Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [2] => [1,1] => 1
[1,1] => [[1],[2]] => [1,1] => [2] => 0
[3] => [[1,2,3]] => [3] => [1,1,1] => 3
[2,1] => [[1,2],[3]] => [2,1] => [2,1] => 2
[1,1,1] => [[1],[2],[3]] => [1,1,1] => [3] => 0
[4] => [[1,2,3,4]] => [4] => [1,1,1,1] => 6
[3,1] => [[1,2,3],[4]] => [3,1] => [2,1,1] => 5
[2,2] => [[1,2],[3,4]] => [2,2] => [1,2,1] => 4
[2,1,1] => [[1,2],[3],[4]] => [2,1,1] => [3,1] => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => [4] => 0
[5] => [[1,2,3,4,5]] => [5] => [1,1,1,1,1] => 10
[4,1] => [[1,2,3,4],[5]] => [4,1] => [2,1,1,1] => 9
[3,2] => [[1,2,3],[4,5]] => [3,2] => [1,2,1,1] => 8
[3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => [3,1,1] => 7
[2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => [2,2,1] => 6
[2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => [4,1] => 4
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [5] => 0
[6] => [[1,2,3,4,5,6]] => [6] => [1,1,1,1,1,1] => 15
[5,1] => [[1,2,3,4,5],[6]] => [5,1] => [2,1,1,1,1] => 14
[4,2] => [[1,2,3,4],[5,6]] => [4,2] => [1,2,1,1,1] => 13
[4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => [3,1,1,1] => 12
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => [1,1,2,1,1] => 12
[3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => [2,2,1,1] => 11
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => [4,1,1] => 9
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => [1,2,2,1] => 9
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => [3,2,1] => 8
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => [5,1] => 5
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => [6] => 0
[7] => [[1,2,3,4,5,6,7]] => [7] => [1,1,1,1,1,1,1] => 21
[6,1] => [[1,2,3,4,5,6],[7]] => [6,1] => [2,1,1,1,1,1] => 20
[5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => [1,2,1,1,1,1] => 19
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,1,1] => [3,1,1,1,1] => 18
[4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => [1,1,2,1,1,1] => 18
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1] => [2,2,1,1,1] => 17
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => [4,1,1,1] => 15
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1] => [2,1,2,1,1] => 16
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => [1,2,2,1,1] => 15
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => [3,2,1,1] => 14
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1] => [5,1,1] => 11
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => [2,2,2,1] => 12
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => [4,2,1] => 10
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => [6,1] => 6
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => [7] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [8] => [1,1,1,1,1,1,1,1] => 28
[7,1] => [[1,2,3,4,5,6,7],[8]] => [7,1] => [2,1,1,1,1,1,1] => 27
[6,2] => [[1,2,3,4,5,6],[7,8]] => [6,2] => [1,2,1,1,1,1,1] => 26
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [6,1,1] => [3,1,1,1,1,1] => 25
[5,3] => [[1,2,3,4,5],[6,7,8]] => [5,3] => [1,1,2,1,1,1,1] => 25
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [5,2,1] => [2,2,1,1,1,1] => 24
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [5,1,1,1] => [4,1,1,1,1] => 22
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [1,1,1,2,1,1,1] => 24
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [4,3,1] => [2,1,2,1,1,1] => 23
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [4,2,2] => [1,2,2,1,1,1] => 22
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [4,2,1,1] => [3,2,1,1,1] => 21
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [4,1,1,1,1] => [5,1,1,1] => 18
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [3,3,2] => [1,2,1,2,1,1] => 21
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [3,3,1,1] => [3,1,2,1,1] => 20
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [3,2,2,1] => [2,2,2,1,1] => 19
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [3,2,1,1,1] => [4,2,1,1] => 17
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [3,1,1,1,1,1] => [6,1,1] => 13
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [1,2,2,2,1] => 16
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [2,2,2,1,1] => [3,2,2,1] => 15
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [2,2,1,1,1,1] => [5,2,1] => 12
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [2,1,1,1,1,1,1] => [7,1] => 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] => 0
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [3,1,1,1,1,1,1] => [7,1,1] => 15
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [2,2,1,1,1,1,1] => [6,2,1] => 14
[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] => [8,1] => 8
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 45
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [7,2,1] => [2,2,1,1,1,1,1,1] => 41
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [5,4,1] => [2,1,1,2,1,1,1,1] => 39
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [3,3,2,1,1] => [3,2,1,2,1,1] => 31
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [3,2,2,2,1] => [2,2,2,2,1,1] => 29
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [3,1,1,1,1,1,1,1] => [8,1,1] => 17
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [2,1,1,1,1,1,1,1,1] => [9,1] => 9
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 66
[5,4,3] => [[1,2,3,4,5],[6,7,8,9],[10,11,12]] => [5,4,3] => [1,1,2,1,1,2,1,1,1,1] => 56
[5,3,2,1,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]] => [5,3,2,1,1] => [3,2,1,2,1,1,1,1] => 52
[5,2,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]] => [5,2,2,2,1] => [2,2,2,2,1,1,1,1] => 50
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Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.
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
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
conjugate
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