Identifier
Values
{{1}} => [1] => [1] => ([],1) => 0
{{1,2}} => [2,1] => [1,2] => ([],2) => 0
{{1},{2}} => [1,2] => [2,1] => ([(0,1)],2) => 1
{{1,2},{3}} => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
{{1,3},{2}} => [3,2,1] => [1,2,3] => ([],3) => 0
{{1},{2,3}} => [1,3,2] => [2,1,3] => ([(1,2)],3) => 1
{{1,2,3,4}} => [2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
{{1,2},{3,4}} => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
{{1,3},{2},{4}} => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
{{1,4},{2,3}} => [4,3,2,1] => [1,2,3,4] => ([],4) => 0
{{1},{2,4},{3}} => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4) => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,2,3,5},{4}} => [2,3,5,4,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [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
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,2,3,4,5] => ([],5) => 0
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5) => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
{{1,2},{3,6},{4,5}} => [2,1,6,5,4,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
{{1,2},{3,6},{4},{5}} => [2,1,6,4,5,3] => [3,2,1,6,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 2
{{1,3},{2},{4,6},{5}} => [3,2,1,6,5,4] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,4},{2,3},{5,6}} => [4,3,2,1,6,5] => [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
{{1,5},{2,4},{3},{6}} => [5,4,3,2,1,6] => [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
{{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([],6) => 0
{{1},{2,6},{3,5},{4}} => [1,6,5,4,3,2] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
{{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
{{1},{2,6},{3},{4},{5}} => [1,6,3,4,5,2] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 2
{{1},{2},{3},{4},{5,6}} => [1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
{{1,2,3,4},{5,7},{6}} => [2,3,4,1,7,6,5] => [3,4,5,2,1,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,2,3,6,7},{4,5}} => [2,3,6,5,4,7,1] => [3,4,7,6,5,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 4
{{1,2,3,7},{4,6},{5}} => [2,3,7,6,5,4,1] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
{{1,2,5,6,7},{3,4}} => [2,5,4,3,6,7,1] => [3,6,5,4,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 4
{{1,2},{3,4},{5,6},{7}} => [2,1,4,3,6,5,7] => [3,2,5,4,7,6,1] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3
{{1,2},{3,7},{4,5,6}} => [2,1,7,5,6,4,3] => [3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 2
{{1,2},{3,7},{4,6},{5}} => [2,1,7,6,5,4,3] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 2
{{1,2},{3,7},{4},{5,6}} => [2,1,7,4,6,5,3] => [3,2,1,6,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 2
{{1,2},{3,7},{4},{5},{6}} => [2,1,7,4,5,6,3] => [3,2,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => 2
{{1,3,7},{2},{4,5},{6}} => [3,2,7,5,4,6,1] => [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,3,6},{2},{4,7},{5}} => [3,2,6,7,5,1,4] => [4,3,7,1,2,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 3
{{1,3},{2},{4,7},{5,6}} => [3,2,1,7,6,5,4] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,3},{2},{4,7},{5},{6}} => [3,2,1,7,5,6,4] => [4,3,2,1,7,5,6] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,4,5,6,7},{2,3}} => [4,3,2,5,6,7,1] => [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 4
{{1,4,5,7},{2,3,6}} => [4,3,6,5,7,2,1] => [5,4,7,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 4
{{1,4,6},{2,3,7},{5}} => [4,3,7,6,5,1,2] => [5,4,1,2,3,7,6] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,4},{2,3},{5,7},{6}} => [4,3,2,1,7,6,5] => [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
{{1,7},{2,3,4,5,6}} => [7,3,4,5,6,2,1] => [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) => 3
{{1,6},{2,3},{4,5},{7}} => [6,3,2,5,4,1,7] => [7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
{{1,4},{2,7},{3,5,6}} => [4,7,5,1,6,3,2] => [5,1,7,3,2,4,6] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 3
{{1,5},{2,4},{3},{6,7}} => [5,4,3,2,1,7,6] => [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
{{1,7},{2,4},{3,5},{6}} => [7,4,5,2,3,6,1] => [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
{{1,7},{2,4,6},{3},{5}} => [7,4,3,6,5,2,1] => [1,6,5,2,3,4,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,6},{2,5},{3,4},{7}} => [6,5,4,3,2,1,7] => [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
{{1,7},{2,6},{3,4,5}} => [7,6,4,5,3,2,1] => [1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1,7},{2,6},{3,4},{5}} => [7,6,4,3,5,2,1] => [1,2,7,6,3,4,5] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,7},{2,6},{3,5},{4}} => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([],7) => 0
{{1},{2,7},{3,5,6},{4}} => [1,7,5,4,6,3,2] => [2,1,7,6,3,4,5] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1},{2,7},{3,6},{4,5}} => [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 1
{{1,7},{2},{3},{4,6},{5}} => [7,2,3,6,5,4,1] => [1,4,5,2,3,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
{{1},{2,7},{3},{4},{5,6}} => [1,7,3,4,6,5,2] => [2,1,5,6,3,4,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
{{1},{2},{3,7},{4},{5},{6}} => [1,2,7,4,5,6,3] => [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
{{1},{2},{3},{4},{5,7},{6}} => [1,2,3,4,7,6,5] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.