Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Images
[1] => [[1]] => [1] => [1]
[2] => [[1,2]] => [2] => [1,1]
[1,1] => [[1],[2]] => [1,1] => [2]
[3] => [[1,2,3]] => [3] => [1,1,1]
[2,1] => [[1,2],[3]] => [2,1] => [1,2]
[1,1,1] => [[1],[2],[3]] => [1,1,1] => [3]
[4] => [[1,2,3,4]] => [4] => [1,1,1,1]
[3,1] => [[1,2,3],[4]] => [3,1] => [1,1,2]
[2,2] => [[1,2],[3,4]] => [2,2] => [1,2,1]
[2,1,1] => [[1,2],[3],[4]] => [2,1,1] => [1,3]
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => [4]
[5] => [[1,2,3,4,5]] => [5] => [1,1,1,1,1]
[4,1] => [[1,2,3,4],[5]] => [4,1] => [1,1,1,2]
[3,2] => [[1,2,3],[4,5]] => [3,2] => [1,1,2,1]
[3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => [1,1,3]
[2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => [1,2,2]
[2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => [1,4]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [5]
[6] => [[1,2,3,4,5,6]] => [6] => [1,1,1,1,1,1]
[5,1] => [[1,2,3,4,5],[6]] => [5,1] => [1,1,1,1,2]
[4,2] => [[1,2,3,4],[5,6]] => [4,2] => [1,1,1,2,1]
[4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => [1,1,1,3]
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => [1,1,2,1,1]
[3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => [1,1,2,2]
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => [1,1,4]
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => [1,2,2,1]
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => [1,2,3]
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => [1,5]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => [6]
[7] => [[1,2,3,4,5,6,7]] => [7] => [1,1,1,1,1,1,1]
[6,1] => [[1,2,3,4,5,6],[7]] => [6,1] => [1,1,1,1,1,2]
[5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => [1,1,1,1,2,1]
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,1,1] => [1,1,1,1,3]
[4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => [1,1,1,2,1,1]
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1] => [1,1,1,2,2]
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => [1,1,1,4]
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1] => [1,1,2,1,2]
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => [1,1,2,2,1]
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => [1,1,2,3]
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1] => [1,1,5]
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => [1,2,2,2]
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => [1,2,4]
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => [1,6]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => [7]
[8] => [[1,2,3,4,5,6,7,8]] => [8] => [1,1,1,1,1,1,1,1]
[7,1] => [[1,2,3,4,5,6,7],[8]] => [7,1] => [1,1,1,1,1,1,2]
[6,2] => [[1,2,3,4,5,6],[7,8]] => [6,2] => [1,1,1,1,1,2,1]
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [6,1,1] => [1,1,1,1,1,3]
[5,3] => [[1,2,3,4,5],[6,7,8]] => [5,3] => [1,1,1,1,2,1,1]
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [5,2,1] => [1,1,1,1,2,2]
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [5,1,1,1] => [1,1,1,1,4]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [1,1,1,2,1,1,1]
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [4,3,1] => [1,1,1,2,1,2]
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [4,2,2] => [1,1,1,2,2,1]
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [4,2,1,1] => [1,1,1,2,3]
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [4,1,1,1,1] => [1,1,1,5]
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [3,3,2] => [1,1,2,1,2,1]
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [3,3,1,1] => [1,1,2,1,3]
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [3,2,2,1] => [1,1,2,2,2]
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [3,2,1,1,1] => [1,1,2,4]
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [3,1,1,1,1,1] => [1,1,6]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [1,2,2,2,1]
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [2,2,2,1,1] => [1,2,2,3]
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [2,2,1,1,1,1] => [1,2,5]
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [2,1,1,1,1,1,1] => [1,7]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,1] => [8]
[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => [1,1,1,1,1,1,1,1,1]
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [8,1] => [1,1,1,1,1,1,1,2]
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [7,2] => [1,1,1,1,1,1,2,1]
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [7,1,1] => [1,1,1,1,1,1,3]
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [6,3] => [1,1,1,1,1,2,1,1]
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [6,2,1] => [1,1,1,1,1,2,2]
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [6,1,1,1] => [1,1,1,1,1,4]
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [5,4] => [1,1,1,1,2,1,1,1]
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [5,3,1] => [1,1,1,1,2,1,2]
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [5,2,2] => [1,1,1,1,2,2,1]
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [5,2,1,1] => [1,1,1,1,2,3]
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [5,1,1,1,1] => [1,1,1,1,5]
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [4,4,1] => [1,1,1,2,1,1,2]
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [4,3,2] => [1,1,1,2,1,2,1]
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [4,3,1,1] => [1,1,1,2,1,3]
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [4,2,2,1] => [1,1,1,2,2,2]
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [4,2,1,1,1] => [1,1,1,2,4]
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [4,1,1,1,1,1] => [1,1,1,6]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => [1,1,2,1,2,1,1]
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [3,3,2,1] => [1,1,2,1,2,2]
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [3,3,1,1,1] => [1,1,2,1,4]
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [3,2,2,2] => [1,1,2,2,2,1]
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [3,2,2,1,1] => [1,1,2,2,3]
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [3,2,1,1,1,1] => [1,1,2,5]
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [3,1,1,1,1,1,1] => [1,1,7]
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [2,2,2,2,1] => [1,2,2,2,2]
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [2,2,2,1,1,1] => [1,2,2,4]
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [2,2,1,1,1,1,1] => [1,2,6]
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [2,1,1,1,1,1,1,1] => [1,8]
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [1,1,1,1,1,1,1,1,1] => [9]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10] => [1,1,1,1,1,1,1,1,1,1]
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [9,1] => [1,1,1,1,1,1,1,1,2]
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [8,2] => [1,1,1,1,1,1,1,2,1]
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [8,1,1] => [1,1,1,1,1,1,1,3]
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [7,3] => [1,1,1,1,1,1,2,1,1]
>>> Load all 197 entries. <<<Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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 $i$.
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
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 $I$.
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 $I$.
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