Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Images
[1] => [[1]] => [1] => [1] => [[1]]
[2] => [[1,2]] => [1,2] => [2,1] => [[1],[2]]
[1,1] => [[1],[2]] => [2,1] => [1,2] => [[1,2]]
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
[2,1] => [[1,2],[3]] => [3,1,2] => [1,3,2] => [[1,2],[3]]
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => [[1,2,3]]
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[1,3],[2,4],[5],[6]]
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,6,5,4,3] => [[1,2,3],[4],[5],[6]]
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[1,4],[2,5],[3,6]]
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,3,2,6,5,4] => [[1,2,4],[3,5],[6]]
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => [[1,2,3,4],[5],[6]]
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => [[1,3,5],[2,4,6]]
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,2,4,3,6,5] => [[1,2,3,5],[4,6]]
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7]]
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7]]
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7]]
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => [[1,2,3],[4],[5],[6],[7]]
[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => [[1,4],[2,5],[3,6],[7]]
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => [1,3,2,7,6,5,4] => [[1,2,4],[3,5],[6],[7]]
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [[1,2,3,4],[5],[6],[7]]
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [1,4,3,2,7,6,5] => [[1,2,5],[3,6],[4,7]]
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => [2,1,4,3,7,6,5] => [[1,3,5],[2,4,6],[7]]
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => [[1,2,3,5],[4,6],[7]]
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => [[1,2,3,4,5],[6],[7]]
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => [[1,2,4,6],[3,5,7]]
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => [[1,2,3,4,5,6],[7]]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7]]
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8]]
[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8]]
[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => [2,1,8,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7],[8]]
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => [1,2,8,7,6,5,4,3] => [[1,2,3],[4],[5],[6],[7],[8]]
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => [3,2,1,8,7,6,5,4] => [[1,4],[2,5],[3,6],[7],[8]]
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [8,6,7,1,2,3,4,5] => [1,3,2,8,7,6,5,4] => [[1,2,4],[3,5],[6],[7],[8]]
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => [1,2,3,8,7,6,5,4] => [[1,2,3,4],[5],[6],[7],[8]]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [[1,5],[2,6],[3,7],[4,8]]
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => [1,4,3,2,8,7,6,5] => [[1,2,5],[3,6],[4,7],[8]]
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => [2,1,4,3,8,7,6,5] => [[1,3,5],[2,4,6],[7],[8]]
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [8,7,5,6,1,2,3,4] => [1,2,4,3,8,7,6,5] => [[1,2,3,5],[4,6],[7],[8]]
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => [1,2,3,4,8,7,6,5] => [[1,2,3,4,5],[6],[7],[8]]
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [7,8,4,5,6,1,2,3] => [2,1,5,4,3,8,7,6] => [[1,3,6],[2,4,7],[5,8]]
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => [[1,2,3,6],[4,7],[5,8]]
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [8,6,7,4,5,1,2,3] => [1,3,2,5,4,8,7,6] => [[1,2,4,6],[3,5,7],[8]]
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [8,7,6,4,5,1,2,3] => [1,2,3,5,4,8,7,6] => [[1,2,3,4,6],[5,7],[8]]
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => [1,2,3,4,5,8,7,6] => [[1,2,3,4,5,6],[7],[8]]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [[1,3,5,7],[2,4,6,8]]
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => [1,2,4,3,6,5,8,7] => [[1,2,3,5,7],[4,6,8]]
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => [[1,2,3,4,5,7],[6,8]]
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => [[1,2,3,4,5,6,7],[8]]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8]]
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [8,9,1,2,3,4,5,6,7] => [2,1,9,8,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7],[8],[9]]
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => [1,2,9,8,7,6,5,4,3] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [7,8,9,1,2,3,4,5,6] => [3,2,1,9,8,7,6,5,4] => [[1,4],[2,5],[3,6],[7],[8],[9]]
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [9,7,8,1,2,3,4,5,6] => [1,3,2,9,8,7,6,5,4] => [[1,2,4],[3,5],[6],[7],[8],[9]]
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => [1,2,3,9,8,7,6,5,4] => [[1,2,3,4],[5],[6],[7],[8],[9]]
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [6,7,8,9,1,2,3,4,5] => [4,3,2,1,9,8,7,6,5] => [[1,5],[2,6],[3,7],[4,8],[9]]
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [9,6,7,8,1,2,3,4,5] => [1,4,3,2,9,8,7,6,5] => [[1,2,5],[3,6],[4,7],[8],[9]]
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [8,9,6,7,1,2,3,4,5] => [2,1,4,3,9,8,7,6,5] => [[1,3,5],[2,4,6],[7],[8],[9]]
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [9,8,6,7,1,2,3,4,5] => [1,2,4,3,9,8,7,6,5] => [[1,2,3,5],[4,6],[7],[8],[9]]
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => [[1,2,3,4,5],[6],[7],[8],[9]]
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [9,5,6,7,8,1,2,3,4] => [1,5,4,3,2,9,8,7,6] => [[1,2,6],[3,7],[4,8],[5,9]]
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [8,9,5,6,7,1,2,3,4] => [2,1,5,4,3,9,8,7,6] => [[1,3,6],[2,4,7],[5,8],[9]]
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [9,8,5,6,7,1,2,3,4] => [1,2,5,4,3,9,8,7,6] => [[1,2,3,6],[4,7],[5,8],[9]]
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,1,2,3,4] => [1,3,2,5,4,9,8,7,6] => [[1,2,4,6],[3,5,7],[8],[9]]
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,1,2,3,4] => [1,2,3,5,4,9,8,7,6] => [[1,2,3,4,6],[5,7],[8],[9]]
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,1,2,3,4] => [1,2,3,4,5,9,8,7,6] => [[1,2,3,4,5,6],[7],[8],[9]]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => [[1,4,7],[2,5,8],[3,6,9]]
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [9,7,8,4,5,6,1,2,3] => [1,3,2,6,5,4,9,8,7] => [[1,2,4,7],[3,5,8],[6,9]]
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [9,8,7,4,5,6,1,2,3] => [1,2,3,6,5,4,9,8,7] => [[1,2,3,4,7],[5,8],[6,9]]
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [8,9,6,7,4,5,1,2,3] => [2,1,4,3,6,5,9,8,7] => [[1,3,5,7],[2,4,6,8],[9]]
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [9,8,6,7,4,5,1,2,3] => [1,2,4,3,6,5,9,8,7] => [[1,2,3,5,7],[4,6,8],[9]]
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [9,8,7,6,4,5,1,2,3] => [1,2,3,4,6,5,9,8,7] => [[1,2,3,4,5,7],[6,8],[9]]
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,1,2,3] => [1,2,3,4,5,6,9,8,7] => [[1,2,3,4,5,6,7],[8],[9]]
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,3,4,1,2] => [1,3,2,5,4,7,6,9,8] => [[1,2,4,6,8],[3,5,7,9]]
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,3,4,1,2] => [1,2,3,5,4,7,6,9,8] => [[1,2,3,4,6,8],[5,7,9]]
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,3,4,1,2] => [1,2,3,4,5,7,6,9,8] => [[1,2,3,4,5,6,8],[7,9]]
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,9,8] => [[1,2,3,4,5,6,7,8],[9]]
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9]]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => [1,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => [2,1,10,9,8,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => [1,2,10,9,8,7,6,5,4,3] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [8,9,10,1,2,3,4,5,6,7] => [3,2,1,10,9,8,7,6,5,4] => [[1,4],[2,5],[3,6],[7],[8],[9],[10]]
>>> Load all 143 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
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
Robinson-Schensted recording tableau
Description
Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.
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