Identifier
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Images
[1] => [[1],[]] => [[1],[]]
[1,1] => [[1,1],[]] => [[2],[]]
[2] => [[2],[]] => [[1,1],[]]
[1,1,1] => [[1,1,1],[]] => [[3],[]]
[1,2] => [[2,1],[]] => [[2,1],[]]
[2,1] => [[2,2],[1]] => [[2,2],[1]]
[3] => [[3],[]] => [[1,1,1],[]]
[1,1,1,1] => [[1,1,1,1],[]] => [[4],[]]
[1,1,2] => [[2,1,1],[]] => [[3,1],[]]
[1,2,1] => [[2,2,1],[1]] => [[3,2],[1]]
[1,3] => [[3,1],[]] => [[2,1,1],[]]
[2,1,1] => [[2,2,2],[1,1]] => [[3,3],[2]]
[2,2] => [[3,2],[1]] => [[2,2,1],[1]]
[3,1] => [[3,3],[2]] => [[2,2,2],[1,1]]
[4] => [[4],[]] => [[1,1,1,1],[]]
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [[5],[]]
[1,1,1,2] => [[2,1,1,1],[]] => [[4,1],[]]
[1,1,2,1] => [[2,2,1,1],[1]] => [[4,2],[1]]
[1,1,3] => [[3,1,1],[]] => [[3,1,1],[]]
[1,2,1,1] => [[2,2,2,1],[1,1]] => [[4,3],[2]]
[1,2,2] => [[3,2,1],[1]] => [[3,2,1],[1]]
[1,3,1] => [[3,3,1],[2]] => [[3,2,2],[1,1]]
[1,4] => [[4,1],[]] => [[2,1,1,1],[]]
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [[4,4],[3]]
[2,1,2] => [[3,2,2],[1,1]] => [[3,3,1],[2]]
[2,2,1] => [[3,3,2],[2,1]] => [[3,3,2],[2,1]]
[2,3] => [[4,2],[1]] => [[2,2,1,1],[1]]
[3,1,1] => [[3,3,3],[2,2]] => [[3,3,3],[2,2]]
[3,2] => [[4,3],[2]] => [[2,2,2,1],[1,1]]
[4,1] => [[4,4],[3]] => [[2,2,2,2],[1,1,1]]
[5] => [[5],[]] => [[1,1,1,1,1],[]]
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [[6],[]]
[1,1,1,1,2] => [[2,1,1,1,1],[]] => [[5,1],[]]
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [[5,2],[1]]
[1,1,1,3] => [[3,1,1,1],[]] => [[4,1,1],[]]
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [[5,3],[2]]
[1,1,2,2] => [[3,2,1,1],[1]] => [[4,2,1],[1]]
[1,1,3,1] => [[3,3,1,1],[2]] => [[4,2,2],[1,1]]
[1,1,4] => [[4,1,1],[]] => [[3,1,1,1],[]]
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [[5,4],[3]]
[1,2,1,2] => [[3,2,2,1],[1,1]] => [[4,3,1],[2]]
[1,2,2,1] => [[3,3,2,1],[2,1]] => [[4,3,2],[2,1]]
[1,2,3] => [[4,2,1],[1]] => [[3,2,1,1],[1]]
[1,3,1,1] => [[3,3,3,1],[2,2]] => [[4,3,3],[2,2]]
[1,3,2] => [[4,3,1],[2]] => [[3,2,2,1],[1,1]]
[1,4,1] => [[4,4,1],[3]] => [[3,2,2,2],[1,1,1]]
[1,5] => [[5,1],[]] => [[2,1,1,1,1],[]]
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [[5,5],[4]]
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [[4,4,1],[3]]
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [[4,4,2],[3,1]]
[2,1,3] => [[4,2,2],[1,1]] => [[3,3,1,1],[2]]
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [[4,4,3],[3,2]]
[2,2,2] => [[4,3,2],[2,1]] => [[3,3,2,1],[2,1]]
[2,3,1] => [[4,4,2],[3,1]] => [[3,3,2,2],[2,1,1]]
[2,4] => [[5,2],[1]] => [[2,2,1,1,1],[1]]
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [[4,4,4],[3,3]]
[3,1,2] => [[4,3,3],[2,2]] => [[3,3,3,1],[2,2]]
[3,2,1] => [[4,4,3],[3,2]] => [[3,3,3,2],[2,2,1]]
[3,3] => [[5,3],[2]] => [[2,2,2,1,1],[1,1]]
[4,1,1] => [[4,4,4],[3,3]] => [[3,3,3,3],[2,2,2]]
[4,2] => [[5,4],[3]] => [[2,2,2,2,1],[1,1,1]]
[5,1] => [[5,5],[4]] => [[2,2,2,2,2],[1,1,1,1]]
[6] => [[6],[]] => [[1,1,1,1,1,1],[]]
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [[7],[]]
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [[6,1],[]]
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [[6,2],[1]]
[1,1,1,1,3] => [[3,1,1,1,1],[]] => [[5,1,1],[]]
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [[6,3],[2]]
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [[5,2,1],[1]]
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [[5,2,2],[1,1]]
[1,1,1,4] => [[4,1,1,1],[]] => [[4,1,1,1],[]]
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [[6,4],[3]]
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [[5,3,1],[2]]
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [[5,3,2],[2,1]]
[1,1,2,3] => [[4,2,1,1],[1]] => [[4,2,1,1],[1]]
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [[5,3,3],[2,2]]
[1,1,3,2] => [[4,3,1,1],[2]] => [[4,2,2,1],[1,1]]
[1,1,4,1] => [[4,4,1,1],[3]] => [[4,2,2,2],[1,1,1]]
[1,1,5] => [[5,1,1],[]] => [[3,1,1,1,1],[]]
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [[6,5],[4]]
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [[5,4,1],[3]]
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [[5,4,2],[3,1]]
[1,2,1,3] => [[4,2,2,1],[1,1]] => [[4,3,1,1],[2]]
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [[5,4,3],[3,2]]
[1,2,2,2] => [[4,3,2,1],[2,1]] => [[4,3,2,1],[2,1]]
[1,2,3,1] => [[4,4,2,1],[3,1]] => [[4,3,2,2],[2,1,1]]
[1,2,4] => [[5,2,1],[1]] => [[3,2,1,1,1],[1]]
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [[5,4,4],[3,3]]
[1,3,1,2] => [[4,3,3,1],[2,2]] => [[4,3,3,1],[2,2]]
[1,3,2,1] => [[4,4,3,1],[3,2]] => [[4,3,3,2],[2,2,1]]
[1,3,3] => [[5,3,1],[2]] => [[3,2,2,1,1],[1,1]]
[1,4,1,1] => [[4,4,4,1],[3,3]] => [[4,3,3,3],[2,2,2]]
[1,4,2] => [[5,4,1],[3]] => [[3,2,2,2,1],[1,1,1]]
[1,5,1] => [[5,5,1],[4]] => [[3,2,2,2,2],[1,1,1,1]]
[1,6] => [[6,1],[]] => [[2,1,1,1,1,1],[]]
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [[6,6],[5]]
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [[5,5,1],[4]]
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [[5,5,2],[4,1]]
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [[4,4,1,1],[3]]
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [[5,5,3],[4,2]]
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [[4,4,2,1],[3,1]]
>>> Load all 216 entries. <<<
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [[4,4,2,2],[3,1,1]]
[2,1,4] => [[5,2,2],[1,1]] => [[3,3,1,1,1],[2]]
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [[5,5,4],[4,3]]
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [[4,4,3,1],[3,2]]
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [[4,4,3,2],[3,2,1]]
[2,2,3] => [[5,3,2],[2,1]] => [[3,3,2,1,1],[2,1]]
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [[4,4,3,3],[3,2,2]]
[2,3,2] => [[5,4,2],[3,1]] => [[3,3,2,2,1],[2,1,1]]
[2,4,1] => [[5,5,2],[4,1]] => [[3,3,2,2,2],[2,1,1,1]]
[2,5] => [[6,2],[1]] => [[2,2,1,1,1,1],[1]]
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [[5,5,5],[4,4]]
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [[4,4,4,1],[3,3]]
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [[4,4,4,2],[3,3,1]]
[3,1,3] => [[5,3,3],[2,2]] => [[3,3,3,1,1],[2,2]]
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [[4,4,4,3],[3,3,2]]
[3,2,2] => [[5,4,3],[3,2]] => [[3,3,3,2,1],[2,2,1]]
[3,3,1] => [[5,5,3],[4,2]] => [[3,3,3,2,2],[2,2,1,1]]
[3,4] => [[6,3],[2]] => [[2,2,2,1,1,1],[1,1]]
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [[4,4,4,4],[3,3,3]]
[4,1,2] => [[5,4,4],[3,3]] => [[3,3,3,3,1],[2,2,2]]
[4,2,1] => [[5,5,4],[4,3]] => [[3,3,3,3,2],[2,2,2,1]]
[4,3] => [[6,4],[3]] => [[2,2,2,2,1,1],[1,1,1]]
[5,1,1] => [[5,5,5],[4,4]] => [[3,3,3,3,3],[2,2,2,2]]
[5,2] => [[6,5],[4]] => [[2,2,2,2,2,1],[1,1,1,1]]
[6,1] => [[6,6],[5]] => [[2,2,2,2,2,2],[1,1,1,1,1]]
[7] => [[7],[]] => [[1,1,1,1,1,1,1],[]]
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => [[8],[]]
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => [[7,1],[]]
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => [[6,1,1],[]]
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [[7,3],[2]]
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => [[6,2,1],[1]]
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]] => [[6,2,2],[1,1]]
[1,1,1,1,4] => [[4,1,1,1,1],[]] => [[5,1,1,1],[]]
[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => [[6,3,1],[2]]
[1,1,1,5] => [[5,1,1,1],[]] => [[4,1,1,1,1],[]]
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => [[6,4,1],[3]]
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]] => [[5,3,1,1],[2]]
[1,1,2,4] => [[5,2,1,1],[1]] => [[4,2,1,1,1],[1]]
[1,1,6] => [[6,1,1],[]] => [[3,1,1,1,1,1],[]]
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [[6,5,1],[4]]
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]] => [[5,4,1,1],[3]]
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [[5,4,4,1],[3,3]]
[1,3,1,3] => [[5,3,3,1],[2,2]] => [[4,3,3,1,1],[2,2]]
[1,7] => [[7,1],[]] => [[2,1,1,1,1,1,1],[]]
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [[7,7],[6]]
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [[6,6,1],[5]]
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => [[6,6,2],[5,1]]
[2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]] => [[5,5,1,1],[4]]
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => [[6,6,3],[5,2]]
[2,1,1,4] => [[5,2,2,2],[1,1,1]] => [[4,4,1,1,1],[3]]
[2,1,5] => [[6,2,2],[1,1]] => [[3,3,1,1,1,1],[2]]
[2,6] => [[7,2],[1]] => [[2,2,1,1,1,1,1],[1]]
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [[6,6,6],[5,5]]
[3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]] => [[5,5,5,1],[4,4]]
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => [[4,4,4,1,1],[3,3]]
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]] => [[5,5,5,5],[4,4,4]]
[5,1,1,1] => [[5,5,5,5],[4,4,4]] => [[4,4,4,4,4],[3,3,3,3]]
[5,1,2] => [[6,5,5],[4,4]] => [[3,3,3,3,3,1],[2,2,2,2]]
[6,1,1] => [[6,6,6],[5,5]] => [[3,3,3,3,3,3],[2,2,2,2,2]]
[6,2] => [[7,6],[5]] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
[7,1] => [[7,7],[6]] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
[8] => [[8],[]] => [[1,1,1,1,1,1,1,1],[]]
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => [[9],[]]
[1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => [[8,1],[]]
[1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => [[7,1,1],[]]
[1,1,1,1,1,2,2] => [[3,2,1,1,1,1,1],[1]] => [[7,2,1],[1]]
[1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => [[6,1,1,1],[]]
[1,1,1,1,5] => [[5,1,1,1,1],[]] => [[5,1,1,1,1],[]]
[1,1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1,1],[1,1,1,1]] => [[8,5],[4]]
[1,1,1,6] => [[6,1,1,1],[]] => [[4,1,1,1,1,1],[]]
[1,1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1,1],[1,1,1,1,1]] => [[8,6],[5]]
[1,1,2,1,4] => [[5,2,2,1,1],[1,1]] => [[5,3,1,1,1],[2]]
[1,1,3,1,1,1,1] => [[3,3,3,3,3,1,1],[2,2,2,2]] => [[7,5,5],[4,4]]
[1,1,7] => [[7,1,1],[]] => [[3,1,1,1,1,1,1],[]]
[1,2,1,1,1,1,2] => [[3,2,2,2,2,2,1],[1,1,1,1,1]] => [[7,6,1],[5]]
[1,2,1,1,1,3] => [[4,2,2,2,2,1],[1,1,1,1]] => [[6,5,1,1],[4]]
[1,2,2,1,1,1,1] => [[3,3,3,3,3,2,1],[2,2,2,2,1]] => [[7,6,5],[5,4]]
[1,4,1,1,1,1] => [[4,4,4,4,4,1],[3,3,3,3]] => [[6,5,5,5],[4,4,4]]
[1,8] => [[8,1],[]] => [[2,1,1,1,1,1,1,1],[]]
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [[8,8],[7]]
[2,1,1,1,1,1,2] => [[3,2,2,2,2,2,2],[1,1,1,1,1,1]] => [[7,7,1],[6]]
[2,1,1,1,1,2,1] => [[3,3,2,2,2,2,2],[2,1,1,1,1,1]] => [[7,7,2],[6,1]]
[2,1,1,1,1,3] => [[4,2,2,2,2,2],[1,1,1,1,1]] => [[6,6,1,1],[5]]
[2,3,1,1,1,1] => [[4,4,4,4,4,2],[3,3,3,3,1]] => [[6,6,5,5],[5,4,4]]
[2,7] => [[8,2],[1]] => [[2,2,1,1,1,1,1,1],[1]]
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]] => [[7,7,7],[6,6]]
[3,1,1,1,1,2] => [[4,3,3,3,3,3],[2,2,2,2,2]] => [[6,6,6,1],[5,5]]
[3,1,1,1,3] => [[5,3,3,3,3],[2,2,2,2]] => [[5,5,5,1,1],[4,4]]
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]] => [[5,5,5,5,5],[4,4,4,4]]
[5,1,1,2] => [[6,5,5,5],[4,4,4]] => [[4,4,4,4,4,1],[3,3,3,3]]
[7,1,1] => [[7,7,7],[6,6]] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
[7,2] => [[8,7],[6]] => [[2,2,2,2,2,2,2,1],[1,1,1,1,1,1]]
[8,1] => [[8,8],[7]] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
[9] => [[9],[]] => [[1,1,1,1,1,1,1,1,1],[]]
[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => [[10],[]]
[1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => [[9,1],[]]
[1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => [[8,1,1],[]]
[1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => [[7,1,1,1],[]]
[1,1,1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1,1,1],[1,1,1]] => [[9,4],[3]]
[1,1,1,1,1,3,1,1] => [[3,3,3,1,1,1,1,1],[2,2]] => [[8,3,3],[2,2]]
[1,1,1,1,1,4,1] => [[4,4,1,1,1,1,1],[3]] => [[7,2,2,2],[1,1,1]]
[1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => [[6,1,1,1,1],[]]
[1,1,1,1,6] => [[6,1,1,1,1],[]] => [[5,1,1,1,1,1],[]]
[1,1,1,7] => [[7,1,1,1],[]] => [[4,1,1,1,1,1,1],[]]
[1,1,8] => [[8,1,1],[]] => [[3,1,1,1,1,1,1,1],[]]
[1,9] => [[9,1],[]] => [[2,1,1,1,1,1,1,1,1],[]]
[2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]] => [[9,9],[8]]
[2,1,1,1,1,1,1,2] => [[3,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [[8,8,1],[7]]
[2,1,1,1,1,1,3] => [[4,2,2,2,2,2,2],[1,1,1,1,1,1]] => [[7,7,1,1],[6]]
[9,1] => [[9,9],[8]] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
[10] => [[10],[]] => [[1,1,1,1,1,1,1,1,1,1],[]]
[1,1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1,1],[]] => [[10,1],[]]
[2,1,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1,1]] => [[10,10],[9]]
[11] => [[11],[]] => [[1,1,1,1,1,1,1,1,1,1,1],[]]
[1,1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1,1],[]] => [[9,1,1],[]]
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
conjugate
Description
The conjugate of the skew partition.
The conjugate of a skew partition $\lambda$ is the skew partition $\lambda^*$ whose Ferrers diagram is obtained from the Ferrers diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.