Identifier
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Images
=>
Cc0028;cc-rep-2Cc0014;cc-rep-3
[1]=>[1]=>[[1],[]]=>([],1) [1,1]=>[2]=>[[2],[]]=>([(0,1)],2) [2]=>[1,1]=>[[1,1],[]]=>([(0,1)],2) [1,1,1]=>[3]=>[[3],[]]=>([(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]=>[1,1,1]=>[[1,1,1],[]]=>([(0,2),(2,1)],3) [1,1,1,1]=>[4]=>[[4],[]]=>([(0,3),(2,1),(3,2)],4) [1,1,2]=>[3,1]=>[[3,3],[2]]=>([(0,3),(1,2),(2,3)],4) [1,2,1]=>[2,2]=>[[3,2],[1]]=>([(0,3),(1,2),(1,3)],4) [1,3]=>[2,1,1]=>[[2,2,2],[1,1]]=>([(0,3),(1,2),(2,3)],4) [2,1,1]=>[1,3]=>[[3,1],[]]=>([(0,2),(0,3),(3,1)],4) [2,2]=>[1,2,1]=>[[2,2,1],[1]]=>([(0,3),(1,2),(1,3)],4) [3,1]=>[1,1,2]=>[[2,1,1],[]]=>([(0,2),(0,3),(3,1)],4) [4]=>[1,1,1,1]=>[[1,1,1,1],[]]=>([(0,3),(2,1),(3,2)],4) [1,1,1,1,1]=>[5]=>[[5],[]]=>([(0,4),(2,3),(3,1),(4,2)],5) [1,1,1,2]=>[4,1]=>[[4,4],[3]]=>([(0,4),(1,2),(2,3),(3,4)],5) [1,1,2,1]=>[3,2]=>[[4,3],[2]]=>([(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]=>[2,3]=>[[4,2],[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]=>[2,1,2]=>[[3,2,2],[1,1]]=>([(0,4),(1,2),(1,3),(3,4)],5) [1,4]=>[2,1,1,1]=>[[2,2,2,2],[1,1,1]]=>([(0,4),(1,2),(2,3),(3,4)],5) [2,1,1,1]=>[1,4]=>[[4,1],[]]=>([(0,2),(0,4),(3,1),(4,3)],5) [2,1,2]=>[1,3,1]=>[[3,3,1],[2]]=>([(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]=>[1,2,1,1]=>[[2,2,2,1],[1,1]]=>([(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]=>[1,1,2,1]=>[[2,2,1,1],[1]]=>([(0,4),(1,2),(1,4),(2,3)],5) [4,1]=>[1,1,1,2]=>[[2,1,1,1],[]]=>([(0,2),(0,4),(3,1),(4,3)],5) [5]=>[1,1,1,1,1]=>[[1,1,1,1,1],[]]=>([(0,4),(2,3),(3,1),(4,2)],5) [1,1,1,1,1,1]=>[6]=>[[6],[]]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6) [1,1,1,1,2]=>[5,1]=>[[5,5],[4]]=>([(0,5),(1,4),(2,5),(3,2),(4,3)],6) [1,1,1,2,1]=>[4,2]=>[[5,4],[3]]=>([(0,4),(1,3),(1,5),(2,5),(4,2)],6) [1,1,1,3]=>[4,1,1]=>[[4,4,4],[3,3]]=>([(0,3),(1,4),(2,5),(3,5),(4,2)],6) [1,1,2,1,1]=>[3,3]=>[[5,3],[2]]=>([(0,3),(1,4),(1,5),(3,5),(4,2)],6) [1,1,2,2]=>[3,2,1]=>[[4,4,3],[3,2]]=>([(0,4),(1,4),(1,5),(2,3),(3,5)],6) [1,1,3,1]=>[3,1,2]=>[[4,3,3],[2,2]]=>([(0,4),(1,2),(1,3),(3,5),(4,5)],6) [1,1,4]=>[3,1,1,1]=>[[3,3,3,3],[2,2,2]]=>([(0,3),(1,4),(2,5),(3,5),(4,2)],6) [1,2,1,1,1]=>[2,4]=>[[5,2],[1]]=>([(0,5),(1,4),(1,5),(3,2),(4,3)],6) [1,2,1,2]=>[2,3,1]=>[[4,4,2],[3,1]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6) [1,2,2,1]=>[2,2,2]=>[[4,3,2],[2,1]]=>([(0,4),(1,4),(1,5),(2,3),(2,5)],6) [1,2,3]=>[2,2,1,1]=>[[3,3,3,2],[2,2,1]]=>([(0,4),(1,4),(1,5),(2,3),(3,5)],6) [1,3,1,1]=>[2,1,3]=>[[4,2,2],[1,1]]=>([(0,5),(1,3),(1,4),(3,5),(4,2)],6) [1,3,2]=>[2,1,2,1]=>[[3,3,2,2],[2,1,1]]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6) [1,4,1]=>[2,1,1,2]=>[[3,2,2,2],[1,1,1]]=>([(0,5),(1,2),(1,4),(3,5),(4,3)],6) [1,5]=>[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) [2,1,1,1,1]=>[1,5]=>[[5,1],[]]=>([(0,2),(0,5),(3,4),(4,1),(5,3)],6) [2,1,1,2]=>[1,4,1]=>[[4,4,1],[3]]=>([(0,5),(1,2),(1,4),(3,5),(4,3)],6) [2,1,2,1]=>[1,3,2]=>[[4,3,1],[2]]=>([(0,4),(0,5),(1,2),(1,3),(3,5)],6) [2,1,3]=>[1,3,1,1]=>[[3,3,3,1],[2,2]]=>([(0,4),(1,2),(1,3),(3,5),(4,5)],6) [2,2,1,1]=>[1,2,3]=>[[4,2,1],[1]]=>([(0,3),(0,5),(1,4),(1,5),(4,2)],6) [2,2,2]=>[1,2,2,1]=>[[3,3,2,1],[2,1]]=>([(0,4),(1,4),(1,5),(2,3),(2,5)],6) [2,3,1]=>[1,2,1,2]=>[[3,2,2,1],[1,1]]=>([(0,4),(0,5),(1,2),(1,3),(3,5)],6) [2,4]=>[1,2,1,1,1]=>[[2,2,2,2,1],[1,1,1]]=>([(0,4),(1,3),(1,5),(2,5),(4,2)],6) [3,1,1,1]=>[1,1,4]=>[[4,1,1],[]]=>([(0,4),(0,5),(3,2),(4,3),(5,1)],6) [3,1,2]=>[1,1,3,1]=>[[3,3,1,1],[2]]=>([(0,5),(1,3),(1,4),(3,5),(4,2)],6) [3,2,1]=>[1,1,2,2]=>[[3,2,1,1],[1]]=>([(0,3),(0,5),(1,4),(1,5),(4,2)],6) [3,3]=>[1,1,2,1,1]=>[[2,2,2,1,1],[1,1]]=>([(0,3),(1,4),(1,5),(3,5),(4,2)],6) [4,1,1]=>[1,1,1,3]=>[[3,1,1,1],[]]=>([(0,4),(0,5),(3,2),(4,3),(5,1)],6) [4,2]=>[1,1,1,2,1]=>[[2,2,1,1,1],[1]]=>([(0,5),(1,4),(1,5),(3,2),(4,3)],6) [5,1]=>[1,1,1,1,2]=>[[2,1,1,1,1],[]]=>([(0,2),(0,5),(3,4),(4,1),(5,3)],6) [6]=>[1,1,1,1,1,1]=>[[1,1,1,1,1,1],[]]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6) [1,1,1,1,1,1,1]=>[7]=>[[7],[]]=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) [1,1,1,1,1,2]=>[6,1]=>[[6,6],[5]]=>([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) [1,1,1,1,2,1]=>[5,2]=>[[6,5],[4]]=>([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7) [1,1,1,1,3]=>[5,1,1]=>[[5,5,5],[4,4]]=>([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) [1,1,1,2,1,1]=>[4,3]=>[[6,4],[3]]=>([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7) [1,1,1,2,2]=>[4,2,1]=>[[5,5,4],[4,3]]=>([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) [1,1,1,3,1]=>[4,1,2]=>[[5,4,4],[3,3]]=>([(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]=>[3,4]=>[[6,3],[2]]=>([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7) [1,1,2,1,2]=>[3,3,1]=>[[5,5,3],[4,2]]=>([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) [1,1,2,2,1]=>[3,2,2]=>[[5,4,3],[3,2]]=>([(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]=>[3,1,3]=>[[5,3,3],[2,2]]=>([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) [1,1,3,2]=>[3,1,2,1]=>[[4,4,3,3],[3,2,2]]=>([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) [1,1,4,1]=>[3,1,1,2]=>[[4,3,3,3],[2,2,2]]=>([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) [1,1,5]=>[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) [1,2,1,1,1,1]=>[2,5]=>[[6,2],[1]]=>([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7) [1,2,1,1,2]=>[2,4,1]=>[[5,5,2],[4,1]]=>([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) [1,2,1,2,1]=>[2,3,2]=>[[5,4,2],[3,1]]=>([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7) [1,2,1,3]=>[2,3,1,1]=>[[4,4,4,2],[3,3,1]]=>([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) [1,2,2,1,1]=>[2,2,3]=>[[5,3,2],[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]=>[2,2,1,2]=>[[4,3,3,2],[2,2,1]]=>([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) [1,2,4]=>[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) [1,3,1,1,1]=>[2,1,4]=>[[5,2,2],[1,1]]=>([(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]=>[2,1,2,2]=>[[4,3,2,2],[2,1,1]]=>([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) [1,3,3]=>[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) [1,4,1,1]=>[2,1,1,3]=>[[4,2,2,2],[1,1,1]]=>([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) [1,4,2]=>[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) [1,5,1]=>[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) [1,6]=>[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) [2,1,1,1,1,1]=>[1,6]=>[[6,1],[]]=>([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) [2,1,1,1,2]=>[1,5,1]=>[[5,5,1],[4]]=>([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) [2,1,1,2,1]=>[1,4,2]=>[[5,4,1],[3]]=>([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) [2,1,1,3]=>[1,4,1,1]=>[[4,4,4,1],[3,3]]=>([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) [2,1,2,1,1]=>[1,3,3]=>[[5,3,1],[2]]=>([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) [2,1,2,2]=>[1,3,2,1]=>[[4,4,3,1],[3,2]]=>([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) [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]=>[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) [2,2,1,1,1]=>[1,2,4]=>[[5,2,1],[1]]=>([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) [2,2,1,2]=>[1,2,3,1]=>[[4,4,2,1],[3,1]]=>([(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]=>[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) [2,3,1,1]=>[1,2,1,3]=>[[4,2,2,1],[1,1]]=>([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) [2,3,2]=>[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) [2,4,1]=>[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) [2,5]=>[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) [3,1,1,1,1]=>[1,1,5]=>[[5,1,1],[]]=>([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) [3,1,1,2]=>[1,1,4,1]=>[[4,4,1,1],[3]]=>([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) [3,1,2,1]=>[1,1,3,2]=>[[4,3,1,1],[2]]=>([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) [3,1,3]=>[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) [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]=>[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) [3,3,1]=>[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) [3,4]=>[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) [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]=>[1,1,1,3,1]=>[[3,3,1,1,1],[2]]=>([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) [4,2,1]=>[1,1,1,2,2]=>[[3,2,1,1,1],[1]]=>([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) [4,3]=>[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) [5,1,1]=>[1,1,1,1,3]=>[[3,1,1,1,1],[]]=>([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) [5,2]=>[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) [6,1]=>[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) [7]=>[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) [1,1,1,1,1,1,1,1]=>[8]=>[[8],[]]=>([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) [1,1,1,1,1,1,2]=>[7,1]=>[[7,7],[6]]=>([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8) [1,1,1,1,1,3]=>[6,1,1]=>[[6,6,6],[5,5]]=>([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8) [1,1,1,1,4]=>[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) [1,1,1,2,1,1,1]=>[4,4]=>[[7,4],[3]]=>([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8) [1,1,1,2,1,2]=>[4,3,1]=>[[6,6,4],[5,3]]=>([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8) [1,1,1,2,2,1]=>[4,2,2]=>[[6,5,4],[4,3]]=>([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8) [1,1,1,2,3]=>[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) [1,1,1,3,1,1]=>[4,1,3]=>[[6,4,4],[3,3]]=>([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8) [1,1,1,3,2]=>[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) [1,1,1,4,1]=>[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) [1,1,1,5]=>[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) [1,1,2,1,1,2]=>[3,4,1]=>[[6,6,3],[5,2]]=>([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8) [1,1,2,1,2,1]=>[3,3,2]=>[[6,5,3],[4,2]]=>([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8) [1,1,2,1,3]=>[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) [1,1,2,2,1,1]=>[3,2,3]=>[[6,4,3],[3,2]]=>([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8) [1,1,2,2,2]=>[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) [1,1,2,3,1]=>[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) [1,1,2,4]=>[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) [1,1,3,1,2]=>[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) [1,1,3,2,1]=>[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) [1,1,3,3]=>[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) [1,1,4,1,1]=>[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) [1,1,4,2]=>[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) [1,1,6]=>[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) [1,2,1,1,2,1]=>[2,4,2]=>[[6,5,2],[4,1]]=>([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8) [1,2,1,1,3]=>[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) [1,2,1,2,1,1]=>[2,3,3]=>[[6,4,2],[3,1]]=>([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8) [1,2,1,2,2]=>[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) [1,2,1,3,1]=>[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) [1,2,1,4]=>[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) [1,2,2,1,2]=>[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) [1,2,2,2,1]=>[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) [1,2,2,3]=>[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) [1,2,3,1,1]=>[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) [1,2,3,2]=>[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) [1,2,4,1]=>[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) [1,2,5]=>[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) [1,3,1,2,1]=>[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) [1,3,1,3]=>[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) [1,3,2,1,1]=>[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) [1,3,2,2]=>[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) [1,3,3,1]=>[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) [1,3,4]=>[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) [1,4,1,2]=>[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) [1,4,2,1]=>[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) [1,4,3]=>[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) [1,5,2]=>[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) [1,6,1]=>[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) [1,7]=>[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) [2,1,1,1,1,1,1]=>[1,7]=>[[7,1],[]]=>([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) [2,1,1,2,1,1]=>[1,4,3]=>[[6,4,1],[3]]=>([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8) [2,1,1,2,2]=>[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) [2,1,1,3,1]=>[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) [2,1,1,4]=>[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) [2,1,2,1,2]=>[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) [2,1,2,2,1]=>[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) [2,1,2,3]=>[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) [2,1,3,1,1]=>[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) [2,1,3,2]=>[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) [2,1,4,1]=>[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) [2,2,1,2,1]=>[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) [2,2,1,3]=>[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) [2,2,2,1,1]=>[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) [2,2,2,2]=>[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) [2,2,3,1]=>[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) [2,2,4]=>[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) [2,3,1,2]=>[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) [2,3,2,1]=>[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) [2,3,3]=>[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) [2,4,2]=>[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) [2,5,1]=>[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) [2,6]=>[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) [3,1,1,1,1,1]=>[1,1,6]=>[[6,1,1],[]]=>([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) [3,1,2,1,1]=>[1,1,3,3]=>[[5,3,1,1],[2]]=>([(0,6),(0,7),(1,4),(1,5),(4,7),(5,3),(6,2)],8) [3,1,2,2]=>[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) [3,1,3,1]=>[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) [3,2,1,2]=>[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) [3,2,2,1]=>[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) [3,2,3]=>[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) [3,3,2]=>[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) [3,4,1]=>[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) [3,5]=>[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) [4,1,1,1,1]=>[1,1,1,5]=>[[5,1,1,1],[]]=>([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) [4,1,2,1]=>[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) [4,2,2]=>[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) [4,3,1]=>[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) [4,4]=>[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) [5,1,1,1]=>[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) [5,2,1]=>[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) [5,3]=>[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) [6,1,1]=>[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) [6,2]=>[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) [7,1]=>[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) [8]=>[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) [1,1,1,1,1,1,1,1,1]=>[9]=>[[9],[]]=>([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) [1,1,1,1,1,1,1,2]=>[8,1]=>[[8,8],[7]]=>([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9) [1,1,1,1,1,1,3]=>[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) [1,1,1,1,1,4]=>[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) [1,1,1,1,5]=>[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]=>[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) [1,1,7]=>[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) [1,8]=>[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) [2,1,1,1,1,1,1,1]=>[1,8]=>[[8,1],[]]=>([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) [3,1,1,1,1,1,1]=>[1,1,7]=>[[7,1,1],[]]=>([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) [4,1,1,1,1,1]=>[1,1,1,6]=>[[6,1,1,1],[]]=>([(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]=>[1,1,1,1,1,4]=>[[4,1,1,1,1,1],[]]=>([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) [7,1,1]=>[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) [8,1]=>[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) [9]=>[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) [1,1,1,1,1,1,1,1,1,1]=>[10]=>[[10],[]]=>([(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]=>[9,1]=>[[9,9],[8]]=>([(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]=>[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) [1,1,1,1,1,1,4]=>[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) [1,1,1,1,1,5]=>[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) [1,1,1,1,6]=>[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) [1,1,1,7]=>[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) [1,1,8]=>[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) [1,9]=>[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) [2,1,1,1,1,1,1,1,1]=>[1,9]=>[[9,1],[]]=>([(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]=>[1,1,8]=>[[8,1,1],[]]=>([(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]=>[1,1,1,7]=>[[7,1,1,1],[]]=>([(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]=>[1,1,1,1,6]=>[[6,1,1,1,1],[]]=>([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) [6,1,1,1,1]=>[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) [7,1,1,1]=>[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) [8,1,1]=>[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) [9,1]=>[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) [10]=>[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)
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 $I$.
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.