Identifier
Mp00039:
Integer compositions
—complement⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Images
=>
Cc0028;cc-rep-2Cc0029;cc-rep-3
[1]=>[1]=>[[1],[]]=>([],1)
[1,1]=>[2]=>[[2],[]]=>([],1)
[2]=>[1,1]=>[[1,1],[]]=>([],1)
[1,1,1]=>[3]=>[[3],[]]=>([],1)
[1,2]=>[2,1]=>[[2,2],[1]]=>([],1)
[2,1]=>[1,2]=>[[2,1],[]]=>([],1)
[3]=>[1,1,1]=>[[1,1,1],[]]=>([],1)
[1,1,1,1]=>[4]=>[[4],[]]=>([],1)
[1,1,2]=>[3,1]=>[[3,3],[2]]=>([],1)
[1,2,1]=>[2,2]=>[[3,2],[1]]=>([(0,1)],2)
[1,3]=>[2,1,1]=>[[2,2,2],[1,1]]=>([],1)
[2,1,1]=>[1,3]=>[[3,1],[]]=>([],1)
[2,2]=>[1,2,1]=>[[2,2,1],[1]]=>([(0,1)],2)
[3,1]=>[1,1,2]=>[[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]=>[4,1]=>[[4,4],[3]]=>([],1)
[1,1,2,1]=>[3,2]=>[[4,3],[2]]=>([(0,1)],2)
[1,1,3]=>[3,1,1]=>[[3,3,3],[2,2]]=>([],1)
[1,2,1,1]=>[2,3]=>[[4,2],[1]]=>([(0,1)],2)
[1,2,2]=>[2,2,1]=>[[3,3,2],[2,1]]=>([(0,2),(2,1)],3)
[1,3,1]=>[2,1,2]=>[[3,2,2],[1,1]]=>([(0,1)],2)
[1,4]=>[2,1,1,1]=>[[2,2,2,2],[1,1,1]]=>([],1)
[2,1,1,1]=>[1,4]=>[[4,1],[]]=>([],1)
[2,1,2]=>[1,3,1]=>[[3,3,1],[2]]=>([(0,1)],2)
[2,2,1]=>[1,2,2]=>[[3,2,1],[1]]=>([(0,2),(2,1)],3)
[2,3]=>[1,2,1,1]=>[[2,2,2,1],[1,1]]=>([(0,1)],2)
[3,1,1]=>[1,1,3]=>[[3,1,1],[]]=>([],1)
[3,2]=>[1,1,2,1]=>[[2,2,1,1],[1]]=>([(0,1)],2)
[4,1]=>[1,1,1,2]=>[[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]=>[5,1]=>[[5,5],[4]]=>([],1)
[1,1,1,2,1]=>[4,2]=>[[5,4],[3]]=>([(0,1)],2)
[1,1,1,3]=>[4,1,1]=>[[4,4,4],[3,3]]=>([],1)
[1,1,2,1,1]=>[3,3]=>[[5,3],[2]]=>([(0,2),(2,1)],3)
[1,1,2,2]=>[3,2,1]=>[[4,4,3],[3,2]]=>([(0,2),(2,1)],3)
[1,1,3,1]=>[3,1,2]=>[[4,3,3],[2,2]]=>([(0,1)],2)
[1,1,4]=>[3,1,1,1]=>[[3,3,3,3],[2,2,2]]=>([],1)
[1,2,1,1,1]=>[2,4]=>[[5,2],[1]]=>([(0,1)],2)
[1,2,1,2]=>[2,3,1]=>[[4,4,2],[3,1]]=>([(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]=>[2,2,1,1]=>[[3,3,3,2],[2,2,1]]=>([(0,2),(2,1)],3)
[1,3,1,1]=>[2,1,3]=>[[4,2,2],[1,1]]=>([(0,1)],2)
[1,3,2]=>[2,1,2,1]=>[[3,3,2,2],[2,1,1]]=>([(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]=>[2,1,1,1,1]=>[[2,2,2,2,2],[1,1,1,1]]=>([],1)
[2,1,1,1,1]=>[1,5]=>[[5,1],[]]=>([],1)
[2,1,1,2]=>[1,4,1]=>[[4,4,1],[3]]=>([(0,1)],2)
[2,1,2,1]=>[1,3,2]=>[[4,3,1],[2]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[2,1,3]=>[1,3,1,1]=>[[3,3,3,1],[2,2]]=>([(0,1)],2)
[2,2,1,1]=>[1,2,3]=>[[4,2,1],[1]]=>([(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]=>[1,2,1,2]=>[[3,2,2,1],[1,1]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[2,4]=>[1,2,1,1,1]=>[[2,2,2,2,1],[1,1,1]]=>([(0,1)],2)
[3,1,1,1]=>[1,1,4]=>[[4,1,1],[]]=>([],1)
[3,1,2]=>[1,1,3,1]=>[[3,3,1,1],[2]]=>([(0,1)],2)
[3,2,1]=>[1,1,2,2]=>[[3,2,1,1],[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]=>[1,1,1,3]=>[[3,1,1,1],[]]=>([],1)
[4,2]=>[1,1,1,2,1]=>[[2,2,1,1,1],[1]]=>([(0,1)],2)
[5,1]=>[1,1,1,1,2]=>[[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]=>[6,1]=>[[6,6],[5]]=>([],1)
[1,1,1,1,2,1]=>[5,2]=>[[6,5],[4]]=>([(0,1)],2)
[1,1,1,1,3]=>[5,1,1]=>[[5,5,5],[4,4]]=>([],1)
[1,1,1,2,1,1]=>[4,3]=>[[6,4],[3]]=>([(0,2),(2,1)],3)
[1,1,1,2,2]=>[4,2,1]=>[[5,5,4],[4,3]]=>([(0,2),(2,1)],3)
[1,1,1,3,1]=>[4,1,2]=>[[5,4,4],[3,3]]=>([(0,1)],2)
[1,1,1,4]=>[4,1,1,1]=>[[4,4,4,4],[3,3,3]]=>([],1)
[1,1,2,1,1,1]=>[3,4]=>[[6,3],[2]]=>([(0,2),(2,1)],3)
[1,1,2,1,2]=>[3,3,1]=>[[5,5,3],[4,2]]=>([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
[1,1,2,2,1]=>[3,2,2]=>[[5,4,3],[3,2]]=>([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
[1,1,2,3]=>[3,2,1,1]=>[[4,4,4,3],[3,3,2]]=>([(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]=>[3,1,2,1]=>[[4,4,3,3],[3,2,2]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[1,1,4,1]=>[3,1,1,2]=>[[4,3,3,3],[2,2,2]]=>([(0,1)],2)
[1,1,5]=>[3,1,1,1,1]=>[[3,3,3,3,3],[2,2,2,2]]=>([],1)
[1,2,1,1,1,1]=>[2,5]=>[[6,2],[1]]=>([(0,1)],2)
[1,2,1,1,2]=>[2,4,1]=>[[5,5,2],[4,1]]=>([(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]=>[2,3,1,1]=>[[4,4,4,2],[3,3,1]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[1,2,2,1,1]=>[2,2,3]=>[[5,3,2],[2,1]]=>([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
[1,2,2,2]=>[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)
[1,2,3,1]=>[2,2,1,2]=>[[4,3,3,2],[2,2,1]]=>([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
[1,2,4]=>[2,2,1,1,1]=>[[3,3,3,3,2],[2,2,2,1]]=>([(0,2),(2,1)],3)
[1,3,1,1,1]=>[2,1,4]=>[[5,2,2],[1,1]]=>([(0,1)],2)
[1,3,1,2]=>[2,1,3,1]=>[[4,4,2,2],[3,1,1]]=>([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
[1,3,2,1]=>[2,1,2,2]=>[[4,3,2,2],[2,1,1]]=>([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
[1,3,3]=>[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)
[1,4,1,1]=>[2,1,1,3]=>[[4,2,2,2],[1,1,1]]=>([(0,1)],2)
[1,4,2]=>[2,1,1,2,1]=>[[3,3,2,2,2],[2,1,1,1]]=>([(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]=>[2,1,1,1,1,1]=>[[2,2,2,2,2,2],[1,1,1,1,1]]=>([],1)
[2,1,1,1,1,1]=>[1,6]=>[[6,1],[]]=>([],1)
[2,1,1,1,2]=>[1,5,1]=>[[5,5,1],[4]]=>([(0,1)],2)
[2,1,1,2,1]=>[1,4,2]=>[[5,4,1],[3]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[2,1,1,3]=>[1,4,1,1]=>[[4,4,4,1],[3,3]]=>([(0,1)],2)
[2,1,2,1,1]=>[1,3,3]=>[[5,3,1],[2]]=>([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
[2,1,2,2]=>[1,3,2,1]=>[[4,4,3,1],[3,2]]=>([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
[2,1,3,1]=>[1,3,1,2]=>[[4,3,3,1],[2,2]]=>([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
[2,1,4]=>[1,3,1,1,1]=>[[3,3,3,3,1],[2,2,2]]=>([(0,1)],2)
[2,2,1,1,1]=>[1,2,4]=>[[5,2,1],[1]]=>([(0,2),(2,1)],3)
[2,2,1,2]=>[1,2,3,1]=>[[4,4,2,1],[3,1]]=>([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
[2,2,2,1]=>[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)
[2,2,3]=>[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)
[2,3,1,1]=>[1,2,1,3]=>[[4,2,2,1],[1,1]]=>([(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]=>[1,2,1,1,2]=>[[3,2,2,2,1],[1,1,1]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[2,5]=>[1,2,1,1,1,1]=>[[2,2,2,2,2,1],[1,1,1,1]]=>([(0,1)],2)
[3,1,1,1,1]=>[1,1,5]=>[[5,1,1],[]]=>([],1)
[3,1,1,2]=>[1,1,4,1]=>[[4,4,1,1],[3]]=>([(0,1)],2)
[3,1,2,1]=>[1,1,3,2]=>[[4,3,1,1],[2]]=>([(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]=>[1,1,2,3]=>[[4,2,1,1],[1]]=>([(0,2),(2,1)],3)
[3,2,2]=>[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)
[3,3,1]=>[1,1,2,1,2]=>[[3,2,2,1,1],[1,1]]=>([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
[3,4]=>[1,1,2,1,1,1]=>[[2,2,2,2,1,1],[1,1,1]]=>([(0,2),(2,1)],3)
[4,1,1,1]=>[1,1,1,4]=>[[4,1,1,1],[]]=>([],1)
[4,1,2]=>[1,1,1,3,1]=>[[3,3,1,1,1],[2]]=>([(0,1)],2)
[4,2,1]=>[1,1,1,2,2]=>[[3,2,1,1,1],[1]]=>([(0,2),(2,1)],3)
[4,3]=>[1,1,1,2,1,1]=>[[2,2,2,1,1,1],[1,1]]=>([(0,2),(2,1)],3)
[5,1,1]=>[1,1,1,1,3]=>[[3,1,1,1,1],[]]=>([],1)
[5,2]=>[1,1,1,1,2,1]=>[[2,2,1,1,1,1],[1]]=>([(0,1)],2)
[6,1]=>[1,1,1,1,1,2]=>[[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)
[1,1,1,2,1,1,1]=>[4,4]=>[[7,4],[3]]=>([(0,3),(2,1),(3,2)],4)
[1,1,1,2,1,2]=>[4,3,1]=>[[6,6,4],[5,3]]=>([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
[1,1,1,2,2,1]=>[4,2,2]=>[[6,5,4],[4,3]]=>([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
[1,1,1,2,3]=>[4,2,1,1]=>[[5,5,5,4],[4,4,3]]=>([(0,2),(2,1)],3)
[1,1,1,3,1,1]=>[4,1,3]=>[[6,4,4],[3,3]]=>([(0,2),(2,1)],3)
[1,1,1,3,2]=>[4,1,2,1]=>[[5,5,4,4],[4,3,3]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[1,1,1,4,1]=>[4,1,1,2]=>[[5,4,4,4],[3,3,3]]=>([(0,1)],2)
[1,1,1,5]=>[4,1,1,1,1]=>[[4,4,4,4,4],[3,3,3,3]]=>([],1)
[1,1,2,1,1,2]=>[3,4,1]=>[[6,6,3],[5,2]]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
[1,1,2,1,2,1]=>[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)
[1,1,2,1,3]=>[3,3,1,1]=>[[5,5,5,3],[4,4,2]]=>([(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]=>[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)
[1,1,2,3,1]=>[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)
[1,1,2,4]=>[3,2,1,1,1]=>[[4,4,4,4,3],[3,3,3,2]]=>([(0,2),(2,1)],3)
[1,1,3,1,2]=>[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)
[1,1,3,2,1]=>[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)
[1,1,3,3]=>[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)
[1,1,4,1,1]=>[3,1,1,3]=>[[5,3,3,3],[2,2,2]]=>([(0,2),(2,1)],3)
[1,1,4,2]=>[3,1,1,2,1]=>[[4,4,3,3,3],[3,2,2,2]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[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)
[1,2,1,1,3]=>[2,4,1,1]=>[[5,5,5,2],[4,4,1]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[1,2,1,2,1,1]=>[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)
[1,2,1,2,2]=>[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)
[1,2,1,3,1]=>[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)
[1,2,1,4]=>[2,3,1,1,1]=>[[4,4,4,4,2],[3,3,3,1]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[1,2,2,1,2]=>[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)
[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]=>[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)
[1,2,3,1,1]=>[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)
[1,2,3,2]=>[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)
[1,2,4,1]=>[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)
[1,2,5]=>[2,2,1,1,1,1]=>[[3,3,3,3,3,2],[2,2,2,2,1]]=>([(0,2),(2,1)],3)
[1,3,1,2,1]=>[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)
[1,3,1,3]=>[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)
[1,3,2,1,1]=>[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)
[1,3,2,2]=>[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)
[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]=>[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)
[1,4,1,2]=>[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)
[1,4,2,1]=>[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)
[1,4,3]=>[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)
[1,5,2]=>[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)
[1,6,1]=>[2,1,1,1,1,2]=>[[3,2,2,2,2,2],[1,1,1,1,1]]=>([(0,1)],2)
[2,1,1,1,1,1,1]=>[1,7]=>[[7,1],[]]=>([],1)
[2,1,1,2,1,1]=>[1,4,3]=>[[6,4,1],[3]]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
[2,1,1,2,2]=>[1,4,2,1]=>[[5,5,4,1],[4,3]]=>([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
[2,1,1,3,1]=>[1,4,1,2]=>[[5,4,4,1],[3,3]]=>([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
[2,1,1,4]=>[1,4,1,1,1]=>[[4,4,4,4,1],[3,3,3]]=>([(0,1)],2)
[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]=>[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)
[2,1,2,3]=>[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)
[2,1,3,1,1]=>[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)
[2,1,3,2]=>[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)
[2,1,4,1]=>[1,3,1,1,2]=>[[4,3,3,3,1],[2,2,2]]=>([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
[2,2,1,2,1]=>[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)
[2,2,1,3]=>[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)
[2,2,2,1,1]=>[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)
[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]=>[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)
[2,2,4]=>[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)
[2,3,1,2]=>[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)
[2,3,2,1]=>[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)
[2,3,3]=>[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)
[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]=>[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)
[2,6]=>[1,2,1,1,1,1,1]=>[[2,2,2,2,2,2,1],[1,1,1,1,1]]=>([(0,1)],2)
[3,1,1,1,1,1]=>[1,1,6]=>[[6,1,1],[]]=>([],1)
[3,1,2,1,1]=>[1,1,3,3]=>[[5,3,1,1],[2]]=>([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
[3,1,2,2]=>[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)
[3,1,3,1]=>[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)
[3,2,1,2]=>[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)
[3,2,2,1]=>[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)
[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)
[3,3,2]=>[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)
[3,4,1]=>[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)
[3,5]=>[1,1,2,1,1,1,1]=>[[2,2,2,2,2,1,1],[1,1,1,1]]=>([(0,2),(2,1)],3)
[4,1,1,1,1]=>[1,1,1,5]=>[[5,1,1,1],[]]=>([],1)
[4,1,2,1]=>[1,1,1,3,2]=>[[4,3,1,1,1],[2]]=>([(0,1),(0,2),(1,3),(2,3)],4)
[4,2,2]=>[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)
[4,3,1]=>[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)
[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]=>[1,1,1,1,4]=>[[4,1,1,1,1],[]]=>([],1)
[5,2,1]=>[1,1,1,1,2,2]=>[[3,2,1,1,1,1],[1]]=>([(0,2),(2,1)],3)
[5,3]=>[1,1,1,1,2,1,1]=>[[2,2,2,1,1,1,1],[1,1]]=>([(0,2),(2,1)],3)
[6,1,1]=>[1,1,1,1,1,3]=>[[3,1,1,1,1,1],[]]=>([],1)
[6,2]=>[1,1,1,1,1,2,1]=>[[2,2,1,1,1,1,1],[1]]=>([(0,1)],2)
[7,1]=>[1,1,1,1,1,1,2]=>[[2,1,1,1,1,1,1],[]]=>([],1)
[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)
[2,1,1,1,1,1,1,1]=>[1,8]=>[[8,1],[]]=>([],1)
[3,1,1,1,1,1,1]=>[1,1,7]=>[[7,1,1],[]]=>([],1)
[4,1,1,1,1,1]=>[1,1,1,6]=>[[6,1,1,1],[]]=>([],1)
[5,1,1,1,1]=>[1,1,1,1,5]=>[[5,1,1,1,1],[]]=>([],1)
[6,1,1,1]=>[1,1,1,1,1,4]=>[[4,1,1,1,1,1],[]]=>([],1)
[7,1,1]=>[1,1,1,1,1,1,3]=>[[3,1,1,1,1,1,1],[]]=>([],1)
[8,1]=>[1,1,1,1,1,1,1,2]=>[[2,1,1,1,1,1,1,1],[]]=>([],1)
[9]=>[1,1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1,1],[]]=>([],1)
[1,1,1,1,1,1,1,1,1,1]=>[10]=>[[10],[]]=>([],1)
[2,1,1,1,1,1,1,1,1]=>[1,9]=>[[9,1],[]]=>([],1)
[3,1,1,1,1,1,1,1]=>[1,1,8]=>[[8,1,1],[]]=>([],1)
[4,1,1,1,1,1,1]=>[1,1,1,7]=>[[7,1,1,1],[]]=>([],1)
[5,1,1,1,1,1]=>[1,1,1,1,6]=>[[6,1,1,1,1],[]]=>([],1)
[6,1,1,1,1]=>[1,1,1,1,1,5]=>[[5,1,1,1,1,1],[]]=>([],1)
[7,1,1,1]=>[1,1,1,1,1,1,4]=>[[4,1,1,1,1,1,1],[]]=>([],1)
[8,1,1]=>[1,1,1,1,1,1,1,3]=>[[3,1,1,1,1,1,1,1],[]]=>([],1)
[9,1]=>[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
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
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.
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.
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.
searching the database
Sorry, this map was not found in the database.