Identifier
Values
[[1]] => [[1]] => [1] => ([],1) => 0
[[1,2]] => [[1,2]] => [2] => ([],2) => 0
[[1],[2]] => [[1,2]] => [2] => ([],2) => 0
[[1,2,3]] => [[1,2,3]] => [3] => ([],3) => 0
[[1,2],[3]] => [[1,2,3]] => [3] => ([],3) => 0
[[1,2,3,4]] => [[1,2,3,4]] => [4] => ([],4) => 0
[[1,2,3],[4]] => [[1,2,3,4]] => [4] => ([],4) => 0
[[1,2],[3,4]] => [[1,2,3,4]] => [4] => ([],4) => 0
[[1,2,3,4,5]] => [[1,2,3,4,5]] => [5] => ([],5) => 0
[[1,2,3,5],[4]] => [[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2,3,4],[5]] => [[1,2,3,4,5]] => [5] => ([],5) => 0
[[1,2,5],[3,4]] => [[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2,3],[4,5]] => [[1,2,3,4,5]] => [5] => ([],5) => 0
[[1,2,5],[3],[4]] => [[1,2,3],[4],[5]] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,3],[4],[5]] => [[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2],[3,4],[5]] => [[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2],[3],[4],[5]] => [[1,2,3],[4],[5]] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,3,4,5,6]] => [[1,2,3,4,5,6]] => [6] => ([],6) => 0
[[1,2,3,5,6],[4]] => [[1,2,3,4,6],[5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,4,5],[6]] => [[1,2,3,4,5,6]] => [6] => ([],6) => 0
[[1,2,5,6],[3,4]] => [[1,2,3,4],[5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,5],[4,6]] => [[1,2,3,4,6],[5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,4],[5,6]] => [[1,2,3,4,5,6]] => [6] => ([],6) => 0
[[1,2,5,6],[3],[4]] => [[1,2,3,6],[4],[5]] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3,5],[4],[6]] => [[1,2,3,4,6],[5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,5],[3,4,6]] => [[1,2,3,4,6],[5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3],[4,5,6]] => [[1,2,3,4,5,6]] => [6] => ([],6) => 0
[[1,2,5],[3,6],[4]] => [[1,2,3,6],[4],[5]] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4,6],[5]] => [[1,2,3,4,6],[5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,5],[3,4],[6]] => [[1,2,3,4],[5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,5],[3],[4],[6]] => [[1,2,3,6],[4],[5]] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2],[3,4],[5,6]] => [[1,2,3,4],[5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2],[3,6],[4],[5]] => [[1,2,3,6],[4],[5]] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3,4,5,6,7]] => [[1,2,3,4,5,6,7]] => [7] => ([],7) => 0
[[1,2,3,5,6,7],[4]] => [[1,2,3,4,6,7],[5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4,5,6],[7]] => [[1,2,3,4,5,6,7]] => [7] => ([],7) => 0
[[1,2,5,6,7],[3,4]] => [[1,2,3,4,7],[5,6]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,5,6],[4,7]] => [[1,2,3,4,6,7],[5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4,5],[6,7]] => [[1,2,3,4,5,6,7]] => [7] => ([],7) => 0
[[1,2,5,6,7],[3],[4]] => [[1,2,3,6,7],[4],[5]] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,5,6],[4],[7]] => [[1,2,3,4,6,7],[5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5,6],[3,4,7]] => [[1,2,3,4,7],[5,6]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,5],[4,6,7]] => [[1,2,3,4,6,7],[5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4],[5,6,7]] => [[1,2,3,4,5,6,7]] => [7] => ([],7) => 0
[[1,2,5,6],[3,7],[4]] => [[1,2,3,6,7],[4],[5]] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3,5],[4,7],[6]] => [[1,2,3,4,6,7],[5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5,6],[3,4],[7]] => [[1,2,3,4,7],[5,6]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5,6],[3],[4],[7]] => [[1,2,3,6,7],[4],[5]] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,5],[3,6,7],[4]] => [[1,2,3,6,7],[4],[5]] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,3],[4,6,7],[5]] => [[1,2,3,4,6,7],[5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5],[3,4,7],[6]] => [[1,2,3,4,7],[5,6]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5],[3,4],[6,7]] => [[1,2,3,4,7],[5,6]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,5],[3,7],[4],[6]] => [[1,2,3,6,7],[4],[5]] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
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 threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
catabolism
Description
Remove the first row of the standard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.