Identifier
Values
[1] => [[1]] => [1] => 1
[2] => [[1,2]] => [2] => 2
[1,1] => [[1],[2]] => [1,1] => 1
[3] => [[1,2,3]] => [3] => 3
[2,1] => [[1,2],[3]] => [2,1] => 1
[1,1,1] => [[1],[2],[3]] => [1,1,1] => 1
[4] => [[1,2,3,4]] => [4] => 4
[3,1] => [[1,2,3],[4]] => [3,1] => 1
[2,2] => [[1,2],[3,4]] => [2,2] => 2
[2,1,1] => [[1,2],[3],[4]] => [2,1,1] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => 1
[5] => [[1,2,3,4,5]] => [5] => 5
[4,1] => [[1,2,3,4],[5]] => [4,1] => 1
[3,2] => [[1,2,3],[4,5]] => [3,2] => 2
[3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 1
[2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => 1
[2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => 1
[6] => [[1,2,3,4,5,6]] => [6] => 6
[5,1] => [[1,2,3,4,5],[6]] => [5,1] => 1
[4,2] => [[1,2,3,4],[5,6]] => [4,2] => 2
[4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 1
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => 3
[3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 1
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 1
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 2
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => 1
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => 1
[7] => [[1,2,3,4,5,6,7]] => [7] => 7
[6,1] => [[1,2,3,4,5,6],[7]] => [6,1] => 1
[5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => 2
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,1,1] => 1
[4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => 3
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1] => 1
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => 1
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1] => 1
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => 2
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => 1
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1] => 1
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => 1
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => 1
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => 1
[8] => [[1,2,3,4,5,6,7,8]] => [8] => 8
[7,1] => [[1,2,3,4,5,6,7],[8]] => [7,1] => 1
[6,2] => [[1,2,3,4,5,6],[7,8]] => [6,2] => 2
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [6,1,1] => 1
[5,3] => [[1,2,3,4,5],[6,7,8]] => [5,3] => 3
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [5,2,1] => 1
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [5,1,1,1] => 1
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => 4
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [4,3,1] => 1
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [4,2,2] => 2
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [4,2,1,1] => 1
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [4,1,1,1,1] => 1
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [3,3,2] => 2
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [3,3,1,1] => 1
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [3,2,2,1] => 1
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [3,2,1,1,1] => 1
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [3,1,1,1,1,1] => 1
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => 2
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [2,2,2,1,1] => 1
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [2,2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,1] => 1
[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => 9
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [8,1] => 1
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [7,2] => 2
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [7,1,1] => 1
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [6,3] => 3
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [6,2,1] => 1
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [6,1,1,1] => 1
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [5,4] => 4
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [5,3,1] => 1
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [5,2,2] => 2
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [5,2,1,1] => 1
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [5,1,1,1,1] => 1
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [4,4,1] => 1
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [4,3,2] => 2
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [4,3,1,1] => 1
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [4,2,2,1] => 1
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [4,2,1,1,1] => 1
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [4,1,1,1,1,1] => 1
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => 3
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [3,3,2,1] => 1
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [3,3,1,1,1] => 1
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [3,2,2,2] => 2
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [3,2,2,1,1] => 1
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [3,2,1,1,1,1] => 1
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [3,1,1,1,1,1,1] => 1
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [2,2,2,2,1] => 1
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [2,1,1,1,1,1,1,1] => 1
[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] => 1
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Description
The smallest part of an integer composition.
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
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.