Identifier
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Images
[1] => [[1],[]] => ([],1)
[1,1] => [[1,1],[]] => ([(0,1)],2)
[2] => [[2],[]] => ([(0,1)],2)
[1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3)
[1,2] => [[2,1],[]] => ([(0,1),(0,2)],3)
[2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3)
[3] => [[3],[]] => ([(0,2),(2,1)],3)
[1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4)
[1,1,2] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4)
[1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4)
[1,3] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4)
[2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4)
[2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4)
[3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4)
[4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4)
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5)
[1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5)
[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5)
[1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5)
[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5)
[1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5)
[1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5)
[1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5)
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5)
[2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5)
[2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5)
[2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5)
[3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5)
[3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5)
[4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5)
[5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5)
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
[1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
[1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
[1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
[1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
[1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
[1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
[1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
[1,4,1] => [[4,4,1],[3]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
[1,5] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
[2,1,3] => [[4,2,2],[1,1]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
[2,2,2] => [[4,3,2],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
[2,3,1] => [[4,4,2],[3,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
[2,4] => [[5,2],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
[3,1,2] => [[4,3,3],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
[3,2,1] => [[4,4,3],[3,2]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
[3,3] => [[5,3],[2]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
[4,1,1] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
[4,2] => [[5,4],[3]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
[5,1] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
[6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
[1,1,1,1,3] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
[1,1,1,4] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7)
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7)
[1,1,5] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7)
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
[1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7)
[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
[1,2,4] => [[5,2,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7)
[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7)
[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)
[1,3,3] => [[5,3,1],[2]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7)
[1,4,2] => [[5,4,1],[3]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7)
[1,5,1] => [[5,5,1],[4]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7)
[1,6] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7)
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7)
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
>>> Load all 256 entries. <<<
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
[2,1,4] => [[5,2,2],[1,1]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
[2,2,3] => [[5,3,2],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
[2,3,2] => [[5,4,2],[3,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
[2,4,1] => [[5,5,2],[4,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
[2,5] => [[6,2],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7)
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
[3,1,3] => [[5,3,3],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7)
[3,2,2] => [[5,4,3],[3,2]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
[3,3,1] => [[5,5,3],[4,2]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
[3,4] => [[6,3],[2]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
[4,1,2] => [[5,4,4],[3,3]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7)
[4,2,1] => [[5,5,4],[4,3]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
[4,3] => [[6,4],[3]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
[5,1,1] => [[5,5,5],[4,4]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
[5,2] => [[6,5],[4]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
[6,1] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
[7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => ([(0,7),(1,6),(1,7),(3,5),(4,3),(5,2),(6,4)],8)
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => ([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => ([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
[1,1,1,1,4] => [[4,1,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => ([(0,6),(0,7),(1,3),(1,4),(4,7),(5,2),(6,5)],8)
[1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]] => ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(5,4)],8)
[1,1,1,3,2] => [[4,3,1,1,1],[2]] => ([(0,4),(0,7),(1,3),(1,6),(3,7),(5,2),(6,5)],8)
[1,1,1,5] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => ([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => ([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8)
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
[1,1,2,2,2] => [[4,3,2,1,1],[2,1]] => ([(0,6),(0,7),(1,3),(1,6),(2,4),(2,7),(4,5)],8)
[1,1,2,3,1] => [[4,4,2,1,1],[3,1]] => ([(0,7),(1,3),(1,6),(2,4),(2,6),(3,7),(4,5)],8)
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => ([(0,3),(0,4),(1,5),(1,6),(4,7),(5,7),(6,2)],8)
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
[1,1,3,3] => [[5,3,1,1],[2]] => ([(0,6),(0,7),(1,4),(1,5),(4,7),(5,3),(6,2)],8)
[1,1,6] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => ([(0,6),(1,3),(1,7),(2,7),(4,5),(5,2),(6,4)],8)
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => ([(0,4),(0,7),(1,2),(1,6),(3,7),(5,3),(6,5)],8)
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]] => ([(0,4),(0,7),(1,3),(1,6),(2,5),(2,6),(5,7)],8)
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => ([(0,6),(1,3),(1,7),(2,4),(2,5),(4,6),(5,7)],8)
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => ([(0,6),(0,7),(1,4),(1,6),(2,3),(2,5),(5,7)],8)
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
[1,2,2,3] => [[5,3,2,1],[2,1]] => ([(0,6),(0,7),(1,3),(1,6),(2,4),(2,7),(4,5)],8)
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => ([(0,5),(1,4),(1,6),(2,3),(2,6),(4,7),(5,7)],8)
[1,2,3,2] => [[5,4,2,1],[3,1]] => ([(0,4),(0,7),(1,3),(1,6),(2,5),(2,6),(5,7)],8)
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => ([(0,3),(0,5),(1,4),(1,6),(2,7),(5,7),(6,2)],8)
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7)],8)
[1,3,1,3] => [[5,3,3,1],[2,2]] => ([(0,3),(0,4),(1,5),(1,6),(4,7),(5,7),(6,2)],8)
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => ([(0,6),(0,7),(1,5),(2,3),(2,4),(4,6),(5,7)],8)
[1,3,2,2] => [[5,4,3,1],[3,2]] => ([(0,6),(0,7),(1,4),(1,6),(2,3),(2,5),(5,7)],8)
[1,3,3,1] => [[5,5,3,1],[4,2]] => ([(0,6),(1,5),(1,7),(2,3),(2,4),(4,7),(5,6)],8)
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
[1,4,1,2] => [[5,4,4,1],[3,3]] => ([(0,3),(0,5),(1,4),(1,6),(2,7),(5,7),(6,2)],8)
[1,4,2,1] => [[5,5,4,1],[4,3]] => ([(0,6),(1,6),(1,7),(2,3),(2,4),(4,5),(5,7)],8)
[1,4,3] => [[6,4,1],[3]] => ([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8)
[1,7] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => ([(0,7),(1,2),(1,6),(3,7),(4,5),(5,3),(6,4)],8)
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => ([(0,7),(1,6),(2,3),(2,6),(3,5),(4,7),(5,4)],8)
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => ([(0,7),(1,3),(1,6),(2,4),(2,6),(4,5),(5,7)],8)
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => ([(0,7),(1,6),(2,3),(2,4),(3,7),(4,5),(5,6)],8)
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
[2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]] => ([(0,6),(1,5),(1,7),(2,3),(2,4),(4,7),(5,6)],8)
[2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]] => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
[2,1,2,3] => [[5,3,2,2],[2,1,1]] => ([(0,7),(1,3),(1,6),(2,4),(2,6),(3,7),(4,5)],8)
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
[2,1,3,2] => [[5,4,2,2],[3,1,1]] => ([(0,6),(1,3),(1,7),(2,4),(2,5),(4,6),(5,7)],8)
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => ([(0,6),(1,6),(1,7),(2,3),(2,4),(4,5),(5,7)],8)
[2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]] => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
[2,2,1,3] => [[5,3,3,2],[2,2,1]] => ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
[2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]] => ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
[2,2,2,2] => [[5,4,3,2],[3,2,1]] => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
[2,2,3,1] => [[5,5,3,2],[4,2,1]] => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
[2,3,1,2] => [[5,4,4,2],[3,3,1]] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7)],8)
[2,3,2,1] => [[5,5,4,2],[4,3,1]] => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
[2,3,3] => [[6,4,2],[3,1]] => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
[2,4,1,1] => [[5,5,5,2],[4,4,1]] => ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
[2,4,2] => [[6,5,2],[4,1]] => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => ([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
[3,1,2,2] => [[5,4,3,3],[3,2,2]] => ([(0,5),(1,4),(1,6),(2,3),(2,6),(4,7),(5,7)],8)
[3,1,3,1] => [[5,5,3,3],[4,2,2]] => ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
[3,2,1,2] => [[5,4,4,3],[3,3,2]] => ([(0,6),(0,7),(1,5),(2,3),(2,4),(4,6),(5,7)],8)
[3,2,2,1] => [[5,5,4,3],[4,3,2]] => ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
[3,2,3] => [[6,4,3],[3,2]] => ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
[3,3,1,1] => [[5,5,5,3],[4,4,2]] => ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
[3,3,2] => [[6,5,3],[4,2]] => ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)
[3,4,1] => [[6,6,3],[5,2]] => ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]] => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
[4,1,1,2] => [[5,4,4,4],[3,3,3]] => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
[4,1,2,1] => [[5,5,4,4],[4,3,3]] => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
[4,1,3] => [[6,4,4],[3,3]] => ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)
[4,2,1,1] => [[5,5,5,4],[4,4,3]] => ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
[4,2,2] => [[6,5,4],[4,3]] => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
[4,3,1] => [[6,6,4],[5,3]] => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
[4,4] => [[7,4],[3]] => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
[5,1,1,1] => [[5,5,5,5],[4,4,4]] => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
[6,1,1] => [[6,6,6],[5,5]] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
[7,1] => [[7,7],[6]] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
[8] => [[8],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
[1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
[1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
[1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
[1,1,1,1,5] => [[5,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
[1,1,1,6] => [[6,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
[1,1,7] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
[1,8] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]] => ([(0,7),(1,3),(2,8),(3,8),(4,6),(5,4),(6,2),(7,5)],9)
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]] => ([(0,7),(1,6),(2,8),(3,8),(4,5),(5,3),(6,4),(7,2)],9)
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]] => ([(0,7),(1,6),(2,8),(3,8),(4,2),(5,3),(6,4),(7,5)],9)
[6,1,1,1] => [[6,6,6,6],[5,5,5]] => ([(0,7),(1,6),(2,8),(3,8),(4,5),(5,3),(6,4),(7,2)],9)
[7,1,1] => [[7,7,7],[6,6]] => ([(0,7),(1,3),(2,8),(3,8),(4,6),(5,4),(6,2),(7,5)],9)
[8,1] => [[8,8],[7]] => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
[9] => [[9],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
[1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
[1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10)
[1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10)
[1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
[1,1,1,1,6] => [[6,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
[1,1,1,7] => [[7,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10)
[1,1,8] => [[8,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10)
[1,9] => [[9,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
[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]] => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]] => ([(0,8),(1,3),(2,9),(3,9),(4,5),(5,7),(6,4),(7,2),(8,6)],10)
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]] => ([(0,8),(1,7),(2,9),(3,9),(4,6),(5,4),(6,3),(7,5),(8,2)],10)
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]] => ([(0,7),(1,8),(2,9),(3,9),(4,5),(5,2),(6,3),(7,6),(8,4)],10)
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]] => ([(0,7),(1,8),(2,9),(3,9),(4,5),(5,2),(6,3),(7,6),(8,4)],10)
[7,1,1,1] => [[7,7,7,7],[6,6,6]] => ([(0,8),(1,7),(2,9),(3,9),(4,6),(5,4),(6,3),(7,5),(8,2)],10)
[8,1,1] => [[8,8,8],[7,7]] => ([(0,8),(1,3),(2,9),(3,9),(4,5),(5,7),(6,4),(7,2),(8,6)],10)
[9,1] => [[9,9],[8]] => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
[10] => [[10],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
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
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.