Identifier
Values
[1,0] => [1] => [.,.] => ([],1) => 0
[1,0,1,0] => [1,2] => [.,[.,.]] => ([(0,1)],2) => 1
[1,1,0,0] => [2,1] => [[.,.],.] => ([(0,1)],2) => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,2,5,4,6,3,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,2,5,4,7,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [1,3,2,4,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,2,7,5,6,4] => [.,[[.,.],[[[.,.],[.,.]],.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [1,3,4,2,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,2,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [1,3,4,6,5,2,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,6,5,7,2] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,7,5,4,6,2] => [.,[[.,.],[[[.,.],[.,.]],.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,7,5,6,4,2] => [.,[[.,.],[[[.,.],[.,.]],.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [1,7,4,5,3,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,7,4,5,6,3,2] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [1,7,4,6,5,3,2] => [.,[[[[.,.],.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,0,0,1,1,0,0] => [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [4,2,3,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [4,2,3,1,6,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [4,2,3,6,5,1,7] => [[[.,.],[.,.]],[[.,.],[.,.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [4,2,3,6,5,7,1] => [[[.,.],[.,.]],[[.,.],[.,.]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [7,2,3,5,4,6,1] => [[[.,.],[.,[[.,.],[.,.]]]],.] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [7,2,3,5,6,4,1] => [[[.,.],[.,[[.,.],[.,.]]]],.] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [7,2,6,4,5,3,1] => [[[.,.],[[[.,.],[.,.]],.]],.] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [6,3,2,4,5,7,1] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [6,3,2,5,4,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [6,3,2,5,4,7,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [6,3,4,2,5,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [6,3,4,2,5,7,1] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [6,3,4,5,2,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [6,3,4,5,2,7,1] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,3,5,4,2,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [6,3,5,4,2,7,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [7,6,3,5,4,2,1] => [[[[[.,.],.],[[.,.],.]],.],.] => ([(0,5),(1,4),(2,3),(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
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
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..