Identifier
Values
{{1}} => [1] => [1] => ([],1) => 0
{{1,2}} => [2,1] => [2,1] => ([(0,1)],2) => 1
{{1},{2}} => [1,2] => [1,2] => ([],2) => 0
{{1,2,3}} => [2,3,1] => [2,1,3] => ([(1,2)],3) => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
{{1},{2},{3}} => [1,2,3] => [1,2,3] => ([],3) => 0
{{1,2,3,4}} => [2,3,4,1] => [2,1,3,4] => ([(2,3)],4) => 1
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4) => 1
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4) => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => ([],4) => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [2,1,3,4,5] => ([(3,4)],5) => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
{{1,3,5},{2,4}} => [3,4,5,2,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5) => 1
{{1,5},{2,4},{3}} => [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
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => ([],5) => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => 1
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => 1
{{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => 1
{{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3},{4},{5,6}} => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => 1
{{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,3,5},{2,4,6}} => [3,4,5,6,1,2] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
{{1,3},{2},{4,6},{5}} => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 2
{{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => [1,3,2,4,5,6] => ([(4,5)],6) => 1
{{1},{2,3},{4,5},{6}} => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => 1
{{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => [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) => 3
{{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
{{1,6},{2,5},{3,4}} => [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
{{1},{2,5},{3,4},{6}} => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6) => 1
{{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => [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
{{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => 0
{{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 1
{{1,2,3,4,5},{6,7}} => [2,3,4,5,1,7,6] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1,2,3,4},{5},{6},{7}} => [2,3,4,1,5,6,7] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1,2,3,5,6,7},{4}} => [2,3,5,4,6,7,1] => [5,2,3,4,1,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,2,3},{4},{5,6,7}} => [2,3,1,4,6,7,5] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => 1
{{1,2},{3,4,5,6,7}} => [2,1,4,5,6,7,3] => [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1,2},{3,4,5},{6,7}} => [2,1,4,5,3,7,6] => [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7) => 1
{{1,2},{3,4},{5,7},{6}} => [2,1,4,3,7,6,5] => [7,2,1,4,3,6,5] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3
{{1,2,5},{3,6,7},{4}} => [2,5,6,4,1,7,3] => [2,1,5,4,3,6,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 2
{{1,2},{3,5,6},{4},{7}} => [2,1,5,4,6,3,7] => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 3
{{1,2},{3,5},{4},{6,7}} => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 2
{{1,2},{3},{4,5},{6},{7}} => [2,1,3,5,4,6,7] => [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
{{1,2},{3,7},{4,6},{5}} => [2,1,7,6,5,4,3] => [7,6,5,2,1,4,3] => ([(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),(4,5),(4,6),(5,6)],7) => 5
{{1,2},{3},{4},{5},{6,7}} => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => 1
{{1,3,4,5,7},{2},{6}} => [3,2,4,5,7,6,1] => [3,2,1,4,5,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7) => 2
{{1,3,5,7},{2,4,6}} => [3,4,5,6,7,2,1] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 2
{{1,3,5,7},{2,4},{6}} => [3,4,5,2,7,6,1] => [3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 2
{{1,3,5},{2,4},{6},{7}} => [3,4,5,2,1,6,7] => [3,4,5,2,1,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,3},{2,4,6},{5,7}} => [3,4,1,6,7,2,5] => [3,4,1,2,6,5,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
{{1,3,5,7},{2},{4,6}} => [3,2,5,6,7,4,1] => [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 2
{{1,3,5,7},{2},{4},{6}} => [3,2,5,4,7,6,1] => [3,2,1,5,4,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 2
{{1,3},{2,6},{4,5},{7}} => [3,6,1,5,4,2,7] => [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
{{1,3},{2},{4,5},{6,7}} => [3,2,1,5,4,7,6] => [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
{{1,3},{2},{4,6},{5},{7}} => [3,2,1,6,5,4,7] => [3,6,2,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,3},{2},{4},{5,7},{6}} => [3,2,1,4,7,6,5] => [3,2,1,4,7,6,5] => ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => 2
{{1,4,7},{2,3,5,6}} => [4,3,5,7,6,2,1] => [4,3,2,1,5,7,6] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,4,7},{2,3,6},{5}} => [4,3,6,7,5,2,1] => [4,3,2,1,6,7,5] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,4,7},{2,3},{5,6}} => [4,3,2,7,6,5,1] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => [1,3,2,4,5,6,7] => ([(5,6)],7) => 1
{{1},{2,3,5},{4},{6,7}} => [1,3,5,4,2,7,6] => [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
{{1,7},{2,3},{4,6},{5}} => [7,3,2,6,5,4,1] => [7,6,3,2,5,4,1] => ([(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},{2,3},{4},{5,6},{7}} => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7) => 1
{{1},{2,3},{4},{5},{6},{7}} => [1,3,2,4,5,6,7] => [1,3,4,5,6,2,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1,4,7},{2,5,6},{3}} => [4,5,3,7,6,2,1] => [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
{{1,4,7},{2,6},{3},{5}} => [4,6,3,7,5,2,1] => [4,3,2,1,6,5,7] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1,5},{2,4},{3},{6,7}} => [5,4,3,2,1,7,6] => [5,4,7,6,3,2,1] => ([(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},{2,4,6},{3,5},{7}} => [1,4,5,6,3,2,7] => [1,4,3,2,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 2
{{1},{2,4,6},{3},{5},{7}} => [1,4,3,6,5,2,7] => [1,4,3,2,6,5,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 2
{{1,7},{2,4},{3},{5,6}} => [7,4,3,2,6,5,1] => [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},{2,4},{3},{5,7},{6}} => [1,4,3,2,7,6,5] => [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},{2,4},{3},{5},{6},{7}} => [1,4,3,2,5,6,7] => [1,4,5,6,3,2,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1},{2},{3,4,5},{6},{7}} => [1,2,4,5,3,6,7] => [1,2,4,3,5,6,7] => ([(5,6)],7) => 1
{{1},{2,6},{3,4},{5,7}} => [1,6,4,3,7,2,5] => [6,7,1,4,3,2,5] => ([(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}} => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
{{1,5},{2},{3,6},{4},{7}} => [5,2,6,4,1,3,7] => [5,6,7,2,1,4,3] => ([(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},{2,5},{3,7},{4},{6}} => [1,5,7,4,2,6,3] => [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
{{1,7},{2,6},{3,5},{4}} => [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
{{1},{2,6},{3,5},{4},{7}} => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
{{1},{2},{3,5,7},{4,6}} => [1,2,5,6,7,4,3] => [5,4,1,2,3,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1},{2},{3,5},{4},{6},{7}} => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7) => 2
{{1},{2},{3},{4,5,6,7}} => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1},{2,6},{3},{4},{5},{7}} => [1,6,3,4,5,2,7] => [1,6,3,4,5,2,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1},{2},{3},{4,6},{5},{7}} => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
{{1},{2},{3},{4},{5,6},{7}} => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1},{2},{3},{4},{5},{6},{7}} => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],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 permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.