- FindStat

Identifier

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,2] => [1,5] => [[5,1],[]] => ([],1)

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5] => [1,1,1,1,2] => [[2,1,1,1,1],[]] => ([],1)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([],1)

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

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

[1,1,1,1,1,2] => [1,6] => [[6,1],[]] => ([],1)

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

[1,1,1,1,3] => [1,1,5] => [[5,1,1],[]] => ([],1)

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

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

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

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

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

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

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

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

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

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

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

[1,1,5] => [1,1,1,1,3] => [[3,1,1,1,1],[]] => ([],1)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,6] => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([],1)

[2,1,1,1,1,1] => [6,1] => [[6,6],[5]] => ([],1)

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

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

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

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

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

>>> Load all 236 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6,1] => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([],1)

[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] => [[8],[]] => ([],1)

[1,1,1,1,1,1,2] => [1,7] => [[7,1],[]] => ([],1)

[1,1,1,1,1,3] => [1,1,6] => [[6,1,1],[]] => ([],1)

[1,1,1,1,4] => [1,1,1,5] => [[5,1,1,1],[]] => ([],1)

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

[1,1,1,5] => [1,1,1,1,4] => [[4,1,1,1,1],[]] => ([],1)

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

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

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

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

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

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

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

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

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

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

[1,1,6] => [1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => ([],1)

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

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

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

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

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

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

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

[1,2,2,2,1] => [2,2,2,2] => [[5,4,3,2],[3,2,1]] => ([(0,4),(0,7),(1,10),(3,8),(4,9),(5,2),(6,1),(6,8),(7,3),(7,9),(8,10),(9,6),(10,5)],11)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,7] => [1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => ([],1)

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

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

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

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

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

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

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

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

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

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

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

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

[2,2,2,2] => [1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => ([(0,6),(1,10),(2,9),(3,8),(4,3),(4,9),(5,1),(5,8),(6,7),(7,2),(7,4),(8,10),(9,5)],11)

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

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

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

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

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

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

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

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

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

[2,6] => [1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => ([(0,1)],2)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6,2] => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => ([(0,1)],2)

[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] => [[9],[]] => ([],1)

[1,1,1,1,1,1,1,2] => [1,8] => [[8,1],[]] => ([],1)

[1,1,1,1,1,1,3] => [1,1,7] => [[7,1,1],[]] => ([],1)

[1,1,1,1,1,4] => [1,1,1,6] => [[6,1,1,1],[]] => ([],1)

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

[1,1,1,6] => [1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => ([],1)

[1,1,7] => [1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => ([],1)

[1,8] => [1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => ([],1)

[9] => [1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([],1)

[1,1,1,1,1,1,1,1,1,1] => [10] => [[10],[]] => ([],1)

[1,1,1,1,1,1,1,1,2] => [1,9] => [[9,1],[]] => ([],1)

[1,1,1,1,1,1,1,3] => [1,1,8] => [[8,1,1],[]] => ([],1)

[1,1,1,1,1,1,4] => [1,1,1,7] => [[7,1,1,1],[]] => ([],1)

[1,1,1,1,1,5] => [1,1,1,1,6] => [[6,1,1,1,1],[]] => ([],1)

[1,1,1,1,6] => [1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => ([],1)

[1,1,1,7] => [1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => ([],1)

[1,1,8] => [1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => ([],1)

[1,9] => [1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => ([],1)

[10] => [1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => ([],1)

Map

conjugate

Description

The conjugate of a composition.
The conjugate of a composition

$C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of

$C$ .
Equivalently, the ribbon shape corresponding to the conjugate of

$C$ is the conjugate of the ribbon shape of

$C$ .

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

dominating sublattice

Description

Return the sublattice of the dominance order induced by the support of the expansion of the skew Schur function into Schur functions.
Consider the expansion of the skew Schur function

$s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions

$\nu$ with

$c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example

$\lambda = (5^2,4^2,1)$ and

$\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.