Identifier
Values
[1,0] => [[1],[2]] => [1,1] => [2] => 2
[1,0,1,0] => [[1,3],[2,4]] => [1,2,1] => [2,2] => 2
[1,1,0,0] => [[1,2],[3,4]] => [2,2] => [1,2,1] => 2
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [1,2,2,1] => [2,2,2] => 2
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [1,3,2] => [2,1,2,1] => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [2,3,1] => [1,2,1,2] => 2
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [2,2,2] => [1,2,2,1] => 2
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [3,3] => [1,1,2,1,1] => 2
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [1,2,2,2,1] => [2,2,2,2] => 2
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [1,2,3,2] => [2,2,1,2,1] => 2
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [1,3,3,1] => [2,1,2,1,2] => 2
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [1,3,2,2] => [2,1,2,2,1] => 2
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [1,4,3] => [2,1,1,2,1,1] => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [2,3,2,1] => [1,2,1,2,2] => 2
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [2,4,2] => [1,2,1,1,2,1] => 2
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [2,2,3,1] => [1,2,2,1,2] => 2
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [2,2,2,2] => [1,2,2,2,1] => 2
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [2,3,3] => [1,2,1,2,1,1] => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [3,4,1] => [1,1,2,1,1,2] => 2
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [3,3,2] => [1,1,2,1,2,1] => 2
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [3,2,3] => [1,1,2,2,1,1] => 2
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [1,1,1,2,1,1,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [1,3,2,3,1] => [2,1,2,2,1,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => [3,4,3] => [1,1,2,1,1,2,1,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [1,2,2,2,2,2,1] => [2,2,2,2,2,2] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [[1,3,5,7,8,9],[2,4,6,10,11,12]] => [1,2,2,4,3] => [2,2,2,1,1,2,1,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,6,7,11],[2,4,8,9,10,12]] => [1,2,4,4,1] => [2,2,1,1,2,1,1,2] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [[1,3,5,6,7,9],[2,4,8,10,11,12]] => [1,2,4,2,3] => [2,2,1,1,2,2,1,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,4,7,8,11],[2,5,6,9,10,12]] => [1,3,4,3,1] => [2,1,2,1,1,2,1,2] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [[1,3,4,6,8,9],[2,5,7,10,11,12]] => [1,3,2,3,3] => [2,1,2,2,1,2,1,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [[1,3,4,5,9,11],[2,6,7,8,10,12]] => [1,4,4,2,1] => [2,1,1,2,1,1,2,2] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [[1,3,4,5,7,11],[2,6,8,9,10,12]] => [1,4,2,4,1] => [2,1,1,2,2,1,1,2] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [[1,3,4,5,7,9],[2,6,8,10,11,12]] => [1,4,2,2,3] => [2,1,1,2,2,2,1,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [[1,3,4,5,6,7],[2,8,9,10,11,12]] => [1,6,5] => [2,1,1,1,1,2,1,1,1,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [[1,2,3,7,9,11],[4,5,6,8,10,12]] => [3,4,2,2,1] => [1,1,2,1,1,2,2,2] => 2
[1,1,1,0,0,0,1,1,1,0,0,0] => [[1,2,3,7,8,9],[4,5,6,10,11,12]] => [3,6,3] => [1,1,2,1,1,1,1,2,1,1] => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [[1,2,3,6,8,11],[4,5,7,9,10,12]] => [3,3,2,3,1] => [1,1,2,1,2,2,1,2] => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [[1,2,3,5,9,11],[4,6,7,8,10,12]] => [3,2,4,2,1] => [1,1,2,2,1,1,2,2] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [[1,2,3,5,7,11],[4,6,8,9,10,12]] => [3,2,2,4,1] => [1,1,2,2,2,1,1,2] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [[1,2,3,5,7,9],[4,6,8,10,11,12]] => [3,2,2,2,3] => [1,1,2,2,2,2,1,1] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [[1,2,3,5,6,7],[4,8,9,10,11,12]] => [3,4,5] => [1,1,2,1,1,2,1,1,1,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [[1,2,3,4,5,11],[6,7,8,9,10,12]] => [5,6,1] => [1,1,1,1,2,1,1,1,1,2] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [[1,2,3,4,5,9],[6,7,8,10,11,12]] => [5,4,3] => [1,1,1,1,2,1,1,2,1,1] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [[1,2,3,4,5,7],[6,8,9,10,11,12]] => [5,2,5] => [1,1,1,1,2,2,1,1,1,1] => 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 part of an integer composition.
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
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 $i$.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.