Identifier
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Images
=>
Cc0005;cc-rep-0Cc0028;cc-rep-1
[1,0]=>[[1],[]]=>[1]=>[1] [1,0,1,0]=>[[1,1],[]]=>[1,1]=>[1,1] [1,1,0,0]=>[[2],[]]=>[2]=>[2] [1,0,1,0,1,0]=>[[1,1,1],[]]=>[1,1,1]=>[1,1,1] [1,0,1,1,0,0]=>[[2,1],[]]=>[2,1]=>[1,2] [1,1,0,0,1,0]=>[[2,2],[1]]=>[1,2]=>[2,1] [1,1,0,1,0,0]=>[[3],[]]=>[3]=>[3] [1,1,1,0,0,0]=>[[2,2],[]]=>[2,2]=>[2,2] [1,0,1,0,1,0,1,0]=>[[1,1,1,1],[]]=>[1,1,1,1]=>[1,1,1,1] [1,0,1,0,1,1,0,0]=>[[2,1,1],[]]=>[2,1,1]=>[1,1,2] [1,0,1,1,0,0,1,0]=>[[2,2,1],[1]]=>[1,2,1]=>[1,2,1] [1,0,1,1,0,1,0,0]=>[[3,1],[]]=>[3,1]=>[1,3] [1,0,1,1,1,0,0,0]=>[[2,2,1],[]]=>[2,2,1]=>[1,2,2] [1,1,0,0,1,0,1,0]=>[[2,2,2],[1,1]]=>[1,1,2]=>[2,1,1] [1,1,0,0,1,1,0,0]=>[[3,2],[1]]=>[2,2]=>[2,2] [1,1,0,1,0,0,1,0]=>[[3,3],[2]]=>[1,3]=>[3,1] [1,1,0,1,0,1,0,0]=>[[4],[]]=>[4]=>[4] [1,1,0,1,1,0,0,0]=>[[3,3],[1]]=>[2,3]=>[3,2] [1,1,1,0,0,0,1,0]=>[[2,2,2],[1]]=>[1,2,2]=>[2,2,1] [1,1,1,0,0,1,0,0]=>[[3,2],[]]=>[3,2]=>[2,3] [1,1,1,0,1,0,0,0]=>[[2,2,2],[]]=>[2,2,2]=>[2,2,2] [1,1,1,1,0,0,0,0]=>[[3,3],[]]=>[3,3]=>[3,3] [1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1],[]]=>[1,1,1,1,1]=>[1,1,1,1,1] [1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1],[]]=>[2,1,1,1]=>[1,1,1,2] [1,0,1,0,1,1,0,0,1,0]=>[[2,2,1,1],[1]]=>[1,2,1,1]=>[1,1,2,1] [1,0,1,0,1,1,0,1,0,0]=>[[3,1,1],[]]=>[3,1,1]=>[1,1,3] [1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1],[]]=>[2,2,1,1]=>[1,1,2,2] [1,0,1,1,0,0,1,0,1,0]=>[[2,2,2,1],[1,1]]=>[1,1,2,1]=>[1,2,1,1] [1,0,1,1,0,0,1,1,0,0]=>[[3,2,1],[1]]=>[2,2,1]=>[1,2,2] [1,0,1,1,0,1,0,0,1,0]=>[[3,3,1],[2]]=>[1,3,1]=>[1,3,1] [1,0,1,1,0,1,0,1,0,0]=>[[4,1],[]]=>[4,1]=>[1,4] [1,0,1,1,0,1,1,0,0,0]=>[[3,3,1],[1]]=>[2,3,1]=>[1,3,2] [1,0,1,1,1,0,0,0,1,0]=>[[2,2,2,1],[1]]=>[1,2,2,1]=>[1,2,2,1] [1,0,1,1,1,0,0,1,0,0]=>[[3,2,1],[]]=>[3,2,1]=>[1,2,3] [1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1],[]]=>[2,2,2,1]=>[1,2,2,2] [1,0,1,1,1,1,0,0,0,0]=>[[3,3,1],[]]=>[3,3,1]=>[1,3,3] [1,1,0,0,1,0,1,0,1,0]=>[[2,2,2,2],[1,1,1]]=>[1,1,1,2]=>[2,1,1,1] [1,1,0,0,1,0,1,1,0,0]=>[[3,2,2],[1,1]]=>[2,1,2]=>[2,1,2] [1,1,0,0,1,1,0,0,1,0]=>[[3,3,2],[2,1]]=>[1,2,2]=>[2,2,1] [1,1,0,0,1,1,0,1,0,0]=>[[4,2],[1]]=>[3,2]=>[2,3] [1,1,0,0,1,1,1,0,0,0]=>[[3,3,2],[1,1]]=>[2,2,2]=>[2,2,2] [1,1,0,1,0,0,1,0,1,0]=>[[3,3,3],[2,2]]=>[1,1,3]=>[3,1,1] [1,1,0,1,0,0,1,1,0,0]=>[[4,3],[2]]=>[2,3]=>[3,2] [1,1,0,1,0,1,0,0,1,0]=>[[4,4],[3]]=>[1,4]=>[4,1] [1,1,0,1,0,1,0,1,0,0]=>[[5],[]]=>[5]=>[5] [1,1,0,1,0,1,1,0,0,0]=>[[4,4],[2]]=>[2,4]=>[4,2] [1,1,0,1,1,0,0,0,1,0]=>[[3,3,3],[2,1]]=>[1,2,3]=>[3,2,1] [1,1,0,1,1,0,0,1,0,0]=>[[4,3],[1]]=>[3,3]=>[3,3] [1,1,0,1,1,0,1,0,0,0]=>[[3,3,3],[1,1]]=>[2,2,3]=>[3,2,2] [1,1,0,1,1,1,0,0,0,0]=>[[4,4],[1]]=>[3,4]=>[4,3] [1,1,1,0,0,0,1,0,1,0]=>[[2,2,2,2],[1,1]]=>[1,1,2,2]=>[2,2,1,1] [1,1,1,0,0,0,1,1,0,0]=>[[3,2,2],[1]]=>[2,2,2]=>[2,2,2] [1,1,1,0,0,1,0,0,1,0]=>[[3,3,2],[2]]=>[1,3,2]=>[2,3,1] [1,1,1,0,0,1,0,1,0,0]=>[[4,2],[]]=>[4,2]=>[2,4] [1,1,1,0,0,1,1,0,0,0]=>[[3,3,2],[1]]=>[2,3,2]=>[2,3,2] [1,1,1,0,1,0,0,0,1,0]=>[[2,2,2,2],[1]]=>[1,2,2,2]=>[2,2,2,1] [1,1,1,0,1,0,0,1,0,0]=>[[3,2,2],[]]=>[3,2,2]=>[2,2,3] [1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2],[]]=>[2,2,2,2]=>[2,2,2,2] [1,1,1,0,1,1,0,0,0,0]=>[[3,3,2],[]]=>[3,3,2]=>[2,3,3] [1,1,1,1,0,0,0,0,1,0]=>[[3,3,3],[2]]=>[1,3,3]=>[3,3,1] [1,1,1,1,0,0,0,1,0,0]=>[[4,3],[]]=>[4,3]=>[3,4] [1,1,1,1,0,0,1,0,0,0]=>[[3,3,3],[1]]=>[2,3,3]=>[3,3,2] [1,1,1,1,0,1,0,0,0,0]=>[[4,4],[]]=>[4,4]=>[4,4] [1,1,1,1,1,0,0,0,0,0]=>[[3,3,3],[]]=>[3,3,3]=>[3,3,3] [1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1],[]]=>[1,1,1,1,1,1]=>[1,1,1,1,1,1] [1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1],[]]=>[2,1,1,1,1]=>[1,1,1,1,2] [1,0,1,0,1,0,1,1,0,0,1,0]=>[[2,2,1,1,1],[1]]=>[1,2,1,1,1]=>[1,1,1,2,1] [1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1],[]]=>[3,1,1,1]=>[1,1,1,3] [1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1],[]]=>[2,2,1,1,1]=>[1,1,1,2,2] [1,0,1,0,1,1,0,0,1,0,1,0]=>[[2,2,2,1,1],[1,1]]=>[1,1,2,1,1]=>[1,1,2,1,1] [1,0,1,0,1,1,0,0,1,1,0,0]=>[[3,2,1,1],[1]]=>[2,2,1,1]=>[1,1,2,2] [1,0,1,0,1,1,0,1,0,0,1,0]=>[[3,3,1,1],[2]]=>[1,3,1,1]=>[1,1,3,1] [1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1],[]]=>[4,1,1]=>[1,1,4] [1,0,1,0,1,1,0,1,1,0,0,0]=>[[3,3,1,1],[1]]=>[2,3,1,1]=>[1,1,3,2] [1,0,1,0,1,1,1,0,0,0,1,0]=>[[2,2,2,1,1],[1]]=>[1,2,2,1,1]=>[1,1,2,2,1] [1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1],[]]=>[3,2,1,1]=>[1,1,2,3] [1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1],[]]=>[2,2,2,1,1]=>[1,1,2,2,2] [1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1],[]]=>[3,3,1,1]=>[1,1,3,3] [1,0,1,1,0,0,1,0,1,0,1,0]=>[[2,2,2,2,1],[1,1,1]]=>[1,1,1,2,1]=>[1,2,1,1,1] [1,0,1,1,0,0,1,0,1,1,0,0]=>[[3,2,2,1],[1,1]]=>[2,1,2,1]=>[1,2,1,2] [1,0,1,1,0,0,1,1,0,0,1,0]=>[[3,3,2,1],[2,1]]=>[1,2,2,1]=>[1,2,2,1] [1,0,1,1,0,0,1,1,0,1,0,0]=>[[4,2,1],[1]]=>[3,2,1]=>[1,2,3] [1,0,1,1,0,0,1,1,1,0,0,0]=>[[3,3,2,1],[1,1]]=>[2,2,2,1]=>[1,2,2,2] [1,0,1,1,0,1,0,0,1,0,1,0]=>[[3,3,3,1],[2,2]]=>[1,1,3,1]=>[1,3,1,1] [1,0,1,1,0,1,0,0,1,1,0,0]=>[[4,3,1],[2]]=>[2,3,1]=>[1,3,2] [1,0,1,1,0,1,0,1,0,0,1,0]=>[[4,4,1],[3]]=>[1,4,1]=>[1,4,1] [1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1],[]]=>[5,1]=>[1,5] [1,0,1,1,0,1,0,1,1,0,0,0]=>[[4,4,1],[2]]=>[2,4,1]=>[1,4,2] [1,0,1,1,0,1,1,0,0,0,1,0]=>[[3,3,3,1],[2,1]]=>[1,2,3,1]=>[1,3,2,1] [1,0,1,1,0,1,1,0,0,1,0,0]=>[[4,3,1],[1]]=>[3,3,1]=>[1,3,3] [1,0,1,1,0,1,1,0,1,0,0,0]=>[[3,3,3,1],[1,1]]=>[2,2,3,1]=>[1,3,2,2] [1,0,1,1,0,1,1,1,0,0,0,0]=>[[4,4,1],[1]]=>[3,4,1]=>[1,4,3] [1,0,1,1,1,0,0,0,1,0,1,0]=>[[2,2,2,2,1],[1,1]]=>[1,1,2,2,1]=>[1,2,2,1,1] [1,0,1,1,1,0,0,0,1,1,0,0]=>[[3,2,2,1],[1]]=>[2,2,2,1]=>[1,2,2,2] [1,0,1,1,1,0,0,1,0,0,1,0]=>[[3,3,2,1],[2]]=>[1,3,2,1]=>[1,2,3,1] [1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1],[]]=>[4,2,1]=>[1,2,4] [1,0,1,1,1,0,0,1,1,0,0,0]=>[[3,3,2,1],[1]]=>[2,3,2,1]=>[1,2,3,2] [1,0,1,1,1,0,1,0,0,0,1,0]=>[[2,2,2,2,1],[1]]=>[1,2,2,2,1]=>[1,2,2,2,1] [1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1],[]]=>[3,2,2,1]=>[1,2,2,3] [1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1],[]]=>[2,2,2,2,1]=>[1,2,2,2,2] [1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1],[]]=>[3,3,2,1]=>[1,2,3,3] [1,0,1,1,1,1,0,0,0,0,1,0]=>[[3,3,3,1],[2]]=>[1,3,3,1]=>[1,3,3,1] [1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1],[]]=>[4,3,1]=>[1,3,4] [1,0,1,1,1,1,0,0,1,0,0,0]=>[[3,3,3,1],[1]]=>[2,3,3,1]=>[1,3,3,2] [1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1],[]]=>[4,4,1]=>[1,4,4] [1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1],[]]=>[3,3,3,1]=>[1,3,3,3] [1,1,0,0,1,0,1,0,1,0,1,0]=>[[2,2,2,2,2],[1,1,1,1]]=>[1,1,1,1,2]=>[2,1,1,1,1] [1,1,0,0,1,0,1,0,1,1,0,0]=>[[3,2,2,2],[1,1,1]]=>[2,1,1,2]=>[2,1,1,2] [1,1,0,0,1,0,1,1,0,0,1,0]=>[[3,3,2,2],[2,1,1]]=>[1,2,1,2]=>[2,1,2,1] [1,1,0,0,1,0,1,1,0,1,0,0]=>[[4,2,2],[1,1]]=>[3,1,2]=>[2,1,3] [1,1,0,0,1,0,1,1,1,0,0,0]=>[[3,3,2,2],[1,1,1]]=>[2,2,1,2]=>[2,1,2,2] [1,1,0,0,1,1,0,0,1,0,1,0]=>[[3,3,3,2],[2,2,1]]=>[1,1,2,2]=>[2,2,1,1] [1,1,0,0,1,1,0,0,1,1,0,0]=>[[4,3,2],[2,1]]=>[2,2,2]=>[2,2,2] [1,1,0,0,1,1,0,1,0,0,1,0]=>[[4,4,2],[3,1]]=>[1,3,2]=>[2,3,1] [1,1,0,0,1,1,0,1,0,1,0,0]=>[[5,2],[1]]=>[4,2]=>[2,4] [1,1,0,0,1,1,0,1,1,0,0,0]=>[[4,4,2],[2,1]]=>[2,3,2]=>[2,3,2] [1,1,0,0,1,1,1,0,0,0,1,0]=>[[3,3,3,2],[2,1,1]]=>[1,2,2,2]=>[2,2,2,1] [1,1,0,0,1,1,1,0,0,1,0,0]=>[[4,3,2],[1,1]]=>[3,2,2]=>[2,2,3] [1,1,0,0,1,1,1,0,1,0,0,0]=>[[3,3,3,2],[1,1,1]]=>[2,2,2,2]=>[2,2,2,2] [1,1,0,0,1,1,1,1,0,0,0,0]=>[[4,4,2],[1,1]]=>[3,3,2]=>[2,3,3] [1,1,0,1,0,0,1,0,1,0,1,0]=>[[3,3,3,3],[2,2,2]]=>[1,1,1,3]=>[3,1,1,1] [1,1,0,1,0,0,1,0,1,1,0,0]=>[[4,3,3],[2,2]]=>[2,1,3]=>[3,1,2] [1,1,0,1,0,0,1,1,0,0,1,0]=>[[4,4,3],[3,2]]=>[1,2,3]=>[3,2,1] [1,1,0,1,0,0,1,1,0,1,0,0]=>[[5,3],[2]]=>[3,3]=>[3,3] [1,1,0,1,0,0,1,1,1,0,0,0]=>[[4,4,3],[2,2]]=>[2,2,3]=>[3,2,2] [1,1,0,1,0,1,0,0,1,0,1,0]=>[[4,4,4],[3,3]]=>[1,1,4]=>[4,1,1] [1,1,0,1,0,1,0,0,1,1,0,0]=>[[5,4],[3]]=>[2,4]=>[4,2] [1,1,0,1,0,1,0,1,0,0,1,0]=>[[5,5],[4]]=>[1,5]=>[5,1] [1,1,0,1,0,1,0,1,0,1,0,0]=>[[6],[]]=>[6]=>[6] [1,1,0,1,0,1,0,1,1,0,0,0]=>[[5,5],[3]]=>[2,5]=>[5,2] [1,1,0,1,0,1,1,0,0,0,1,0]=>[[4,4,4],[3,2]]=>[1,2,4]=>[4,2,1] [1,1,0,1,0,1,1,0,0,1,0,0]=>[[5,4],[2]]=>[3,4]=>[4,3] [1,1,0,1,0,1,1,0,1,0,0,0]=>[[4,4,4],[2,2]]=>[2,2,4]=>[4,2,2] [1,1,0,1,0,1,1,1,0,0,0,0]=>[[5,5],[2]]=>[3,5]=>[5,3] [1,1,0,1,1,0,0,0,1,0,1,0]=>[[3,3,3,3],[2,2,1]]=>[1,1,2,3]=>[3,2,1,1] [1,1,0,1,1,0,0,0,1,1,0,0]=>[[4,3,3],[2,1]]=>[2,2,3]=>[3,2,2] [1,1,0,1,1,0,0,1,0,0,1,0]=>[[4,4,3],[3,1]]=>[1,3,3]=>[3,3,1] [1,1,0,1,1,0,0,1,0,1,0,0]=>[[5,3],[1]]=>[4,3]=>[3,4] [1,1,0,1,1,0,0,1,1,0,0,0]=>[[4,4,3],[2,1]]=>[2,3,3]=>[3,3,2] [1,1,0,1,1,0,1,0,0,0,1,0]=>[[3,3,3,3],[2,1,1]]=>[1,2,2,3]=>[3,2,2,1] [1,1,0,1,1,0,1,0,0,1,0,0]=>[[4,3,3],[1,1]]=>[3,2,3]=>[3,2,3] [1,1,0,1,1,0,1,0,1,0,0,0]=>[[3,3,3,3],[1,1,1]]=>[2,2,2,3]=>[3,2,2,2] [1,1,0,1,1,0,1,1,0,0,0,0]=>[[4,4,3],[1,1]]=>[3,3,3]=>[3,3,3] [1,1,0,1,1,1,0,0,0,0,1,0]=>[[4,4,4],[3,1]]=>[1,3,4]=>[4,3,1] [1,1,0,1,1,1,0,0,0,1,0,0]=>[[5,4],[1]]=>[4,4]=>[4,4] [1,1,0,1,1,1,0,0,1,0,0,0]=>[[4,4,4],[2,1]]=>[2,3,4]=>[4,3,2] [1,1,0,1,1,1,0,1,0,0,0,0]=>[[5,5],[1]]=>[4,5]=>[5,4] [1,1,0,1,1,1,1,0,0,0,0,0]=>[[4,4,4],[1,1]]=>[3,3,4]=>[4,3,3] [1,1,1,0,0,0,1,0,1,0,1,0]=>[[2,2,2,2,2],[1,1,1]]=>[1,1,1,2,2]=>[2,2,1,1,1] [1,1,1,0,0,0,1,0,1,1,0,0]=>[[3,2,2,2],[1,1]]=>[2,1,2,2]=>[2,2,1,2] [1,1,1,0,0,0,1,1,0,0,1,0]=>[[3,3,2,2],[2,1]]=>[1,2,2,2]=>[2,2,2,1] [1,1,1,0,0,0,1,1,0,1,0,0]=>[[4,2,2],[1]]=>[3,2,2]=>[2,2,3] [1,1,1,0,0,0,1,1,1,0,0,0]=>[[3,3,2,2],[1,1]]=>[2,2,2,2]=>[2,2,2,2] [1,1,1,0,0,1,0,0,1,0,1,0]=>[[3,3,3,2],[2,2]]=>[1,1,3,2]=>[2,3,1,1] [1,1,1,0,0,1,0,0,1,1,0,0]=>[[4,3,2],[2]]=>[2,3,2]=>[2,3,2] [1,1,1,0,0,1,0,1,0,0,1,0]=>[[4,4,2],[3]]=>[1,4,2]=>[2,4,1] [1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2],[]]=>[5,2]=>[2,5] [1,1,1,0,0,1,0,1,1,0,0,0]=>[[4,4,2],[2]]=>[2,4,2]=>[2,4,2] [1,1,1,0,0,1,1,0,0,0,1,0]=>[[3,3,3,2],[2,1]]=>[1,2,3,2]=>[2,3,2,1] [1,1,1,0,0,1,1,0,0,1,0,0]=>[[4,3,2],[1]]=>[3,3,2]=>[2,3,3] [1,1,1,0,0,1,1,0,1,0,0,0]=>[[3,3,3,2],[1,1]]=>[2,2,3,2]=>[2,3,2,2] [1,1,1,0,0,1,1,1,0,0,0,0]=>[[4,4,2],[1]]=>[3,4,2]=>[2,4,3] [1,1,1,0,1,0,0,0,1,0,1,0]=>[[2,2,2,2,2],[1,1]]=>[1,1,2,2,2]=>[2,2,2,1,1] [1,1,1,0,1,0,0,0,1,1,0,0]=>[[3,2,2,2],[1]]=>[2,2,2,2]=>[2,2,2,2] [1,1,1,0,1,0,0,1,0,0,1,0]=>[[3,3,2,2],[2]]=>[1,3,2,2]=>[2,2,3,1] [1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2],[]]=>[4,2,2]=>[2,2,4] [1,1,1,0,1,0,0,1,1,0,0,0]=>[[3,3,2,2],[1]]=>[2,3,2,2]=>[2,2,3,2] [1,1,1,0,1,0,1,0,0,0,1,0]=>[[2,2,2,2,2],[1]]=>[1,2,2,2,2]=>[2,2,2,2,1] [1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2],[]]=>[3,2,2,2]=>[2,2,2,3] [1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2],[]]=>[2,2,2,2,2]=>[2,2,2,2,2] [1,1,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2],[]]=>[3,3,2,2]=>[2,2,3,3] [1,1,1,0,1,1,0,0,0,0,1,0]=>[[3,3,3,2],[2]]=>[1,3,3,2]=>[2,3,3,1] [1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2],[]]=>[4,3,2]=>[2,3,4] [1,1,1,0,1,1,0,0,1,0,0,0]=>[[3,3,3,2],[1]]=>[2,3,3,2]=>[2,3,3,2] [1,1,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2],[]]=>[4,4,2]=>[2,4,4] [1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2],[]]=>[3,3,3,2]=>[2,3,3,3] [1,1,1,1,0,0,0,0,1,0,1,0]=>[[3,3,3,3],[2,2]]=>[1,1,3,3]=>[3,3,1,1] [1,1,1,1,0,0,0,0,1,1,0,0]=>[[4,3,3],[2]]=>[2,3,3]=>[3,3,2] [1,1,1,1,0,0,0,1,0,0,1,0]=>[[4,4,3],[3]]=>[1,4,3]=>[3,4,1] [1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3],[]]=>[5,3]=>[3,5] [1,1,1,1,0,0,0,1,1,0,0,0]=>[[4,4,3],[2]]=>[2,4,3]=>[3,4,2] [1,1,1,1,0,0,1,0,0,0,1,0]=>[[3,3,3,3],[2,1]]=>[1,2,3,3]=>[3,3,2,1] [1,1,1,1,0,0,1,0,0,1,0,0]=>[[4,3,3],[1]]=>[3,3,3]=>[3,3,3] [1,1,1,1,0,0,1,0,1,0,0,0]=>[[3,3,3,3],[1,1]]=>[2,2,3,3]=>[3,3,2,2] [1,1,1,1,0,0,1,1,0,0,0,0]=>[[4,4,3],[1]]=>[3,4,3]=>[3,4,3] [1,1,1,1,0,1,0,0,0,0,1,0]=>[[4,4,4],[3]]=>[1,4,4]=>[4,4,1] [1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4],[]]=>[5,4]=>[4,5] [1,1,1,1,0,1,0,0,1,0,0,0]=>[[4,4,4],[2]]=>[2,4,4]=>[4,4,2] [1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5],[]]=>[5,5]=>[5,5] [1,1,1,1,0,1,1,0,0,0,0,0]=>[[4,4,4],[1]]=>[3,4,4]=>[4,4,3] [1,1,1,1,1,0,0,0,0,0,1,0]=>[[3,3,3,3],[2]]=>[1,3,3,3]=>[3,3,3,1] [1,1,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3],[]]=>[4,3,3]=>[3,3,4] [1,1,1,1,1,0,0,0,1,0,0,0]=>[[3,3,3,3],[1]]=>[2,3,3,3]=>[3,3,3,2] [1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3],[]]=>[4,4,3]=>[3,4,4] [1,1,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3],[]]=>[3,3,3,3]=>[3,3,3,3] [1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4],[]]=>[4,4,4]=>[4,4,4] [1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1],[]]=>[1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1] [1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1],[]]=>[2,1,1,1,1,1]=>[1,1,1,1,1,2] [1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[2,2,1,1,1,1],[1]]=>[1,2,1,1,1,1]=>[1,1,1,1,2,1] [1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1],[]]=>[3,1,1,1,1]=>[1,1,1,1,3] [1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1],[]]=>[2,2,1,1,1,1]=>[1,1,1,1,2,2] [1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[2,2,2,1,1,1],[1,1]]=>[1,1,2,1,1,1]=>[1,1,1,2,1,1] [1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[3,2,1,1,1],[1]]=>[2,2,1,1,1]=>[1,1,1,2,2] [1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[[3,3,1,1,1],[2]]=>[1,3,1,1,1]=>[1,1,1,3,1] [1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1],[]]=>[4,1,1,1]=>[1,1,1,4] [1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1],[]]=>[3,2,1,1,1]=>[1,1,1,2,3] [1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1],[]]=>[2,2,2,1,1,1]=>[1,1,1,2,2,2] [1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1],[]]=>[3,3,1,1,1]=>[1,1,1,3,3] [1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[2,2,2,2,1,1],[1,1,1]]=>[1,1,1,2,1,1]=>[1,1,2,1,1,1] [1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[3,2,2,1,1],[1,1]]=>[2,1,2,1,1]=>[1,1,2,1,2] [1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[3,3,2,1,1],[2,1]]=>[1,2,2,1,1]=>[1,1,2,2,1] [1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[[4,2,1,1],[1]]=>[3,2,1,1]=>[1,1,2,3] [1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[[3,3,3,1,1],[2,2]]=>[1,1,3,1,1]=>[1,1,3,1,1] [1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>[[4,3,1,1],[2]]=>[2,3,1,1]=>[1,1,3,2] [1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[[4,4,1,1],[3]]=>[1,4,1,1]=>[1,1,4,1] [1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1],[]]=>[5,1,1]=>[1,1,5] [1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1],[]]=>[4,2,1,1]=>[1,1,2,4] [1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1,1],[]]=>[3,2,2,1,1]=>[1,1,2,2,3] [1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1,1],[]]=>[2,2,2,2,1,1]=>[1,1,2,2,2,2] [1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1,1],[]]=>[3,3,2,1,1]=>[1,1,2,3,3] [1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1,1],[]]=>[4,3,1,1]=>[1,1,3,4] [1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1,1],[]]=>[4,4,1,1]=>[1,1,4,4] [1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[2,2,2,2,2,1],[1,1,1,1]]=>[1,1,1,1,2,1]=>[1,2,1,1,1,1] [1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[3,2,2,2,1],[1,1,1]]=>[2,1,1,2,1]=>[1,2,1,1,2] [1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[3,3,2,2,1],[2,1,1]]=>[1,2,1,2,1]=>[1,2,1,2,1] [1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>[[4,2,2,1],[1,1]]=>[3,1,2,1]=>[1,2,1,3] [1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[3,3,3,2,1],[2,2,1]]=>[1,1,2,2,1]=>[1,2,2,1,1] [1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[4,3,2,1],[2,1]]=>[2,2,2,1]=>[1,2,2,2] [1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[[4,4,2,1],[3,1]]=>[1,3,2,1]=>[1,2,3,1] [1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[[5,2,1],[1]]=>[4,2,1]=>[1,2,4] [1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[[3,3,3,3,1],[2,2,2]]=>[1,1,1,3,1]=>[1,3,1,1,1] [1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[[4,3,3,1],[2,2]]=>[2,1,3,1]=>[1,3,1,2] [1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[[4,4,3,1],[3,2]]=>[1,2,3,1]=>[1,3,2,1] [1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>[[5,3,1],[2]]=>[3,3,1]=>[1,3,3] [1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>[[4,4,4,1],[3,3]]=>[1,1,4,1]=>[1,4,1,1] [1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>[[5,4,1],[3]]=>[2,4,1]=>[1,4,2] [1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[[5,5,1],[4]]=>[1,5,1]=>[1,5,1] [1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1],[]]=>[6,1]=>[1,6] [1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1],[]]=>[5,2,1]=>[1,2,5] [1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,1],[]]=>[4,2,2,1]=>[1,2,2,4] [1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,1],[]]=>[3,2,2,2,1]=>[1,2,2,2,3] [1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,1],[]]=>[4,3,2,1]=>[1,2,3,4] [1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,1],[]]=>[3,3,3,2,1]=>[1,2,3,3,3] [1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3,1],[]]=>[5,3,1]=>[1,3,5] [1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4,1],[]]=>[5,4,1]=>[1,4,5] [1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5,1],[]]=>[5,5,1]=>[1,5,5] [1,0,1,1,1,1,0,1,1,0,0,0,0,0]=>[[4,4,4,1],[1]]=>[3,4,4,1]=>[1,4,4,3] [1,0,1,1,1,1,1,0,0,0,1,0,0,0]=>[[3,3,3,3,1],[1]]=>[2,3,3,3,1]=>[1,3,3,3,2] [1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,1],[]]=>[4,4,3,1]=>[1,3,4,4] [1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[2,2,2,2,2,2],[1,1,1,1,1]]=>[1,1,1,1,1,2]=>[2,1,1,1,1,1] [1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[3,2,2,2,2],[1,1,1,1]]=>[2,1,1,1,2]=>[2,1,1,1,2] [1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[3,3,2,2,2],[2,1,1,1]]=>[1,2,1,1,2]=>[2,1,1,2,1] [1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>[[4,2,2,2],[1,1,1]]=>[3,1,1,2]=>[2,1,1,3] [1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[3,3,3,2,2],[2,2,1,1]]=>[1,1,2,1,2]=>[2,1,2,1,1] [1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[4,3,2,2],[2,1,1]]=>[2,2,1,2]=>[2,1,2,2] [1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>[[4,4,2,2],[3,1,1]]=>[1,3,1,2]=>[2,1,3,1] [1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>[[5,2,2],[1,1]]=>[4,1,2]=>[2,1,4] [1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[3,3,3,3,2],[2,2,2,1]]=>[1,1,1,2,2]=>[2,2,1,1,1] [1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[4,3,3,2],[2,2,1]]=>[2,1,2,2]=>[2,2,1,2] [1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[4,4,3,2],[3,2,1]]=>[1,2,2,2]=>[2,2,2,1] [1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[[5,3,2],[2,1]]=>[3,2,2]=>[2,2,3] [1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>[[4,4,4,2],[3,3,1]]=>[1,1,3,2]=>[2,3,1,1] [1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>[[5,4,2],[3,1]]=>[2,3,2]=>[2,3,2] [1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[[5,5,2],[4,1]]=>[1,4,2]=>[2,4,1] [1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[[6,2],[1]]=>[5,2]=>[2,5] [1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[4,4,4,2],[1,1,1]]=>[3,3,3,2]=>[2,3,3,3] [1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[[3,3,3,3,3],[2,2,2,2]]=>[1,1,1,1,3]=>[3,1,1,1,1] [1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[[4,3,3,3],[2,2,2]]=>[2,1,1,3]=>[3,1,1,2] [1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>[[4,4,3,3],[3,2,2]]=>[1,2,1,3]=>[3,1,2,1] [1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[[5,3,3],[2,2]]=>[3,1,3]=>[3,1,3] [1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[[4,4,4,3],[3,3,2]]=>[1,1,2,3]=>[3,2,1,1] [1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[[5,4,3],[3,2]]=>[2,2,3]=>[3,2,2] [1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>[[5,5,3],[4,2]]=>[1,3,3]=>[3,3,1] [1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[[6,3],[2]]=>[4,3]=>[3,4] [1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[[4,4,4,4],[3,3,3]]=>[1,1,1,4]=>[4,1,1,1] [1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>[[5,4,4],[3,3]]=>[2,1,4]=>[4,1,2] [1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[[5,5,4],[4,3]]=>[1,2,4]=>[4,2,1] [1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[[6,4],[3]]=>[3,4]=>[4,3] [1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[[5,5,5],[4,4]]=>[1,1,5]=>[5,1,1] [1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[[6,5],[4]]=>[2,5]=>[5,2] [1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[6,6],[5]]=>[1,6]=>[6,1] [1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7],[]]=>[7]=>[7] [1,1,0,1,0,1,0,1,0,1,1,0,0,0]=>[[6,6],[4]]=>[2,6]=>[6,2] [1,1,0,1,0,1,0,1,1,0,0,1,0,0]=>[[6,5],[3]]=>[3,5]=>[5,3] [1,1,0,1,0,1,0,1,1,0,1,0,0,0]=>[[5,5,5],[3,3]]=>[2,2,5]=>[5,2,2] [1,1,0,1,0,1,0,1,1,1,0,0,0,0]=>[[6,6],[3]]=>[3,6]=>[6,3] [1,1,0,1,0,1,1,0,0,1,0,1,0,0]=>[[6,4],[2]]=>[4,4]=>[4,4] [1,1,0,1,0,1,1,0,0,1,1,0,0,0]=>[[5,5,4],[3,2]]=>[2,3,4]=>[4,3,2] [1,1,0,1,0,1,1,0,1,0,0,1,0,0]=>[[5,4,4],[2,2]]=>[3,2,4]=>[4,2,3] [1,1,0,1,0,1,1,0,1,0,1,0,0,0]=>[[4,4,4,4],[2,2,2]]=>[2,2,2,4]=>[4,2,2,2] [1,1,0,1,0,1,1,0,1,1,0,0,0,0]=>[[5,5,4],[2,2]]=>[3,3,4]=>[4,3,3] [1,1,0,1,0,1,1,1,0,0,0,1,0,0]=>[[6,5],[2]]=>[4,5]=>[5,4] [1,1,0,1,0,1,1,1,0,0,1,0,0,0]=>[[5,5,5],[3,2]]=>[2,3,5]=>[5,3,2] [1,1,0,1,0,1,1,1,0,1,0,0,0,0]=>[[6,6],[2]]=>[4,6]=>[6,4] [1,1,0,1,1,0,0,1,0,1,0,1,0,0]=>[[6,3],[1]]=>[5,3]=>[3,5] [1,1,0,1,1,0,0,1,0,1,1,0,0,0]=>[[5,5,3],[3,1]]=>[2,4,3]=>[3,4,2] [1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>[[5,4,3],[2,1]]=>[3,3,3]=>[3,3,3] [1,1,0,1,1,0,0,1,1,0,1,0,0,0]=>[[4,4,4,3],[2,2,1]]=>[2,2,3,3]=>[3,3,2,2] [1,1,0,1,1,0,0,1,1,1,0,0,0,0]=>[[5,5,3],[2,1]]=>[3,4,3]=>[3,4,3] [1,1,0,1,1,0,1,0,0,1,0,1,0,0]=>[[5,3,3],[1,1]]=>[4,2,3]=>[3,2,4] [1,1,0,1,1,0,1,0,0,1,1,0,0,0]=>[[4,4,3,3],[2,1,1]]=>[2,3,2,3]=>[3,2,3,2] [1,1,0,1,1,0,1,0,1,0,0,1,0,0]=>[[4,3,3,3],[1,1,1]]=>[3,2,2,3]=>[3,2,2,3] [1,1,0,1,1,0,1,1,0,0,0,1,0,0]=>[[5,4,3],[1,1]]=>[4,3,3]=>[3,3,4] [1,1,0,1,1,0,1,1,0,0,1,0,0,0]=>[[4,4,4,3],[2,1,1]]=>[2,3,3,3]=>[3,3,3,2] [1,1,0,1,1,0,1,1,0,1,0,0,0,0]=>[[5,5,3],[1,1]]=>[4,4,3]=>[3,4,4] [1,1,0,1,1,0,1,1,1,0,0,0,0,0]=>[[4,4,4,3],[1,1,1]]=>[3,3,3,3]=>[3,3,3,3] [1,1,0,1,1,1,0,0,0,1,0,1,0,0]=>[[6,4],[1]]=>[5,4]=>[4,5] [1,1,0,1,1,1,0,0,0,1,1,0,0,0]=>[[5,5,4],[3,1]]=>[2,4,4]=>[4,4,2] [1,1,0,1,1,1,0,0,1,0,0,1,0,0]=>[[5,4,4],[2,1]]=>[3,3,4]=>[4,3,3] [1,1,0,1,1,1,0,0,1,1,0,0,0,0]=>[[5,5,4],[2,1]]=>[3,4,4]=>[4,4,3] [1,1,0,1,1,1,0,1,0,0,0,1,0,0]=>[[6,5],[1]]=>[5,5]=>[5,5] [1,1,0,1,1,1,0,1,1,0,0,0,0,0]=>[[5,5,5],[2,1]]=>[3,4,5]=>[5,4,3] [1,1,0,1,1,1,1,0,0,0,1,0,0,0]=>[[4,4,4,4],[2,1,1]]=>[2,3,3,4]=>[4,3,3,2] [1,1,0,1,1,1,1,0,0,1,0,0,0,0]=>[[5,5,4],[1,1]]=>[4,4,4]=>[4,4,4] [1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2],[]]=>[6,2]=>[2,6] [1,1,1,0,0,1,0,1,0,1,1,0,0,0]=>[[5,5,2],[3]]=>[2,5,2]=>[2,5,2] [1,1,1,0,0,1,0,1,1,0,0,1,0,0]=>[[5,4,2],[2]]=>[3,4,2]=>[2,4,3] [1,1,1,0,0,1,0,1,1,0,1,0,0,0]=>[[4,4,4,2],[2,2]]=>[2,2,4,2]=>[2,4,2,2] [1,1,1,0,0,1,0,1,1,1,0,0,0,0]=>[[5,5,2],[2]]=>[3,5,2]=>[2,5,3] [1,1,1,0,0,1,1,0,0,1,0,1,0,0]=>[[5,3,2],[1]]=>[4,3,2]=>[2,3,4] [1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[[4,4,3,2],[2,1]]=>[2,3,3,2]=>[2,3,3,2] [1,1,1,0,0,1,1,0,1,0,0,1,0,0]=>[[4,3,3,2],[1,1]]=>[3,2,3,2]=>[2,3,2,3] [1,1,1,0,0,1,1,0,1,1,0,0,0,0]=>[[4,4,3,2],[1,1]]=>[3,3,3,2]=>[2,3,3,3] [1,1,1,0,0,1,1,1,0,0,0,1,0,0]=>[[5,4,2],[1]]=>[4,4,2]=>[2,4,4] [1,1,1,0,0,1,1,1,1,0,0,0,0,0]=>[[4,4,4,2],[1,1]]=>[3,3,4,2]=>[2,4,3,3] [1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2],[]]=>[5,2,2]=>[2,2,5] [1,1,1,0,1,0,0,1,0,1,1,0,0,0]=>[[4,4,2,2],[2]]=>[2,4,2,2]=>[2,2,4,2] [1,1,1,0,1,0,0,1,1,0,0,1,0,0]=>[[4,3,2,2],[1]]=>[3,3,2,2]=>[2,2,3,3] [1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,2],[]]=>[4,2,2,2]=>[2,2,2,4] [1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2],[]]=>[2,2,2,2,2,2]=>[2,2,2,2,2,2] [1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,2],[]]=>[3,3,2,2,2]=>[2,2,2,3,3] [1,1,1,0,1,0,1,1,0,0,1,0,0,0]=>[[3,3,3,2,2],[1]]=>[2,3,3,2,2]=>[2,2,3,3,2] [1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,2],[]]=>[4,4,2,2]=>[2,2,4,4] [1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[[5,3,2],[]]=>[5,3,2]=>[2,3,5] [1,1,1,0,1,1,0,0,1,0,0,1,0,0]=>[[4,3,3,2],[1]]=>[3,3,3,2]=>[2,3,3,3] [1,1,1,0,1,1,0,0,1,0,1,0,0,0]=>[[3,3,3,3,2],[1,1]]=>[2,2,3,3,2]=>[2,3,3,2,2] [1,1,1,0,1,1,0,0,1,1,0,0,0,0]=>[[4,4,3,2],[1]]=>[3,4,3,2]=>[2,3,4,3] [1,1,1,0,1,1,0,1,0,0,1,0,0,0]=>[[4,4,4,2],[2]]=>[2,4,4,2]=>[2,4,4,2] [1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[[5,5,2],[]]=>[5,5,2]=>[2,5,5] [1,1,1,0,1,1,1,0,0,0,0,0,1,0]=>[[3,3,3,3,2],[2]]=>[1,3,3,3,2]=>[2,3,3,3,1] [1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,2],[]]=>[4,3,3,2]=>[2,3,3,4] [1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3],[]]=>[6,3]=>[3,6] [1,1,1,1,0,0,0,1,0,1,1,0,0,0]=>[[5,5,3],[3]]=>[2,5,3]=>[3,5,2] [1,1,1,1,0,0,0,1,1,0,0,1,0,0]=>[[5,4,3],[2]]=>[3,4,3]=>[3,4,3] [1,1,1,1,0,0,1,0,0,1,0,1,0,0]=>[[5,3,3],[1]]=>[4,3,3]=>[3,3,4] [1,1,1,1,0,0,1,0,0,1,1,0,0,0]=>[[4,4,3,3],[2,1]]=>[2,3,3,3]=>[3,3,3,2] [1,1,1,1,0,0,1,0,1,0,1,0,0,0]=>[[3,3,3,3,3],[1,1,1]]=>[2,2,2,3,3]=>[3,3,2,2,2] [1,1,1,1,0,0,1,0,1,1,0,0,0,0]=>[[4,4,3,3],[1,1]]=>[3,3,3,3]=>[3,3,3,3] [1,1,1,1,0,0,1,1,0,0,0,1,0,0]=>[[5,4,3],[1]]=>[4,4,3]=>[3,4,4] [1,1,1,1,0,0,1,1,0,0,1,0,0,0]=>[[4,4,4,3],[2,1]]=>[2,3,4,3]=>[3,4,3,2] [1,1,1,1,0,0,1,1,0,1,0,0,0,0]=>[[5,5,3],[1]]=>[4,5,3]=>[3,5,4] [1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[[6,4],[]]=>[6,4]=>[4,6] [1,1,1,1,0,1,0,0,1,0,0,1,0,0]=>[[5,4,4],[2]]=>[3,4,4]=>[4,4,3] [1,1,1,1,0,1,0,0,1,0,1,0,0,0]=>[[4,4,4,4],[2,2]]=>[2,2,4,4]=>[4,4,2,2] [1,1,1,1,0,1,0,0,1,1,0,0,0,0]=>[[5,5,4],[2]]=>[3,5,4]=>[4,5,3] [1,1,1,1,0,1,0,1,0,0,0,0,1,0]=>[[5,5,5],[4]]=>[1,5,5]=>[5,5,1] [1,1,1,1,0,1,0,1,0,0,1,0,0,0]=>[[5,5,5],[3]]=>[2,5,5]=>[5,5,2] [1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6],[]]=>[6,6]=>[6,6] [1,1,1,1,0,1,1,0,0,0,0,0,1,0]=>[[4,4,4,4],[3,1]]=>[1,3,4,4]=>[4,4,3,1] [1,1,1,1,0,1,1,0,0,0,0,1,0,0]=>[[5,4,4],[1]]=>[4,4,4]=>[4,4,4] [1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[4,3,3,3],[2]]=>[2,3,3,3]=>[3,3,3,2] [1,1,1,1,1,0,0,0,0,1,1,0,0,0]=>[[4,4,3,3],[2]]=>[2,4,3,3]=>[3,3,4,2] [1,1,1,1,1,0,0,0,1,0,0,0,1,0]=>[[3,3,3,3,3],[2,1]]=>[1,2,3,3,3]=>[3,3,3,2,1] [1,1,1,1,1,0,0,0,1,0,0,1,0,0]=>[[4,3,3,3],[1]]=>[3,3,3,3]=>[3,3,3,3] [1,1,1,1,1,0,0,1,0,0,0,0,1,0]=>[[4,4,4,3],[3]]=>[1,4,4,3]=>[3,4,4,1] [1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3],[]]=>[5,4,3]=>[3,4,5] [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1,1],[]]=>[1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1] [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1],[]]=>[2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2] [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1],[]]=>[3,1,1,1,1,1]=>[1,1,1,1,1,3] [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1,1],[]]=>[2,2,1,1,1,1,1]=>[1,1,1,1,1,2,2] [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1],[]]=>[4,1,1,1,1]=>[1,1,1,1,4] [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1,1],[]]=>[3,2,1,1,1,1]=>[1,1,1,1,2,3] [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1,1],[]]=>[2,2,2,1,1,1,1]=>[1,1,1,1,2,2,2] [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1,1],[]]=>[3,3,1,1,1,1]=>[1,1,1,1,3,3] [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1],[]]=>[5,1,1,1]=>[1,1,1,5] [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1,1],[]]=>[4,2,1,1,1]=>[1,1,1,2,4] [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1,1,1],[]]=>[3,2,2,1,1,1]=>[1,1,1,2,2,3] [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1,1,1],[]]=>[4,3,1,1,1]=>[1,1,1,3,4] [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1],[]]=>[6,1,1]=>[1,1,6] [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1,1],[]]=>[5,2,1,1]=>[1,1,2,5] [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,1,1],[]]=>[4,2,2,1,1]=>[1,1,2,2,4] [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3,1,1],[]]=>[5,3,1,1]=>[1,1,3,5] [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1],[]]=>[7,1]=>[1,7] [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,1],[]]=>[6,2,1]=>[1,2,6] [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2,1],[]]=>[5,2,2,1]=>[1,2,2,5] [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3,1],[]]=>[6,3,1]=>[1,3,6] [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8],[]]=>[8]=>[8] [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2],[]]=>[7,2]=>[2,7] [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,2],[]]=>[6,2,2]=>[2,2,6] [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[[7,3],[]]=>[7,3]=>[3,7] [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9],[]]=>[9]=>[9] [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,2],[]]=>[8,2]=>[2,8] [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[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] [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1,1],[]]=>[2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,2] [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1,1],[]]=>[3,1,1,1,1,1,1]=>[1,1,1,1,1,1,3] [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1,1],[]]=>[4,1,1,1,1,1]=>[1,1,1,1,1,4] [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1,1],[]]=>[5,1,1,1,1]=>[1,1,1,1,5] [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1,1],[]]=>[6,1,1,1]=>[1,1,1,6] [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1,1],[]]=>[7,1,1]=>[1,1,7] [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,1],[]]=>[8,1]=>[1,8] [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[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] [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1,1,1],[]]=>[2,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,2] [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1,1,1],[]]=>[2,2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,2] [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1,1,1],[]]=>[3,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,3] [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1,1,1],[]]=>[3,2,1,1,1,1,1]=>[1,1,1,1,1,2,3] [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1,1,1],[]]=>[4,1,1,1,1,1,1]=>[1,1,1,1,1,1,4] [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1,1,1],[]]=>[4,2,1,1,1,1]=>[1,1,1,1,2,4] [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1,1,1],[]]=>[5,1,1,1,1,1]=>[1,1,1,1,1,5] [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1,1,1],[]]=>[5,2,1,1,1]=>[1,1,1,2,5] [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1,1,1],[]]=>[6,1,1,1,1]=>[1,1,1,1,6] [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,1,1],[]]=>[6,2,1,1]=>[1,1,2,6] [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1,1,1],[]]=>[7,1,1,1]=>[1,1,1,7] [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2,1],[]]=>[7,2,1]=>[1,2,7] [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,1,1],[]]=>[8,1,1]=>[1,1,8] [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9,1],[]]=>[9,1]=>[1,9] [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[10],[]]=>[10]=>[10]
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Map
row lengths
Description
The sequence of row lengths from top to bottom.
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.