Identifier
Values
[1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [1,2] => ([],2) => 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3) => 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[4,2,3,1,5] => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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) => 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
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
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
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].