Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
[1] => [[1]] => [1] =>
[2] => [[1,2]] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => 00
[2,1] => [[1,2],[3]] => [3,1,2] => 10
[1,1,1] => [[1],[2],[3]] => [3,2,1] => 11
[4] => [[1,2,3,4]] => [1,2,3,4] => 000
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => 100
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => 010
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 110
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 111
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0000
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 1000
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0100
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1100
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1010
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 1110
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 1111
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 00000
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 10000
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 01000
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 11000
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 00100
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 10100
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 11100
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 01010
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => 11010
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => 11110
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 11111
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 000000
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => 100000
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => 010000
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => 110000
[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => 001000
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => 101000
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => 111000
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => 100100
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => 010100
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => 110100
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => 111100
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => 101010
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => 111010
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => 111110
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 111111
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => 0000000
[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => 1000000
[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => 0100000
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => 1100000
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => 0010000
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [8,6,7,1,2,3,4,5] => 1010000
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => 1110000
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => 0001000
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => 1001000
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => 0101000
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [8,7,5,6,1,2,3,4] => 1101000
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => 1111000
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [7,8,4,5,6,1,2,3] => 0100100
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => 1100100
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [8,6,7,4,5,1,2,3] => 1010100
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [8,7,6,4,5,1,2,3] => 1110100
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => 1111100
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => 0101010
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => 1101010
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => 1111010
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => 1111110
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => 1111111
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => 00000000
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => 10000000
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [8,9,1,2,3,4,5,6,7] => 01000000
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => 11000000
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [7,8,9,1,2,3,4,5,6] => 00100000
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [9,7,8,1,2,3,4,5,6] => 10100000
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => 11100000
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [6,7,8,9,1,2,3,4,5] => 00010000
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [9,6,7,8,1,2,3,4,5] => 10010000
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [8,9,6,7,1,2,3,4,5] => 01010000
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [9,8,6,7,1,2,3,4,5] => 11010000
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [9,8,7,6,1,2,3,4,5] => 11110000
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [9,5,6,7,8,1,2,3,4] => 10001000
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [8,9,5,6,7,1,2,3,4] => 01001000
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [9,8,5,6,7,1,2,3,4] => 11001000
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,1,2,3,4] => 10101000
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,1,2,3,4] => 11101000
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,1,2,3,4] => 11111000
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => 00100100
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [9,7,8,4,5,6,1,2,3] => 10100100
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [9,8,7,4,5,6,1,2,3] => 11100100
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [8,9,6,7,4,5,1,2,3] => 01010100
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [9,8,6,7,4,5,1,2,3] => 11010100
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [9,8,7,6,4,5,1,2,3] => 11110100
[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] => 11111100
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,3,4,1,2] => 10101010
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,3,4,1,2] => 11101010
[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] => 11111010
[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] => 11111110
[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] => 11111111
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => 000000000
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => 100000000
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => 010000000
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => 110000000
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [8,9,10,1,2,3,4,5,6,7] => 001000000
>>> Load all 145 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
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
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