Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Images
[1] => [[1]] => [1] => [1] =>
[2] => [[1,2]] => [1,2] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 00
[2,1] => [[1,2],[3]] => [3,1,2] => [1,3,2] => 01
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 11
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 000
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [1,2,4,3] => 001
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [1,3,4,2] => 010
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [1,4,3,2] => 011
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 111
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0000
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,5,4] => 0001
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,4,5,3] => 0010
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 0011
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,5,4,2] => 0101
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,5,4,3,2] => 0111
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 1111
[6] => [[1,2,3,4,5,6]] => [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] => [1,2,3,4,6,5] => 00001
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,3,5,6,4] => 00010
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 00011
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,4,5,6,3] => 00100
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,2,4,6,5,3] => 00101
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 00111
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,3,5,6,4,2] => 01010
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,3,6,5,4,2] => 01011
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 01111
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 11111
[7] => [[1,2,3,4,5,6,7]] => [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] => [1,2,3,4,5,7,6] => 000001
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [1,2,3,4,6,7,5] => 000010
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 000011
[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => [1,2,3,5,6,7,4] => 000100
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => [1,2,3,5,7,6,4] => 000101
[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] => 000111
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [1,2,4,5,7,6,3] => 001001
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => [1,2,4,6,7,5,3] => 001010
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => 001011
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 001111
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [1,3,5,7,6,4,2] => 010101
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [1,3,7,6,5,4,2] => 010111
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 011111
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [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] => [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] => [1,2,3,4,5,6,8,7] => 0000001
[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => [1,2,3,4,5,7,8,6] => 0000010
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => 0000011
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => [1,2,3,4,6,7,8,5] => 0000100
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [8,6,7,1,2,3,4,5] => [1,2,3,4,6,8,7,5] => 0000101
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => 0000111
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => 0001000
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => [1,2,3,5,6,8,7,4] => 0001001
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => [1,2,3,5,7,8,6,4] => 0001010
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [8,7,5,6,1,2,3,4] => [1,2,3,5,8,7,6,4] => 0001011
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => 0001111
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [7,8,4,5,6,1,2,3] => [1,2,4,5,7,8,6,3] => 0010010
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => [1,2,4,5,8,7,6,3] => 0010011
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [8,6,7,4,5,1,2,3] => [1,2,4,6,8,7,5,3] => 0010101
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [8,7,6,4,5,1,2,3] => [1,2,4,8,7,6,5,3] => 0010111
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => 0011111
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => 0101010
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => [1,3,5,8,7,6,4,2] => 0101011
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => [1,3,8,7,6,5,4,2] => 0101111
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => 0111111
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [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] => [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] => [1,2,3,4,5,6,7,9,8] => 00000001
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,9,7] => 00000010
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => 00000011
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [7,8,9,1,2,3,4,5,6] => [1,2,3,4,5,7,8,9,6] => 00000100
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [9,7,8,1,2,3,4,5,6] => [1,2,3,4,5,7,9,8,6] => 00000101
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => 00000111
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [6,7,8,9,1,2,3,4,5] => [1,2,3,4,6,7,8,9,5] => 00001000
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [9,6,7,8,1,2,3,4,5] => [1,2,3,4,6,7,9,8,5] => 00001001
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [8,9,6,7,1,2,3,4,5] => [1,2,3,4,6,8,9,7,5] => 00001010
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [9,8,6,7,1,2,3,4,5] => [1,2,3,4,6,9,8,7,5] => 00001011
[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] => 00001111
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [9,5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,9,8,4] => 00010001
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [8,9,5,6,7,1,2,3,4] => [1,2,3,5,6,8,9,7,4] => 00010010
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [9,8,5,6,7,1,2,3,4] => [1,2,3,5,6,9,8,7,4] => 00010011
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,1,2,3,4] => [1,2,3,5,7,9,8,6,4] => 00010101
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,1,2,3,4] => [1,2,3,5,9,8,7,6,4] => 00010111
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => 00011111
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => 00100100
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [9,7,8,4,5,6,1,2,3] => [1,2,4,5,7,9,8,6,3] => 00100101
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [9,8,7,4,5,6,1,2,3] => [1,2,4,5,9,8,7,6,3] => 00100111
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [8,9,6,7,4,5,1,2,3] => [1,2,4,6,8,9,7,5,3] => 00101010
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [9,8,6,7,4,5,1,2,3] => [1,2,4,6,9,8,7,5,3] => 00101011
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [9,8,7,6,4,5,1,2,3] => [1,2,4,9,8,7,6,5,3] => 00101111
[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] => [1,2,9,8,7,6,5,4,3] => 00111111
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,3,4,1,2] => [1,3,5,7,9,8,6,4,2] => 01010101
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,3,4,1,2] => [1,3,5,9,8,7,6,4,2] => 01010111
[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] => [1,3,9,8,7,6,5,4,2] => 01011111
[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] => [1,9,8,7,6,5,4,3,2] => 01111111
[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] => [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] => [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] => [1,2,3,4,5,6,7,8,10,9] => 000000001
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,10,8] => 000000010
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,10,9,8] => 000000011
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [8,9,10,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,9,10,7] => 000000100
>>> Load all 142 entries. <<<
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,10,9,7] => 000000101
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [10,9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,9,8,7] => 000000111
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [7,8,9,10,1,2,3,4,5,6] => [1,2,3,4,5,7,8,9,10,6] => 000001000
[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [10,7,8,9,1,2,3,4,5,6] => [1,2,3,4,5,7,8,10,9,6] => 000001001
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [9,10,7,8,1,2,3,4,5,6] => [1,2,3,4,5,7,9,10,8,6] => 000001010
[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [10,9,7,8,1,2,3,4,5,6] => [1,2,3,4,5,7,10,9,8,6] => 000001011
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [10,9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,10,9,8,7,6] => 000001111
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,6,7,8,9,10,5] => 000010000
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [10,6,7,8,9,1,2,3,4,5] => [1,2,3,4,6,7,8,10,9,5] => 000010001
[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [9,10,6,7,8,1,2,3,4,5] => [1,2,3,4,6,7,9,10,8,5] => 000010010
[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [10,9,6,7,8,1,2,3,4,5] => [1,2,3,4,6,7,10,9,8,5] => 000010011
[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [10,8,9,6,7,1,2,3,4,5] => [1,2,3,4,6,8,10,9,7,5] => 000010101
[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => [10,9,8,6,7,1,2,3,4,5] => [1,2,3,4,6,10,9,8,7,5] => 000010111
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,1,2,3,4,5] => [1,2,3,4,10,9,8,7,6,5] => 000011111
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [9,10,5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,9,10,8,4] => 000100010
[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [10,9,5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,10,9,8,4] => 000100011
[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [8,9,10,5,6,7,1,2,3,4] => [1,2,3,5,6,8,9,10,7,4] => 000100100
[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [10,8,9,5,6,7,1,2,3,4] => [1,2,3,5,6,8,10,9,7,4] => 000100101
[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => [10,9,8,5,6,7,1,2,3,4] => [1,2,3,5,6,10,9,8,7,4] => 000100111
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,1,2,3,4] => [1,2,3,5,7,9,10,8,6,4] => 000101010
[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => [10,9,7,8,5,6,1,2,3,4] => [1,2,3,5,7,10,9,8,6,4] => 000101011
[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => [10,9,8,7,5,6,1,2,3,4] => [1,2,3,5,10,9,8,7,6,4] => 000101111
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,1,2,3,4] => [1,2,3,10,9,8,7,6,5,4] => 000111111
[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [10,7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,10,9,6,3] => 001001001
[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [9,10,7,8,4,5,6,1,2,3] => [1,2,4,5,7,9,10,8,6,3] => 001001010
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [10,9,7,8,4,5,6,1,2,3] => [1,2,4,5,7,10,9,8,6,3] => 001001011
[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => [10,9,8,7,4,5,6,1,2,3] => [1,2,4,5,10,9,8,7,6,3] => 001001111
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [10,8,9,6,7,4,5,1,2,3] => [1,2,4,6,8,10,9,7,5,3] => 001010101
[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => [10,9,8,6,7,4,5,1,2,3] => [1,2,4,6,10,9,8,7,5,3] => 001010111
[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,4,5,1,2,3] => [1,2,4,10,9,8,7,6,5,3] => 001011111
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,1,2,3] => [1,2,10,9,8,7,6,5,4,3] => 001111111
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [1,3,5,7,9,10,8,6,4,2] => 010101010
[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [10,9,7,8,5,6,3,4,1,2] => [1,3,5,7,10,9,8,6,4,2] => 010101011
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [10,9,8,7,5,6,3,4,1,2] => [1,3,5,10,9,8,7,6,4,2] => 010101111
[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,3,4,1,2] => [1,3,10,9,8,7,6,5,4,2] => 010111111
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,1,2] => [1,10,9,8,7,6,5,4,3,2] => 011111111
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 111111111
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => [1,2,3,4,5,6,7,8,9,10,11,12] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [7,8,9,10,11,12,1,2,3,4,5,6] => [1,2,3,4,5,7,8,9,10,11,12,6] => 00000100000
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 11111111111
[] => [] => [] => [] =>
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
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
descent bottoms
Description
The descent bottoms of a permutation as a binary word.