- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions

Images

[1] => [[1]] => [1] => [1]

[2] => [[1,2]] => [2] => [1,1]

[1,1] => [[1],[2]] => [1,1] => [2]

[3] => [[1,2,3]] => [3] => [1,1,1]

[2,1] => [[1,2],[3]] => [2,1] => [1,2]

[1,1,1] => [[1],[2],[3]] => [1,1,1] => [3]

[4] => [[1,2,3,4]] => [4] => [1,1,1,1]

[3,1] => [[1,2,3],[4]] => [3,1] => [1,1,2]

[2,2] => [[1,2],[3,4]] => [2,2] => [1,2,1]

[2,1,1] => [[1,2],[3],[4]] => [2,1,1] => [1,3]

[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => [4]

[5] => [[1,2,3,4,5]] => [5] => [1,1,1,1,1]

[4,1] => [[1,2,3,4],[5]] => [4,1] => [1,1,1,2]

[3,2] => [[1,2,3],[4,5]] => [3,2] => [1,1,2,1]

[3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => [1,1,3]

[2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => [1,2,2]

[2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => [1,4]

[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [5]

[6] => [[1,2,3,4,5,6]] => [6] => [1,1,1,1,1,1]

[5,1] => [[1,2,3,4,5],[6]] => [5,1] => [1,1,1,1,2]

[4,2] => [[1,2,3,4],[5,6]] => [4,2] => [1,1,1,2,1]

[4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => [1,1,1,3]

[3,3] => [[1,2,3],[4,5,6]] => [3,3] => [1,1,2,1,1]

[3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => [1,1,2,2]

[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => [1,1,4]

[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => [1,2,2,1]

[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => [1,2,3]

[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => [1,5]

[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => [6]

[7] => [[1,2,3,4,5,6,7]] => [7] => [1,1,1,1,1,1,1]

[6,1] => [[1,2,3,4,5,6],[7]] => [6,1] => [1,1,1,1,1,2]

[5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => [1,1,1,1,2,1]

[5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,1,1] => [1,1,1,1,3]

[4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => [1,1,1,2,1,1]

[4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1] => [1,1,1,2,2]

[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1] => [1,1,1,4]

[3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1] => [1,1,2,1,2]

[3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => [1,1,2,2,1]

[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1] => [1,1,2,3]

[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1] => [1,1,5]

[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => [1,2,2,2]

[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => [1,2,4]

[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => [1,6]

[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => [7]

[8] => [[1,2,3,4,5,6,7,8]] => [8] => [1,1,1,1,1,1,1,1]

[7,1] => [[1,2,3,4,5,6,7],[8]] => [7,1] => [1,1,1,1,1,1,2]

[6,2] => [[1,2,3,4,5,6],[7,8]] => [6,2] => [1,1,1,1,1,2,1]

[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [6,1,1] => [1,1,1,1,1,3]

[5,3] => [[1,2,3,4,5],[6,7,8]] => [5,3] => [1,1,1,1,2,1,1]

[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [5,2,1] => [1,1,1,1,2,2]

[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [5,1,1,1] => [1,1,1,1,4]

[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [1,1,1,2,1,1,1]

[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [4,3,1] => [1,1,1,2,1,2]

[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [4,2,2] => [1,1,1,2,2,1]

[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [4,2,1,1] => [1,1,1,2,3]

[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [4,1,1,1,1] => [1,1,1,5]

[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [3,3,2] => [1,1,2,1,2,1]

[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [3,3,1,1] => [1,1,2,1,3]

[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [3,2,2,1] => [1,1,2,2,2]

[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [3,2,1,1,1] => [1,1,2,4]

[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [3,1,1,1,1,1] => [1,1,6]

[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [1,2,2,2,1]

[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [2,2,2,1,1] => [1,2,2,3]

[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [2,2,1,1,1,1] => [1,2,5]

[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [2,1,1,1,1,1,1] => [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]

[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => [1,1,1,1,1,1,1,1,1]

[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [8,1] => [1,1,1,1,1,1,1,2]

[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [7,2] => [1,1,1,1,1,1,2,1]

[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [7,1,1] => [1,1,1,1,1,1,3]

[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [6,3] => [1,1,1,1,1,2,1,1]

[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [6,2,1] => [1,1,1,1,1,2,2]

[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [6,1,1,1] => [1,1,1,1,1,4]

[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [5,4] => [1,1,1,1,2,1,1,1]

[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [5,3,1] => [1,1,1,1,2,1,2]

[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [5,2,2] => [1,1,1,1,2,2,1]

[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [5,2,1,1] => [1,1,1,1,2,3]

[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [5,1,1,1,1] => [1,1,1,1,5]

[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [4,4,1] => [1,1,1,2,1,1,2]

[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [4,3,2] => [1,1,1,2,1,2,1]

[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [4,3,1,1] => [1,1,1,2,1,3]

[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [4,2,2,1] => [1,1,1,2,2,2]

[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [4,2,1,1,1] => [1,1,1,2,4]

[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [4,1,1,1,1,1] => [1,1,1,6]

[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => [1,1,2,1,2,1,1]

[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [3,3,2,1] => [1,1,2,1,2,2]

[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [3,3,1,1,1] => [1,1,2,1,4]

[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [3,2,2,2] => [1,1,2,2,2,1]

[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [3,2,2,1,1] => [1,1,2,2,3]

[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [3,2,1,1,1,1] => [1,1,2,5]

[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [3,1,1,1,1,1,1] => [1,1,7]

[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [2,2,2,2,1] => [1,2,2,2,2]

[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [2,2,2,1,1,1] => [1,2,2,4]

[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [2,2,1,1,1,1,1] => [1,2,6]

[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] => [1,8]

[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] => [9]

[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10] => [1,1,1,1,1,1,1,1,1,1]

[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [9,1] => [1,1,1,1,1,1,1,1,2]

[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [8,2] => [1,1,1,1,1,1,1,2,1]

[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [8,1,1] => [1,1,1,1,1,1,1,3]

[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [7,3] => [1,1,1,1,1,1,2,1,1]

>>> Load all 197 entries. <<<

[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [7,2,1] => [1,1,1,1,1,1,2,2]

[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [7,1,1,1] => [1,1,1,1,1,1,4]

[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [6,4] => [1,1,1,1,1,2,1,1,1]

[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [6,3,1] => [1,1,1,1,1,2,1,2]

[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [6,2,2] => [1,1,1,1,1,2,2,1]

[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [6,2,1,1] => [1,1,1,1,1,2,3]

[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [6,1,1,1,1] => [1,1,1,1,1,5]

[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [5,5] => [1,1,1,1,2,1,1,1,1]

[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [5,4,1] => [1,1,1,1,2,1,1,2]

[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [5,3,2] => [1,1,1,1,2,1,2,1]

[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [5,3,1,1] => [1,1,1,1,2,1,3]

[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [5,2,2,1] => [1,1,1,1,2,2,2]

[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => [5,2,1,1,1] => [1,1,1,1,2,4]

[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [5,1,1,1,1,1] => [1,1,1,1,6]

[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [4,4,2] => [1,1,1,2,1,1,2,1]

[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [4,4,1,1] => [1,1,1,2,1,1,3]

[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [4,3,3] => [1,1,1,2,1,2,1,1]

[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [4,3,2,1] => [1,1,1,2,1,2,2]

[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => [4,3,1,1,1] => [1,1,1,2,1,4]

[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [4,2,2,2] => [1,1,1,2,2,2,1]

[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => [4,2,2,1,1] => [1,1,1,2,2,3]

[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => [4,2,1,1,1,1] => [1,1,1,2,5]

[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [4,1,1,1,1,1,1] => [1,1,1,7]

[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [3,3,3,1] => [1,1,2,1,2,1,2]

[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [3,3,2,2] => [1,1,2,1,2,2,1]

[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [3,3,2,1,1] => [1,1,2,1,2,3]

[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => [3,3,1,1,1,1] => [1,1,2,1,5]

[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [3,2,2,2,1] => [1,1,2,2,2,2]

[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => [3,2,2,1,1,1] => [1,1,2,2,4]

[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => [3,2,1,1,1,1,1] => [1,1,2,6]

[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] => [1,1,8]

[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [2,2,2,2,2] => [1,2,2,2,2,1]

[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [2,2,2,2,1,1] => [1,2,2,2,3]

[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [2,2,2,1,1,1,1] => [1,2,2,5]

[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => [2,2,1,1,1,1,1,1] => [1,2,7]

[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] => [1,9]

[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [1,1,1,1,1,1,1,1,1,1] => [10]

[5,4,2] => [[1,2,3,4,5],[6,7,8,9],[10,11]] => [5,4,2] => [1,1,1,1,2,1,1,2,1]

[5,4,1,1] => [[1,2,3,4,5],[6,7,8,9],[10],[11]] => [5,4,1,1] => [1,1,1,1,2,1,1,3]

[5,3,3] => [[1,2,3,4,5],[6,7,8],[9,10,11]] => [5,3,3] => [1,1,1,1,2,1,2,1,1]

[5,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11]] => [5,3,2,1] => [1,1,1,1,2,1,2,2]

[5,3,1,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10],[11]] => [5,3,1,1,1] => [1,1,1,1,2,1,4]

[5,2,2,2] => [[1,2,3,4,5],[6,7],[8,9],[10,11]] => [5,2,2,2] => [1,1,1,1,2,2,2,1]

[5,2,2,1,1] => [[1,2,3,4,5],[6,7],[8,9],[10],[11]] => [5,2,2,1,1] => [1,1,1,1,2,2,3]

[4,4,3] => [[1,2,3,4],[5,6,7,8],[9,10,11]] => [4,4,3] => [1,1,1,2,1,1,2,1,1]

[4,4,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11]] => [4,4,2,1] => [1,1,1,2,1,1,2,2]

[4,4,1,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10],[11]] => [4,4,1,1,1] => [1,1,1,2,1,1,4]

[4,3,3,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11]] => [4,3,3,1] => [1,1,1,2,1,2,1,2]

[4,3,2,2] => [[1,2,3,4],[5,6,7],[8,9],[10,11]] => [4,3,2,2] => [1,1,1,2,1,2,2,1]

[4,3,2,1,1] => [[1,2,3,4],[5,6,7],[8,9],[10],[11]] => [4,3,2,1,1] => [1,1,1,2,1,2,3]

[4,2,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9,10],[11]] => [4,2,2,2,1] => [1,1,1,2,2,2,2]

[3,3,3,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11]] => [3,3,3,2] => [1,1,2,1,2,1,2,1]

[3,3,3,1,1] => [[1,2,3],[4,5,6],[7,8,9],[10],[11]] => [3,3,3,1,1] => [1,1,2,1,2,1,3]

[3,3,2,2,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11]] => [3,3,2,2,1] => [1,1,2,1,2,2,2]

[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]

[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [6,6] => [1,1,1,1,1,2,1,1,1,1,1]

[6,4,2] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => [6,4,2] => [1,1,1,1,1,2,1,1,2,1]

[5,4,3] => [[1,2,3,4,5],[6,7,8,9],[10,11,12]] => [5,4,3] => [1,1,1,1,2,1,1,2,1,1]

[5,4,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]] => [5,4,2,1] => [1,1,1,1,2,1,1,2,2]

[5,4,1,1,1] => [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]] => [5,4,1,1,1] => [1,1,1,1,2,1,1,4]

[5,3,3,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]] => [5,3,3,1] => [1,1,1,1,2,1,2,1,2]

[5,3,2,2] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]] => [5,3,2,2] => [1,1,1,1,2,1,2,2,1]

[5,3,2,1,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]] => [5,3,2,1,1] => [1,1,1,1,2,1,2,3]

[5,2,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]] => [5,2,2,2,1] => [1,1,1,1,2,2,2,2]

[4,4,3,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]] => [4,4,3,1] => [1,1,1,2,1,1,2,1,2]

[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => [4,4,2,2] => [1,1,1,2,1,1,2,2,1]

[4,4,2,1,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]] => [4,4,2,1,1] => [1,1,1,2,1,1,2,3]

[4,3,3,2] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]] => [4,3,3,2] => [1,1,1,2,1,2,1,2,1]

[4,3,3,1,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]] => [4,3,3,1,1] => [1,1,1,2,1,2,1,3]

[4,3,2,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]] => [4,3,2,2,1] => [1,1,1,2,1,2,2,2]

[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => [4,2,2,2,2] => [1,1,1,2,2,2,2,1]

[3,3,3,2,1] => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]] => [3,3,3,2,1] => [1,1,2,1,2,1,2,2]

[3,3,2,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]] => [3,3,2,2,1,1] => [1,1,2,1,2,2,3]

[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [2,2,2,2,2,2] => [1,2,2,2,2,2,1]

[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [1,1,1,1,1,1,1,1,1,1,1,1] => [12]

[5,4,3,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]] => [5,4,3,1] => [1,1,1,1,2,1,1,2,1,2]

[5,4,2,2] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]] => [5,4,2,2] => [1,1,1,1,2,1,1,2,2,1]

[5,4,2,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]] => [5,4,2,1,1] => [1,1,1,1,2,1,1,2,3]

[5,3,3,2] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]] => [5,3,3,2] => [1,1,1,1,2,1,2,1,2,1]

[5,3,3,1,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]] => [5,3,3,1,1] => [1,1,1,1,2,1,2,1,3]

[5,3,2,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]] => [5,3,2,2,1] => [1,1,1,1,2,1,2,2,2]

[4,4,3,2] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]] => [4,4,3,2] => [1,1,1,2,1,1,2,1,2,1]

[4,4,3,1,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]] => [4,4,3,1,1] => [1,1,1,2,1,1,2,1,3]

[4,4,2,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]] => [4,4,2,2,1] => [1,1,1,2,1,1,2,2,2]

[4,3,3,2,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]] => [4,3,3,2,1] => [1,1,1,2,1,2,1,2,2]

[7,7] => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]] => [7,7] => [1,1,1,1,1,1,2,1,1,1,1,1,1]

[5,4,3,2] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]] => [5,4,3,2] => [1,1,1,1,2,1,1,2,1,2,1]

[5,4,3,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]] => [5,4,3,1,1] => [1,1,1,1,2,1,1,2,1,3]

[5,4,2,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]] => [5,4,2,2,1] => [1,1,1,1,2,1,1,2,2,2]

[5,3,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]] => [5,3,3,2,1] => [1,1,1,1,2,1,2,1,2,2]

[4,4,3,2,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]] => [4,4,3,2,1] => [1,1,1,2,1,1,2,1,2,2]

[2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]] => [2,2,2,2,2,2,2] => [1,2,2,2,2,2,2,1]

[5,4,3,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]] => [5,4,3,2,1] => [1,1,1,1,2,1,1,2,1,2,2]

[2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]] => [2,2,2,2,2,2,2,2] => [1,2,2,2,2,2,2,2,1]

[2,2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]] => [2,2,2,2,2,2,2,2,2] => [1,2,2,2,2,2,2,2,2,1]

[2,2,2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20]] => [2,2,2,2,2,2,2,2,2,2] => [1,2,2,2,2,2,2,2,2,2,1]

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.

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

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .