Identifier
Values
[[1,3,5,6],[2,4]] => [[1,2,4,6],[3,5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5,6],[2],[4]] => [[1,2,4,6],[3],[5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5],[2,4,6]] => [[1,2,4,6],[3,5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5],[2,6],[4]] => [[1,2,4,6],[3],[5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5],[2,4],[6]] => [[1,2,4,6],[3,5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5],[2],[4],[6]] => [[1,2,4,6],[3],[5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3],[2,4],[5,6]] => [[1,2,4,6],[3,5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3],[2,6],[4],[5]] => [[1,2,4,6],[3],[5]] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,5,6,7],[2,4]] => [[1,2,4,6,7],[3,5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6,7],[2,5]] => [[1,2,4,5,7],[3,6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6,7],[3,5]] => [[1,2,3,5,7],[4,6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,6,7],[2],[4]] => [[1,2,4,6,7],[3],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6,7],[2],[5]] => [[1,2,4,5,7],[3],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6,7],[3],[5]] => [[1,2,3,5,7],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,7],[2,4,6]] => [[1,2,4,6],[3,5,7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,6],[2,4,7]] => [[1,2,4,6,7],[3,5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6],[2,5,7]] => [[1,2,4,5,7],[3,6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6],[3,5,7]] => [[1,2,3,5,7],[4,6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6,7],[2,5],[3]] => [[1,2,5,7],[3,6],[4]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6,7],[2,5],[4]] => [[1,2,4,5],[3,7],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,6,7],[3,5],[4]] => [[1,2,3,5],[4,7],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6,7],[2,4],[5]] => [[1,2,4,7],[3,5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,7],[2,6],[4]] => [[1,2,4,6],[3,7],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,7],[2,4],[6]] => [[1,2,4,6],[3,5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,6],[2,7],[4]] => [[1,2,4,6,7],[3],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6],[2,7],[5]] => [[1,2,4,5,7],[3],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6],[3,7],[5]] => [[1,2,3,5,7],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,6],[2,4],[7]] => [[1,2,4,6,7],[3,5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6],[2,5],[7]] => [[1,2,4,5,7],[3,6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6],[3,5],[7]] => [[1,2,3,5,7],[4,6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6,7],[2],[3],[5]] => [[1,2,5,7],[3],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6,7],[2],[4],[5]] => [[1,2,4,7],[3],[5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,7],[2],[4],[6]] => [[1,2,4,6],[3],[5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5,6],[2],[4],[7]] => [[1,2,4,6,7],[3],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4,6],[2],[5],[7]] => [[1,2,4,5,7],[3],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4,6],[3],[5],[7]] => [[1,2,3,5,7],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6],[2,5,7],[3]] => [[1,2,5,7],[3,6],[4]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,5,7],[4]] => [[1,2,4,5,7],[3],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,6],[3,5,7],[4]] => [[1,2,3,5,7],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,4,7],[5]] => [[1,2,4,7],[3,5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,6,7],[4]] => [[1,2,4,6,7],[3],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,4,7],[6]] => [[1,2,4,6,7],[3,5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4],[2,5,7],[6]] => [[1,2,4,5,7],[3,6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4],[3,5,7],[6]] => [[1,2,3,5,7],[4,6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,4,6],[7]] => [[1,2,4,6],[3,5,7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,7],[2,4],[5,6]] => [[1,2,4,6],[3,5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6],[2,5],[3,7]] => [[1,2,5,7],[3,6],[4]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,5],[4,7]] => [[1,2,4,5],[3,7],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,6],[3,5],[4,7]] => [[1,2,3,5],[4,7],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,4],[5,7]] => [[1,2,4,7],[3,5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,6],[4,7]] => [[1,2,4,6],[3,7],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,4],[6,7]] => [[1,2,4,6,7],[3,5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4],[2,5],[6,7]] => [[1,2,4,5,7],[3,6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4],[3,5],[6,7]] => [[1,2,3,5,7],[4,6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,7],[2,6],[4],[5]] => [[1,2,4,6],[3],[5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6],[2,7],[3],[5]] => [[1,2,5,7],[3],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,7],[4],[5]] => [[1,2,4,7],[3],[5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,7],[4],[6]] => [[1,2,4,6,7],[3],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,4],[2,7],[5],[6]] => [[1,2,4,5,7],[3],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,4],[3,7],[5],[6]] => [[1,2,3,5,7],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6],[2,5],[3],[7]] => [[1,2,5,7],[3,6],[4]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,5],[4],[7]] => [[1,2,4,5],[3,7],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,6],[3,5],[4],[7]] => [[1,2,3,5],[4,7],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2,4],[5],[7]] => [[1,2,4,7],[3,5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,6],[4],[7]] => [[1,2,4,6],[3,7],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2,4],[6],[7]] => [[1,2,4,6],[3,5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,6],[2],[3],[5],[7]] => [[1,2,5,7],[3],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,6],[2],[4],[5],[7]] => [[1,2,4,7],[3],[5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3,5],[2],[4],[6],[7]] => [[1,2,4,6],[3],[5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3],[2,6],[4,7],[5]] => [[1,2,4,6],[3,7],[5]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4],[2,5],[3,7],[6]] => [[1,2,5,7],[3,6],[4]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3],[2,5],[4,7],[6]] => [[1,2,4,5],[3,7],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2],[3,5],[4,7],[6]] => [[1,2,3,5],[4,7],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3],[2,4],[5,7],[6]] => [[1,2,4,7],[3,5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3],[2,4],[5,6],[7]] => [[1,2,4,6],[3,5],[7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4],[2,7],[3],[5],[6]] => [[1,2,5,7],[3],[4],[6]] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3],[2,7],[4],[5],[6]] => [[1,2,4,7],[3],[5],[6]] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,3],[2,6],[4],[5],[7]] => [[1,2,4,6],[3],[5],[7]] => [3,2,2] => ([(1,6),(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
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
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
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
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.