- FindStat

Identifier

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets

Images

[1] => [1] => [[1],[]] => ([],1)

[1,1] => [1,1] => [[1,1],[]] => ([(0,1)],2)

[2] => [2] => [[2],[]] => ([(0,1)],2)

[1,1,1] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3)

[1,2] => [2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3)

[2,1] => [1,2] => [[2,1],[]] => ([(0,1),(0,2)],3)

[3] => [3] => [[3],[]] => ([(0,2),(2,1)],3)

[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4)

[1,1,2] => [2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4)

[1,2,1] => [1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4)

[1,3] => [3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4)

[2,1,1] => [1,1,2] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4)

[2,2] => [2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4)

[3,1] => [1,3] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4)

[4] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4)

[1,1,1,1,1] => [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] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5)

[1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5)

[1,1,3] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5)

[1,2,1,1] => [1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5)

[1,2,2] => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5)

[1,3,1] => [1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5)

[1,4] => [4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5)

[2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5)

[2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5)

[2,2,1] => [1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5)

[2,3] => [3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5)

[3,1,1] => [1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5)

[3,2] => [2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5)

[4,1] => [1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5)

[5] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5)

[1,1,1,1,1,1] => [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] => [[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)

[1,1,1,2,1] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)

[1,1,1,3] => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)

[1,1,2,1,1] => [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] => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)

[1,1,3,1] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)

[1,1,4] => [4,1,1] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)

[1,2,1,1,1] => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)

[1,2,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)

[1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)

[1,2,3] => [3,2,1] => [[4,4,3],[3,2]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)

[1,3,1,1] => [1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)

[1,3,2] => [2,3,1] => [[4,4,2],[3,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)

[1,4,1] => [1,4,1] => [[4,4,1],[3]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)

[1,5] => [5,1] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)

[2,1,1,1,1] => [1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)

[2,1,1,2] => [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] => [1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)

[2,1,3] => [3,1,2] => [[4,3,3],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)

[2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)

[2,2,2] => [2,2,2] => [[4,3,2],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)

[2,3,1] => [1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)

[2,4] => [4,2] => [[5,4],[3]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)

[3,1,1,1] => [1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)

[3,1,2] => [2,1,3] => [[4,2,2],[1,1]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)

[3,2,1] => [1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)

[3,3] => [3,3] => [[5,3],[2]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)

[4,1,1] => [1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)

[4,2] => [2,4] => [[5,2],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)

[5,1] => [1,5] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)

[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] => [[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] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)

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

[1,1,1,1,3] => [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)

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

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

[1,1,1,3,1] => [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)

[1,1,1,4] => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)

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

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

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

[1,1,2,3] => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7)

[1,1,3,1,1] => [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] => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)

[1,1,4,1] => [1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7)

[1,1,5] => [5,1,1] => [[5,5,5],[4,4]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)

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

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

[1,2,1,2,1] => [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] => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)

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

[1,2,2,2] => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)

[1,2,3,1] => [1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)

[1,2,4] => [4,2,1] => [[5,5,4],[4,3]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)

[1,3,1,1,1] => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)

[1,3,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)

[1,3,2,1] => [1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)

[1,3,3] => [3,3,1] => [[5,5,3],[4,2]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)

[1,4,1,1] => [1,1,4,1] => [[4,4,1,1],[3]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7)

[1,4,2] => [2,4,1] => [[5,5,2],[4,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)

[1,5,1] => [1,5,1] => [[5,5,1],[4]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7)

[1,6] => [6,1] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)

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

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

[2,1,1,3] => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7)

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

[2,1,2,2] => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)

>>> Load all 256 entries. <<<

[2,1,3,1] => [1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7)

[2,1,4] => [4,1,2] => [[5,4,4],[3,3]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7)

[2,2,1,1,1] => [1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)

[2,2,1,2] => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)

[2,2,2,1] => [1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7)

[2,2,3] => [3,2,2] => [[5,4,3],[3,2]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)

[2,3,1,1] => [1,1,3,2] => [[4,3,1,1],[2]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)

[2,3,2] => [2,3,2] => [[5,4,2],[3,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)

[2,4,1] => [1,4,2] => [[5,4,1],[3]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7)

[2,5] => [5,2] => [[6,5],[4]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)

[3,1,1,1,1] => [1,1,1,1,3] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)

[3,1,1,2] => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7)

[3,1,2,1] => [1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)

[3,1,3] => [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] => [1,1,2,3] => [[4,2,1,1],[1]] => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7)

[3,2,2] => [2,2,3] => [[5,3,2],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)

[3,3,1] => [1,3,3] => [[5,3,1],[2]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)

[3,4] => [4,3] => [[6,4],[3]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)

[4,1,1,1] => [1,1,1,4] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)

[4,1,2] => [2,1,4] => [[5,2,2],[1,1]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)

[4,2,1] => [1,2,4] => [[5,2,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)

[4,3] => [3,4] => [[6,3],[2]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)

[5,1,1] => [1,1,5] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)

[5,2] => [2,5] => [[6,2],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)

[6,1] => [1,6] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)

[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] => [[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] => [[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)

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

[1,1,1,1,1,3] => [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)

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

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

[1,1,1,1,4] => [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)

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

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

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

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

[1,1,1,4,1] => [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)

[1,1,1,5] => [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)

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

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

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

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

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

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

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

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

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

[1,1,3,3] => [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)

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

[1,1,6] => [6,1,1] => [[6,6,6],[5,5]] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,3,1,3] => [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)

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

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

[1,3,3,1] => [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)

[1,3,4] => [4,3,1] => [[6,6,4],[5,3]] => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)

[1,4,3] => [3,4,1] => [[6,6,3],[5,2]] => ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)

[1,7] => [7,1] => [[7,7],[6]] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,2,2,2] => [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] => [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)

[2,2,4] => [4,2,2] => [[6,5,4],[4,3]] => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)

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

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

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

[2,3,3] => [3,3,2] => [[6,5,3],[4,2]] => ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)

[2,4,2] => [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] => [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)

[3,1,1,3] => [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)

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

[3,1,3,1] => [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)

[3,1,4] => [4,1,3] => [[6,4,4],[3,3]] => ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)

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

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

[3,2,3] => [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] => [1,1,3,3] => [[5,3,1,1],[2]] => ([(0,6),(0,7),(1,4),(1,5),(4,7),(5,3),(6,2)],8)

[3,3,2] => [2,3,3] => [[6,4,2],[3,1]] => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)

[3,4,1] => [1,4,3] => [[6,4,1],[3]] => ([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8)

[4,1,1,1,1] => [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)

[4,4] => [4,4] => [[7,4],[3]] => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)

[5,1,1,1] => [1,1,1,5] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)

[6,1,1] => [1,1,6] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)

[7,1] => [1,7] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)

[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] => [[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] => [[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)

[1,1,1,1,1,1,3] => [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)

[1,1,1,1,1,4] => [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)

[1,1,1,1,5] => [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)

[1,1,1,6] => [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)

[1,1,7] => [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)

[1,8] => [8,1] => [[8,8],[7]] => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)

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

[3,1,1,1,1,1,1] => [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)

[4,1,1,1,1,1] => [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)

[5,1,1,1,1] => [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)

[6,1,1,1] => [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)

[7,1,1] => [1,1,7] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)

[8,1] => [1,8] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)

[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] => [[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] => [[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)

[1,1,1,1,1,1,1,3] => [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)

[1,1,1,1,1,1,4] => [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)

[1,1,1,1,1,5] => [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)

[1,1,1,1,6] => [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)

[1,1,1,7] => [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)

[1,1,8] => [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)

[1,9] => [9,1] => [[9,9],[8]] => ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)

[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)

[3,1,1,1,1,1,1,1] => [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)

[4,1,1,1,1,1,1] => [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)

[5,1,1,1,1,1] => [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)

[6,1,1,1,1] => [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)

[7,1,1,1] => [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)

[8,1,1] => [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)

[9,1] => [1,9] => [[9,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)

[10] => [10] => [[10],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)

Map

reverse

Description

Return the reversal of a composition.
That is, the composition

$(i_1, i_2, \ldots, i_k)$ is sent to

$(i_k, i_{k-1}, \ldots, i_1)$ .

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

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.