Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Images
[1] => [[1]] => [1]
[2] => [[1,2]] => [2]
[1,1] => [[1],[2]] => [1,1]
[3] => [[1,2,3]] => [3]
[2,1] => [[1,2],[3]] => [2,1]
[1,1,1] => [[1],[2],[3]] => [1,1,1]
[4] => [[1,2,3,4]] => [4]
[3,1] => [[1,2,3],[4]] => [3,1]
[2,2] => [[1,2],[3,4]] => [2,2]
[2,1,1] => [[1,2],[3],[4]] => [2,1,1]
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1]
[5] => [[1,2,3,4,5]] => [5]
[4,1] => [[1,2,3,4],[5]] => [4,1]
[3,2] => [[1,2,3],[4,5]] => [3,2]
[3,1,1] => [[1,2,3],[4],[5]] => [3,1,1]
[2,2,1] => [[1,2],[3,4],[5]] => [2,2,1]
[2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1]
[6] => [[1,2,3,4,5,6]] => [6]
[5,1] => [[1,2,3,4,5],[6]] => [5,1]
[4,2] => [[1,2,3,4],[5,6]] => [4,2]
[4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1]
[3,3] => [[1,2,3],[4,5,6]] => [3,3]
[3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1]
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1]
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2]
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1]
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1]
[7] => [[1,2,3,4,5,6,7]] => [7]
[6,1] => [[1,2,3,4,5,6],[7]] => [6,1]
[5,2] => [[1,2,3,4,5],[6,7]] => [5,2]
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [5,1,1]
[4,3] => [[1,2,3,4],[5,6,7]] => [4,3]
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [4,2,1]
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [4,1,1,1]
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [3,3,1]
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2]
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [3,2,1,1]
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [3,1,1,1,1]
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1]
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1]
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1]
[8] => [[1,2,3,4,5,6,7,8]] => [8]
[7,1] => [[1,2,3,4,5,6,7],[8]] => [7,1]
[6,2] => [[1,2,3,4,5,6],[7,8]] => [6,2]
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [6,1,1]
[5,3] => [[1,2,3,4,5],[6,7,8]] => [5,3]
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [5,2,1]
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [5,1,1,1]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4]
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [4,3,1]
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [4,2,2]
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [4,2,1,1]
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [4,1,1,1,1]
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [3,3,2]
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [3,3,1,1]
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [3,2,2,1]
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [3,2,1,1,1]
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [3,1,1,1,1,1]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2]
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [2,2,2,1,1]
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [2,2,1,1,1,1]
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [2,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,1]
[9] => [[1,2,3,4,5,6,7,8,9]] => [9]
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [8,1]
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [7,2]
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [7,1,1]
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [6,3]
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [6,2,1]
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [6,1,1,1]
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [5,4]
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [5,3,1]
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [5,2,2]
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [5,2,1,1]
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [5,1,1,1,1]
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [4,4,1]
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [4,3,2]
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [4,3,1,1]
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [4,2,2,1]
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [4,2,1,1,1]
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [4,1,1,1,1,1]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3]
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [3,3,2,1]
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [3,3,1,1,1]
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [3,2,2,2]
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [3,2,2,1,1]
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [3,2,1,1,1,1]
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [3,1,1,1,1,1,1]
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [2,2,2,2,1]
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [2,2,2,1,1,1]
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [2,2,1,1,1,1,1]
[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]
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [1,1,1,1,1,1,1,1,1]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10]
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [9,1]
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [8,2]
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [8,1,1]
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [7,3]
>>> Load all 201 entries. <<<
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [7,2,1]
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [7,1,1,1]
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [6,4]
[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [6,3,1]
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [6,2,2]
[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [6,2,1,1]
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [6,1,1,1,1]
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [5,5]
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [5,4,1]
[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [5,3,2]
[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [5,3,1,1]
[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [5,2,2,1]
[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => [5,2,1,1,1]
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [5,1,1,1,1,1]
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [4,4,2]
[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [4,4,1,1]
[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [4,3,3]
[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [4,3,2,1]
[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => [4,3,1,1,1]
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [4,2,2,2]
[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => [4,2,2,1,1]
[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => [4,2,1,1,1,1]
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [4,1,1,1,1,1,1]
[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [3,3,3,1]
[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [3,3,2,2]
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [3,3,2,1,1]
[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => [3,3,1,1,1,1]
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [3,2,2,2,1]
[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => [3,2,2,1,1,1]
[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => [3,2,1,1,1,1,1]
[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]
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [2,2,2,2,2]
[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [2,2,2,2,1,1]
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [2,2,2,1,1,1,1]
[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]
[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]
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [1,1,1,1,1,1,1,1,1,1]
[5,4,2] => [[1,2,3,4,5],[6,7,8,9],[10,11]] => [5,4,2]
[5,4,1,1] => [[1,2,3,4,5],[6,7,8,9],[10],[11]] => [5,4,1,1]
[5,3,3] => [[1,2,3,4,5],[6,7,8],[9,10,11]] => [5,3,3]
[5,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11]] => [5,3,2,1]
[5,3,1,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10],[11]] => [5,3,1,1,1]
[5,2,2,2] => [[1,2,3,4,5],[6,7],[8,9],[10,11]] => [5,2,2,2]
[5,2,2,1,1] => [[1,2,3,4,5],[6,7],[8,9],[10],[11]] => [5,2,2,1,1]
[4,4,3] => [[1,2,3,4],[5,6,7,8],[9,10,11]] => [4,4,3]
[4,4,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11]] => [4,4,2,1]
[4,4,1,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10],[11]] => [4,4,1,1,1]
[4,3,3,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11]] => [4,3,3,1]
[4,3,2,2] => [[1,2,3,4],[5,6,7],[8,9],[10,11]] => [4,3,2,2]
[4,3,2,1,1] => [[1,2,3,4],[5,6,7],[8,9],[10],[11]] => [4,3,2,1,1]
[4,2,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9,10],[11]] => [4,2,2,2,1]
[3,3,3,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11]] => [3,3,3,2]
[3,3,3,1,1] => [[1,2,3],[4,5,6],[7,8,9],[10],[11]] => [3,3,3,1,1]
[3,3,2,2,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11]] => [3,3,2,2,1]
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => [12]
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [6,6]
[6,4,2] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => [6,4,2]
[5,4,3] => [[1,2,3,4,5],[6,7,8,9],[10,11,12]] => [5,4,3]
[5,4,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]] => [5,4,2,1]
[5,4,1,1,1] => [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]] => [5,4,1,1,1]
[5,3,3,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]] => [5,3,3,1]
[5,3,2,2] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]] => [5,3,2,2]
[5,3,2,1,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]] => [5,3,2,1,1]
[5,2,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]] => [5,2,2,2,1]
[4,4,3,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]] => [4,4,3,1]
[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => [4,4,2,2]
[4,4,2,1,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]] => [4,4,2,1,1]
[4,3,3,2] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]] => [4,3,3,2]
[4,3,3,1,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]] => [4,3,3,1,1]
[4,3,2,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]] => [4,3,2,2,1]
[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => [4,2,2,2,2]
[3,3,3,2,1] => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]] => [3,3,3,2,1]
[3,3,2,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]] => [3,3,2,2,1,1]
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [2,2,2,2,2,2]
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [1,1,1,1,1,1,1,1,1,1,1,1]
[5,4,3,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]] => [5,4,3,1]
[5,4,2,2] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]] => [5,4,2,2]
[5,4,2,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]] => [5,4,2,1,1]
[5,3,3,2] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]] => [5,3,3,2]
[5,3,3,1,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]] => [5,3,3,1,1]
[5,3,2,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]] => [5,3,2,2,1]
[4,4,3,2] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]] => [4,4,3,2]
[4,4,3,1,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]] => [4,4,3,1,1]
[4,4,2,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]] => [4,4,2,2,1]
[4,3,3,2,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]] => [4,3,3,2,1]
[7,7] => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]] => [7,7]
[5,4,3,2] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]] => [5,4,3,2]
[5,4,3,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]] => [5,4,3,1,1]
[5,4,2,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]] => [5,4,2,2,1]
[5,3,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]] => [5,3,3,2,1]
[4,4,3,2,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]] => [4,4,3,2,1]
[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]
[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]
[8,8] => [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]] => [8,8]
[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]
[] => [] => [0]
[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]
[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]
[10,10] => [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]] => [10,10]
[9,9] => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]] => [9,9]
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 $i$.