5,6,7] => ([(5,6)],7) => ([(5,6)],7) => 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [3,4,1,2,6,7,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [3,4,1,2,6,5,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [3,4,1,2,5,7,6] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [3,4,1,2,7,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,2,4,3,5,6,7] => ([(5,6)],7) => ([(5,6)],7) => 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [1,4,5,2,3,7,6] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,4,5,2,3,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,2,3,5,4,6,7] => ([(5,6)],7) => ([(5,6)],7) => 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [1,5,6,7,2,3,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,2,3,4,6,5,7] => ([(5,6)],7) => ([(5,6)],7) => 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7) => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [1,6,2,3,4,5,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => ([(5,6)],7) => ([(5,6)],7) => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => ([],7) => ([],7) => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of {1,…,n}, this is the graph with vertices {1,…,n}, where (i,j) is an edge if and only if it is an inversion of the permutation.
Map
connected complement
Description
The componentwise connected complement of a graph.
For a connected graph G, this map returns the complement of G if it is connected, otherwise G itself. If G is not connected, the map is applied to each connected component separately.