Identifier
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Images
=>
Cc0002;cc-rep-0Cc0005;cc-rep-1Cc0028;cc-rep-2
[1]=>[1,0]=>[[1],[]] [2]=>[1,0,1,0]=>[[1,1],[]] [1,1]=>[1,1,0,0]=>[[2],[]] [3]=>[1,0,1,0,1,0]=>[[1,1,1],[]] [2,1]=>[1,0,1,1,0,0]=>[[2,1],[]] [1,1,1]=>[1,1,0,1,0,0]=>[[3],[]] [4]=>[1,0,1,0,1,0,1,0]=>[[1,1,1,1],[]] [3,1]=>[1,0,1,0,1,1,0,0]=>[[2,1,1],[]] [2,2]=>[1,1,1,0,0,0]=>[[2,2],[]] [2,1,1]=>[1,0,1,1,0,1,0,0]=>[[3,1],[]] [1,1,1,1]=>[1,1,0,1,0,1,0,0]=>[[4],[]] [5]=>[1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1],[]] [4,1]=>[1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1],[]] [3,2]=>[1,0,1,1,1,0,0,0]=>[[2,2,1],[]] [3,1,1]=>[1,0,1,0,1,1,0,1,0,0]=>[[3,1,1],[]] [2,2,1]=>[1,1,1,0,0,1,0,0]=>[[3,2],[]] [2,1,1,1]=>[1,0,1,1,0,1,0,1,0,0]=>[[4,1],[]] [1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,0]=>[[5],[]] [6]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1],[]] [5,1]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1],[]] [4,2]=>[1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1],[]] [4,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1],[]] [3,3]=>[1,1,1,0,1,0,0,0]=>[[2,2,2],[]] [3,2,1]=>[1,0,1,1,1,0,0,1,0,0]=>[[3,2,1],[]] [3,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1],[]] [2,2,2]=>[1,1,1,1,0,0,0,0]=>[[3,3],[]] [2,2,1,1]=>[1,1,1,0,0,1,0,1,0,0]=>[[4,2],[]] [2,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1],[]] [1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>[[6],[]] [7]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1],[]] [6,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1],[]] [5,2]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1],[]] [5,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1],[]] [4,3]=>[1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1],[]] [4,2,1]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1],[]] [4,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1],[]] [3,3,1]=>[1,1,1,0,1,0,0,1,0,0]=>[[3,2,2],[]] [3,2,2]=>[1,0,1,1,1,1,0,0,0,0]=>[[3,3,1],[]] [3,2,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1],[]] [3,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1],[]] [2,2,2,1]=>[1,1,1,1,0,0,0,1,0,0]=>[[4,3],[]] [2,2,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2],[]] [2,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1],[]] [1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7],[]] [8]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1,1],[]] [7,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1],[]] [6,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1],[]] [6,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1],[]] [5,3]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1],[]] [5,2,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1],[]] [5,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1],[]] [4,4]=>[1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2],[]] [4,3,1]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1],[]] [4,2,2]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1],[]] [4,2,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1],[]] [4,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1],[]] [3,3,2]=>[1,1,1,0,1,1,0,0,0,0]=>[[3,3,2],[]] [3,3,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2],[]] [3,2,2,1]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1],[]] [3,2,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1],[]] [3,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1],[]] [2,2,2,2]=>[1,1,1,1,0,1,0,0,0,0]=>[[4,4],[]] [2,2,2,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3],[]] [2,2,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2],[]] [2,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1],[]] [1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8],[]] [9]=>[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],[]] [8,1]=>[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],[]] [7,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1,1],[]] [7,1,1]=>[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],[]] [6,3]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1],[]] [6,2,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1,1],[]] [6,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1,1],[]] [5,4]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1],[]] [5,3,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1,1],[]] [5,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1],[]] [5,2,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1,1],[]] [5,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1,1],[]] [4,4,1]=>[1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2],[]] [4,3,2]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1],[]] [4,3,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,1],[]] [4,2,2,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1,1],[]] [4,2,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1,1],[]] [4,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1,1],[]] [3,3,3]=>[1,1,1,1,1,0,0,0,0,0]=>[[3,3,3],[]] [3,3,2,1]=>[1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2],[]] [3,3,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2],[]] [3,2,2,2]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1],[]] [3,2,2,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3,1],[]] [3,2,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,1],[]] [3,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1,1],[]] [2,2,2,2,1]=>[1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4],[]] [2,2,2,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3],[]] [2,2,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2],[]] [2,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,1],[]] [1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9],[]] [10]=>[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],[]] [9,1]=>[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],[]] [8,2]=>[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],[]] [8,1,1]=>[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],[]] [7,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1,1],[]] [7,2,1]=>[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],[]] [7,1,1,1]=>[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],[]] [6,4]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1,1],[]] [6,3,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1,1,1],[]] [6,2,2]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1,1],[]] [6,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1,1,1],[]] [6,1,1,1,1]=>[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],[]] [5,5]=>[1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2],[]] [5,4,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,1],[]] [5,3,2]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1,1],[]] [5,3,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,1,1],[]] [5,2,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1,1,1],[]] [5,2,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1,1,1],[]] [5,1,1,1,1,1]=>[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],[]] [4,4,2]=>[1,1,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2],[]] [4,4,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,2],[]] [4,3,3]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1],[]] [4,3,2,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,1],[]] [4,3,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2,1],[]] [4,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1,1],[]] [4,2,2,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3,1,1],[]] [4,2,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,1,1],[]] [4,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1,1,1],[]] [3,3,3,1]=>[1,1,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3],[]] [3,3,2,2]=>[1,1,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2],[]] [3,3,2,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[[5,3,2],[]] [3,3,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,2],[]] [3,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4,1],[]] [3,2,2,1,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3,1],[]] [3,2,1,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2,1],[]] [3,1,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,1,1],[]] [2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5],[]] [2,2,2,2,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[[6,4],[]] [2,2,2,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[[7,3],[]] [2,2,1,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,2],[]] [2,1,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9,1],[]] [1,1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[10],[]] [10,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1,1,1,1],[]] [9,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1,1,1,1],[]] [9,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1,1,1,1],[]] [8,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1,1,1],[]] [7,4]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1,1,1],[]] [7,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1,1,1],[]] [6,5]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,1],[]] [6,4,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,1,1],[]] [6,3,2]=>[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1,1,1],[]] [5,5,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2],[]] [5,4,2]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,1],[]] [5,4,1,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,2,1],[]] [5,3,3]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1,1],[]] [5,3,2,1]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,1,1],[]] [5,2,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1,1,1],[]] [4,4,3]=>[1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2],[]] [4,4,2,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,2],[]] [4,4,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2,2],[]] [4,3,3,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,1],[]] [4,3,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,1],[]] [4,3,2,1,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[[5,3,2,1],[]] [4,2,2,2,1]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4,1,1],[]] [3,3,3,2]=>[1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3],[]] [3,3,3,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[[5,3,3],[]] [3,3,2,2,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[[5,4,2],[]] [3,3,2,1,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3,2],[]] [3,3,1,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2,2],[]] [3,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5,1],[]] [3,2,2,2,1,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[[6,4,1],[]] [2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[[6,5],[]] [2,2,2,2,1,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[[7,4],[]] [2,2,2,1,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,3],[]] [2,2,1,1,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9,2],[]] [1,1,1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[11],[]] [8,4]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1,1,1,1],[]] [8,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1,1,1,1],[]] [7,5]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,1,1],[]] [6,6]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2],[]] [6,5,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,1],[]] [6,4,2]=>[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,1,1],[]] [6,4,1,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,2,1,1],[]] [6,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1,1,1],[]] [5,5,2]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,2],[]] [5,5,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,2,2],[]] [5,4,3]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,1],[]] [5,4,2,1]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,2,1],[]] [5,3,3,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,1,1],[]] [5,3,2,2]=>[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,1,1],[]] [4,4,4]=>[1,1,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3],[]] [4,4,3,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,2],[]] [4,4,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,2],[]] [4,4,2,1,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[[5,3,2,2],[]] [4,4,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,2,2],[]] [4,3,3,2]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,1],[]] [4,3,3,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[[5,3,3,1],[]] [4,3,2,2,1]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[[5,4,2,1],[]] [4,2,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5,1,1],[]] [3,3,3,3]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4],[]] [3,3,3,2,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3],[]] [3,3,3,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]=>[[6,3,3],[]] [3,3,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[[5,5,2],[]] [3,3,2,2,1,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]=>[[6,4,2],[]] [3,2,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[[6,5,1],[]] [2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6],[]] [2,2,2,2,2,1,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]=>[[7,5],[]] [2,2,2,2,1,1,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[[8,4],[]] [2,2,2,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9,3],[]] [8,5]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,1,1,1],[]] [7,6]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,1],[]] [7,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1,1,1,1],[]] [6,6,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,2],[]] [6,5,2]=>[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,2,1],[]] [6,4,3]=>[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,1,1],[]] [5,5,3]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,2],[]] [5,5,2,1]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,2,2],[]] [5,4,4]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,1],[]] [5,4,3,1]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,2,1],[]] [5,4,2,2]=>[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,2,1],[]] [5,3,3,2]=>[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,1,1],[]] [4,4,4,1]=>[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[[4,3,3,3],[]] [4,4,3,2]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,2],[]] [4,4,3,1,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]=>[[5,3,3,2],[]] [4,4,2,2,1]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]=>[[5,4,2,2],[]] [4,3,3,3]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,1],[]] [4,3,3,2,1]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3,1],[]] [4,3,2,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[[5,5,2,1],[]] [3,3,3,3,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[[5,4,4],[]] [3,3,3,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[[5,5,3],[]] [3,3,3,2,1,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]=>[[6,4,3],[]] [3,3,2,2,2,1]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]=>[[6,5,2],[]] [3,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6,1],[]] [2,2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[[7,6],[]] [2,2,2,2,2,1,1,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[[8,5],[]] [2,2,2,2,1,1,1,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9,4],[]] [8,6]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,1,1],[]] [8,3,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1,1,1,1,1],[]] [7,7]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2],[]] [7,6,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,2,1],[]] [6,6,2]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,2,2],[]] [6,5,3]=>[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,2,1],[]] [6,4,4]=>[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,1,1],[]] [5,5,4]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,2],[]] [5,5,3,1]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,2,2],[]] [5,5,2,2]=>[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,2,2],[]] [5,5,1,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,2,2,2],[]] [5,4,4,1]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[[4,3,3,3,1],[]] [5,4,3,2]=>[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,2,1],[]] [5,3,3,3]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,1,1],[]] [4,4,4,2]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[[4,4,3,3],[]] [4,4,4,1,1]=>[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]=>[[5,3,3,3],[]] [4,4,3,3]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,2],[]] [4,4,3,2,1]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3,2],[]] [4,4,2,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[[5,5,2,2],[]] [4,3,3,3,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[[5,4,4,1],[]] [4,3,3,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[[5,5,3,1],[]] [3,3,3,3,2]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[[5,5,4],[]] [3,3,3,3,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]=>[[6,4,4],[]] [3,3,3,2,2,1]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]=>[[6,5,3],[]] [3,3,2,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6,2],[]] [2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[7,7],[]] [2,2,2,2,2,2,1,1]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0]=>[[8,6],[]] [2,2,2,2,2,1,1,1,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[[9,5],[]] [8,7]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2,1],[]] [7,7,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,2,2],[]] [6,6,3]=>[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,2,2],[]] [6,5,4]=>[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,2,1],[]] [6,3,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,1,1,1],[]] [5,5,5]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3],[]] [5,5,4,1]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]=>[[4,3,3,3,2],[]] [5,5,3,2]=>[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,2,2],[]] [5,4,4,2]=>[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[[4,4,3,3,1],[]] [5,4,3,3]=>[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,2,1],[]] [4,4,4,3]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[[4,4,4,3],[]] [4,4,4,2,1]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3,3],[]] [4,4,3,3,1]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]=>[[5,4,4,2],[]] [4,4,3,2,2]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[[5,5,3,2],[]] [4,3,3,3,2]=>[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[[5,5,4,1],[]] [3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[[5,5,5],[]] [3,3,3,3,2,1]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]=>[[6,5,4],[]] [3,3,3,2,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[[6,6,3],[]] [3,2,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[7,7,1],[]] [2,2,2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[[8,7],[]] [2,2,2,2,2,2,1,1,1]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[[9,6],[]] [8,8]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2,2],[]] [7,3,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,1,1,1,1],[]] [6,6,4]=>[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,2,2],[]] [6,5,5]=>[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3,1],[]] [5,5,5,1]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]=>[[4,3,3,3,3],[]] [5,5,4,2]=>[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]=>[[4,4,3,3,2],[]] [5,5,3,3]=>[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,2,2],[]] [5,5,2,2,2]=>[1,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[[5,5,2,2,2],[]] [5,4,4,3]=>[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[[4,4,4,3,1],[]] [4,4,4,4]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[4,4,4,4],[]] [4,4,4,3,1]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]=>[[5,4,4,3],[]] [4,4,4,2,2]=>[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]=>[[5,5,3,3],[]] [4,4,3,3,2]=>[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]=>[[5,5,4,2],[]] [4,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[[5,5,5,1],[]] [4,3,3,2,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[[6,6,3,1],[]] [3,3,3,3,3,1]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]=>[[6,5,5],[]] [3,3,3,3,2,2]=>[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]=>[[6,6,4],[]] [2,2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[8,8],[]] [2,2,2,2,2,2,2,1,1]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0]=>[[9,7],[]] [9,8]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2,2,1],[]] [8,8,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,2,2,2],[]] [6,6,5]=>[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3,2],[]] [5,5,5,2]=>[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]=>[[4,4,3,3,3],[]] [5,5,4,3]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]=>[[4,4,4,3,2],[]] [5,4,4,4]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[4,4,4,4,1],[]] [4,4,4,4,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]=>[[5,4,4,4],[]] [4,4,4,3,2]=>[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]=>[[5,5,4,3],[]] [4,4,3,3,3]=>[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]=>[[5,5,5,2],[]] [3,3,3,3,3,2]=>[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]=>[[6,6,5],[]] [3,3,3,2,2,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]=>[[7,7,3],[]] [3,2,2,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[8,8,1],[]] [2,2,2,2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[[9,8],[]] [6,5,4,2,1]=>[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3,3,2,1],[]] [6,5,4,3,3,2,1]=>[1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]=>[[7,6,5,3,2,1],[]] [4,4,4,4,3,1]=>[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]=>[[6,5,5,4],[]] [5,4,4,4,3]=>[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]=>[[5,5,5,4,1],[]] [4,4,4,4,2]=>[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]=>[[5,5,4,4],[]] [2,2,2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[9,9],[]] [3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[[6,6,6],[]] [3,3,3,3,3,3,1]=>[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]=>[[7,6,6],[]] [4,3,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[[6,6,6,1],[]] [4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[5,5,5,5],[]] [4,4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[[6,6,6,6],[]] [5,5,5,5]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[[4,4,4,4,4],[]] [6,6,6,6]=>[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]=>[[4,4,4,4,4,4],[]] [6,6,6]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3,3],[]] [7,6,6]=>[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3,3,1],[]] [10,10]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2,2,2,2],[]] [6,6,6,6,6]=>[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]=>[[5,5,5,5,5,5],[]] [5,5,5,5,5,5]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[[6,6,6,6,6],[]] [5,5,5,5,5]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[[5,5,5,5,5],[]] [9,9]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2,2,2],[]] [6,5,5,5]=>[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[[4,4,4,4,4,1],[]] [5,4,4,4,4]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[5,5,5,5,1],[]] [4,4,4,4,4,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[[6,5,5,5],[]] [5,5,5,5,1]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]=>[[5,4,4,4,4],[]]
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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 $\gamma(D)$.