Identifier
Mp00041:
Integer compositions
—conjugate⟶
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]=>[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)
[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
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.
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