Identifier
Values
[1] => [1,0,1,0] => [2,1] => ([(0,1)],2) => 1
[2,1] => [1,0,1,0,1,0] => [2,1,3] => ([(1,2)],3) => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => ([(2,3)],4) => 1
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => ([(0,3),(1,2)],4) => 1
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => ([(3,4)],5) => 1
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [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) => 3
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => 1
[6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 1
[6,4,1,1] => [1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[6,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => 1
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => 1
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[4,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[6,3,2,2] => [1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [4,1,2,5,6,3,7] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 2
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 2
[4,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5,7] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[6,3,3,1,1] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[6,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => 1
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => 1
[5,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[6,5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[6,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [2,1,3,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[6,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => 1
[4,4,4,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [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) => 3
[6,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7) => 1
[5,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
[6,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7) => 1
[5,5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [3,4,1,2,5,7,6] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7) => 1
[6,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7) => 1
[6,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7) => 1
[6,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,5,6,3,4,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2
[5,5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [3,4,1,2,6,7,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2
[5,5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,7,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
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.