Identifier
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Images
=>
Cc0002;cc-rep-0Cc0005;cc-rep-1
[1]=>[1,0]=>[1]=>[1] [2]=>[1,0,1,0]=>[1,1]=>[1,1] [1,1]=>[1,1,0,0]=>[2]=>[2] [3]=>[1,0,1,0,1,0]=>[1,1,1]=>[1,1,1] [2,1]=>[1,0,1,1,0,0]=>[1,2]=>[2,1] [1,1,1]=>[1,1,0,1,0,0]=>[3]=>[3] [4]=>[1,0,1,0,1,0,1,0]=>[1,1,1,1]=>[1,1,1,1] [3,1]=>[1,0,1,0,1,1,0,0]=>[1,1,2]=>[2,1,1] [2,2]=>[1,1,1,0,0,0]=>[3]=>[3] [2,1,1]=>[1,0,1,1,0,1,0,0]=>[1,3]=>[3,1] [1,1,1,1]=>[1,1,0,1,0,1,0,0]=>[4]=>[4] [5]=>[1,0,1,0,1,0,1,0,1,0]=>[1,1,1,1,1]=>[1,1,1,1,1] [4,1]=>[1,0,1,0,1,0,1,1,0,0]=>[1,1,1,2]=>[2,1,1,1] [3,2]=>[1,0,1,1,1,0,0,0]=>[1,3]=>[3,1] [3,1,1]=>[1,0,1,0,1,1,0,1,0,0]=>[1,1,3]=>[3,1,1] [2,2,1]=>[1,1,1,0,0,1,0,0]=>[4]=>[4] [2,1,1,1]=>[1,0,1,1,0,1,0,1,0,0]=>[1,4]=>[4,1] [1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,0]=>[5]=>[5] [6]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,1,1,1,1]=>[1,1,1,1,1,1] [5,1]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,1,1,2]=>[2,1,1,1,1] [4,2]=>[1,0,1,0,1,1,1,0,0,0]=>[1,1,3]=>[3,1,1] [4,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,1,3]=>[3,1,1,1] [3,3]=>[1,1,1,0,1,0,0,0]=>[4]=>[4] [3,2,1]=>[1,0,1,1,1,0,0,1,0,0]=>[1,4]=>[4,1] [3,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,4]=>[4,1,1] [2,2,2]=>[1,1,1,1,0,0,0,0]=>[4]=>[4] [2,2,1,1]=>[1,1,1,0,0,1,0,1,0,0]=>[5]=>[5] [2,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>[1,5]=>[5,1] [1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>[6]=>[6] [7]=>[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] [6,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,1,1,1,2]=>[2,1,1,1,1,1] [5,2]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,1,3]=>[3,1,1,1] [5,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,1,1,3]=>[3,1,1,1,1] [4,3]=>[1,0,1,1,1,0,1,0,0,0]=>[1,4]=>[4,1] [4,2,1]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,4]=>[4,1,1] [4,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,1,4]=>[4,1,1,1] [3,3,1]=>[1,1,1,0,1,0,0,1,0,0]=>[5]=>[5] [3,2,2]=>[1,0,1,1,1,1,0,0,0,0]=>[1,4]=>[4,1] [3,2,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>[1,5]=>[5,1] [3,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,1,5]=>[5,1,1] [2,2,2,1]=>[1,1,1,1,0,0,0,1,0,0]=>[5]=>[5] [2,2,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>[6]=>[6] [2,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,6]=>[6,1] [1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[7]=>[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]=>[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]=>[1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1] [6,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,1,1,3]=>[3,1,1,1,1] [6,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,1,1,1,3]=>[3,1,1,1,1,1] [5,3]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>[1,1,4]=>[4,1,1] [5,2,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,1,4]=>[4,1,1,1] [5,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,1,1,4]=>[4,1,1,1,1] [4,4]=>[1,1,1,0,1,0,1,0,0,0]=>[5]=>[5] [4,3,1]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>[1,5]=>[5,1] [4,2,2]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,4]=>[4,1,1] [4,2,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,1,5]=>[5,1,1] [4,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,1,1,5]=>[5,1,1,1] [3,3,2]=>[1,1,1,0,1,1,0,0,0,0]=>[5]=>[5] [3,3,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>[6]=>[6] [3,2,2,1]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>[1,5]=>[5,1] [3,2,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,6]=>[6,1] [3,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,1,6]=>[6,1,1] [2,2,2,2]=>[1,1,1,1,0,1,0,0,0,0]=>[5]=>[5] [2,2,2,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>[6]=>[6] [2,2,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[7]=>[7] [2,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,7]=>[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]=>[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]=>[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]=>[1,1,1,1,1,1,1,2]=>[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]=>[1,1,1,1,1,3]=>[3,1,1,1,1,1] [7,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1] [6,3]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[1,1,1,4]=>[4,1,1,1] [6,2,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,1,1,4]=>[4,1,1,1,1] [6,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,1,1,1,4]=>[4,1,1,1,1,1] [5,4]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>[1,5]=>[5,1] [5,3,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,1,5]=>[5,1,1] [5,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,1,4]=>[4,1,1,1] [5,2,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,1,1,5]=>[5,1,1,1] [5,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [4,4,1]=>[1,1,1,0,1,0,1,0,0,1,0,0]=>[6]=>[6] [4,3,2]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>[1,5]=>[5,1] [4,3,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,6]=>[6,1] [4,2,2,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[1,1,5]=>[5,1,1] [4,2,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,1,6]=>[6,1,1] [4,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,1,1,6]=>[6,1,1,1] [3,3,3]=>[1,1,1,1,1,0,0,0,0,0]=>[5]=>[5] [3,3,2,1]=>[1,1,1,0,1,1,0,0,0,1,0,0]=>[6]=>[6] [3,3,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[7]=>[7] [3,2,2,2]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>[1,5]=>[5,1] [3,2,2,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,6]=>[6,1] [3,2,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,7]=>[7,1] [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]=>[1,1,7]=>[7,1,1] [2,2,2,2,1]=>[1,1,1,1,0,1,0,0,0,1,0,0]=>[6]=>[6] [2,2,2,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[7]=>[7] [2,2,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[8]=>[8] [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]=>[1,8]=>[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]=>[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]=>[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]=>[1,1,1,1,1,1,1,1,2]=>[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]=>[1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1] [8,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,1,1,1,1,1,3]=>[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]=>[1,1,1,1,4]=>[4,1,1,1,1] [7,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,1,1,1,4]=>[4,1,1,1,1,1] [7,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,1,1,1,1,4]=>[4,1,1,1,1,1,1] [6,4]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,1,5]=>[5,1,1] [6,3,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,1,1,5]=>[5,1,1,1] [6,2,2]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,1,1,4]=>[4,1,1,1,1] [6,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [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]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1] [5,5]=>[1,1,1,0,1,0,1,0,1,0,0,0]=>[6]=>[6] [5,4,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[1,6]=>[6,1] [5,3,2]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,1,5]=>[5,1,1] [5,3,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,1,6]=>[6,1,1] [5,2,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[1,1,1,5]=>[5,1,1,1] [5,2,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,1,1,6]=>[6,1,1,1] [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]=>[1,1,1,1,6]=>[6,1,1,1,1] [4,4,2]=>[1,1,1,0,1,0,1,1,0,0,0,0]=>[6]=>[6] [4,4,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[7]=>[7] [4,3,3]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,5]=>[5,1] [4,3,2,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[1,6]=>[6,1] [4,3,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,7]=>[7,1] [4,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,1,5]=>[5,1,1] [4,2,2,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,1,6]=>[6,1,1] [4,2,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,1,7]=>[7,1,1] [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]=>[1,1,1,7]=>[7,1,1,1] [3,3,3,1]=>[1,1,1,1,1,0,0,0,0,1,0,0]=>[6]=>[6] [3,3,2,2]=>[1,1,1,0,1,1,0,1,0,0,0,0]=>[6]=>[6] [3,3,2,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[7]=>[7] [3,3,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[8]=>[8] [3,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,6]=>[6,1] [3,2,2,1,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[1,7]=>[7,1] [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]=>[1,8]=>[8,1] [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]=>[1,1,8]=>[8,1,1] [2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[6]=>[6] [2,2,2,2,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[7]=>[7] [2,2,2,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[8]=>[8] [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]=>[9]=>[9] [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]=>[1,9]=>[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] [11]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1,1,1] [10,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,1,1,1,1,1,1,1,2]=>[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]=>[1,1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1,1] [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]=>[1,1,1,1,1,1,1,1,3]=>[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]=>[1,1,1,1,1,4]=>[4,1,1,1,1,1] [8,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,1,1,1,1,4]=>[4,1,1,1,1,1,1] [7,4]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,1,1,5]=>[5,1,1,1] [7,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [7,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,1,1,1,4]=>[4,1,1,1,1,1] [6,5]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,6]=>[6,1] [6,4,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[1,1,6]=>[6,1,1] [6,3,2]=>[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,1,1,5]=>[5,1,1,1] [6,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,1,1,6]=>[6,1,1,1] [6,2,2,1]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [6,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,1,1,1,1,6]=>[6,1,1,1,1,1] [5,5,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[7]=>[7] [5,4,2]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,6]=>[6,1] [5,4,1,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[1,7]=>[7,1] [5,3,3]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,5]=>[5,1,1] [5,3,2,1]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[1,1,6]=>[6,1,1] [5,3,1,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,1,7]=>[7,1,1] [5,2,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,1,1,5]=>[5,1,1,1] [5,2,2,1,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,1,1,6]=>[6,1,1,1] [4,4,3]=>[1,1,1,0,1,1,1,0,0,0,0,0]=>[6]=>[6] [4,4,2,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[7]=>[7] [4,4,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[8]=>[8] [4,3,3,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[1,6]=>[6,1] [4,3,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[1,6]=>[6,1] [4,3,2,1,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[1,7]=>[7,1] [4,3,1,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,8]=>[8,1] [4,2,2,2,1]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,1,6]=>[6,1,1] [3,3,3,2]=>[1,1,1,1,1,0,0,1,0,0,0,0]=>[6]=>[6] [3,3,3,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[7]=>[7] [3,3,2,2,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[7]=>[7] [3,3,2,1,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]=>[8]=>[8] [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]=>[9]=>[9] [3,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,6]=>[6,1] [3,2,2,2,1,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[1,7]=>[7,1] [3,2,1,1,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,9]=>[9,1] [3,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,1,9]=>[9,1,1] [2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[7]=>[7] [2,2,2,2,1,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[8]=>[8] [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]=>[9]=>[9] [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]=>[10]=>[10] [2,1,1,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,10]=>[10,1] [1,1,1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[11]=>[11] [10,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1,1,1] [9,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[1,1,1,1,1,1,4]=>[4,1,1,1,1,1,1] [8,4]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [8,3,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1] [8,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,1,1,1,1,4]=>[4,1,1,1,1,1,1] [7,5]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,1,6]=>[6,1,1] [7,4,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[1,1,1,6]=>[6,1,1,1] [7,3,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [7,3,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,1,1,1,6]=>[6,1,1,1,1] [6,6]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[7]=>[7] [6,5,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[1,7]=>[7,1] [6,4,2]=>[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,1,6]=>[6,1,1] [6,4,1,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[1,1,7]=>[7,1,1] [6,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,1,5]=>[5,1,1,1] [6,3,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,1,1,7]=>[7,1,1,1] [6,2,2,2]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [5,5,2]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[7]=>[7] [5,5,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[8]=>[8] [5,4,3]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[1,6]=>[6,1] [5,4,2,1]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[1,7]=>[7,1] [5,4,1,1,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,8]=>[8,1] [5,3,3,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[1,1,6]=>[6,1,1] [5,3,2,2]=>[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[1,1,6]=>[6,1,1] [5,3,2,1,1]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[1,1,7]=>[7,1,1] [5,3,1,1,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,1,8]=>[8,1,1] [5,2,2,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,1,1,6]=>[6,1,1,1] [4,4,4]=>[1,1,1,1,1,0,1,0,0,0,0,0]=>[6]=>[6] [4,4,3,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[7]=>[7] [4,4,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[7]=>[7] [4,4,2,1,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[8]=>[8] [4,4,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[9]=>[9] [4,3,3,2]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[1,6]=>[6,1] [4,3,3,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[1,7]=>[7,1] [4,3,2,2,1]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[1,7]=>[7,1] [4,3,1,1,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,9]=>[9,1] [4,2,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,6]=>[6,1,1] [4,2,2,2,1,1]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[1,1,7]=>[7,1,1] [3,3,3,3]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[6]=>[6] [3,3,3,2,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[7]=>[7] [3,3,3,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]=>[8]=>[8] [3,3,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[7]=>[7] [3,3,2,2,1,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]=>[8]=>[8] [3,3,2,1,1,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[9]=>[9] [3,3,1,1,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[10]=>[10] [3,2,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[1,7]=>[7,1] [3,2,2,2,1,1,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[1,8]=>[8,1] [2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[7]=>[7] [2,2,2,2,2,1,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]=>[8]=>[8] [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]=>[9]=>[9] [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]=>[10]=>[10] [2,2,1,1,1,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[11]=>[11] [2,1,1,1,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,11]=>[11,1] [9,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1] [8,5]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,1,1,6]=>[6,1,1,1] [8,3,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1] [7,6]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,7]=>[7,1] [7,4,2]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,1,1,6]=>[6,1,1,1] [7,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,1,1,5]=>[5,1,1,1,1] [7,2,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1] [6,6,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[8]=>[8] [6,5,2]=>[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[1,7]=>[7,1] [6,4,3]=>[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[1,1,6]=>[6,1,1] [5,5,3]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[7]=>[7] [5,5,2,1]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]=>[8]=>[8] [5,5,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[9]=>[9] [5,4,4]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,6]=>[6,1] [5,4,3,1]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[1,7]=>[7,1] [5,4,2,2]=>[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[1,7]=>[7,1] [5,4,2,1,1]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[1,8]=>[8,1] [5,3,3,2]=>[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[1,1,6]=>[6,1,1] [5,3,3,1,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[1,1,7]=>[7,1,1] [5,3,2,2,1]=>[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[1,1,7]=>[7,1,1] [4,4,4,1]=>[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[7]=>[7] [4,4,3,2]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[7]=>[7] [4,4,3,1,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]=>[8]=>[8] [4,4,2,2,1]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]=>[8]=>[8] [4,4,2,1,1,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]=>[9]=>[9] [4,4,1,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[10]=>[10] [4,3,3,3]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,6]=>[6,1] [4,3,3,2,1]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,7]=>[7,1] [4,3,2,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[1,7]=>[7,1] [4,2,2,2,2,1]=>[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[1,1,7]=>[7,1,1] [3,3,3,3,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[7]=>[7] [3,3,3,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[7]=>[7] [3,3,3,2,1,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]=>[8]=>[8] [3,3,3,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]=>[9]=>[9] [3,3,2,2,2,1]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]=>[8]=>[8] [3,3,1,1,1,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[11]=>[11] [3,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[1,7]=>[7,1] [2,2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[8]=>[8] [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]=>[9]=>[9] [9,5]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,1,1,1,6]=>[6,1,1,1,1] [8,6]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,1,7]=>[7,1,1] [8,4,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,1,1,1,6]=>[6,1,1,1,1] [8,3,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1] [7,7]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[8]=>[8] [7,6,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[1,8]=>[8,1] [7,5,2]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[1,1,7]=>[7,1,1] [6,6,2]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[8]=>[8] [6,6,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[9]=>[9] [6,5,3]=>[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[1,7]=>[7,1] [6,4,4]=>[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,1,6]=>[6,1,1] [6,3,3,2]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[1,1,1,6]=>[6,1,1,1] [5,5,4]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[7]=>[7] [5,5,3,1]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]=>[8]=>[8] [5,5,2,2]=>[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]=>[8]=>[8] [5,5,2,1,1]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[9]=>[9] [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]=>[10]=>[10] [5,4,4,1]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[1,7]=>[7,1] [5,4,3,2]=>[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[1,7]=>[7,1] [5,4,3,1,1]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]=>[1,8]=>[8,1] [5,4,2,2,1]=>[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]=>[1,8]=>[8,1] [5,3,3,3]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,6]=>[6,1,1] [5,3,3,2,1]=>[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,1,7]=>[7,1,1] [4,4,4,2]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[7]=>[7] [4,4,4,1,1]=>[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]=>[8]=>[8] [4,4,3,3]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[7]=>[7] [4,4,3,2,1]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[8]=>[8] [4,4,3,1,1,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,0]=>[9]=>[9] [4,4,2,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[8]=>[8] [4,3,3,3,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[1,7]=>[7,1] [4,3,3,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,7]=>[7,1] [3,3,3,3,2]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[7]=>[7] [3,3,3,3,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]=>[8]=>[8] [3,3,3,2,2,1]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]=>[8]=>[8] [3,3,3,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[10]=>[10] [3,3,2,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]=>[8]=>[8] [3,2,2,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[1,8]=>[8,1] [2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[8]=>[8] [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]=>[9]=>[9] [9,6]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,1,1,7]=>[7,1,1,1] [8,7]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,8]=>[8,1] [8,5,2]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[1,1,1,7]=>[7,1,1,1] [7,7,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[9]=>[9] [7,6,2]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[1,8]=>[8,1] [7,4,4]=>[1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,1,1,6]=>[6,1,1,1] [6,6,3]=>[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]=>[8]=>[8] [6,6,2,1]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,0]=>[9]=>[9] [6,6,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[10]=>[10] [6,5,4]=>[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[1,7]=>[7,1] [6,4,4,1]=>[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[1,1,7]=>[7,1,1] [6,3,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,1,6]=>[6,1,1,1] [5,5,5]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[7]=>[7] [5,5,4,1]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]=>[8]=>[8] [5,5,3,2]=>[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]=>[8]=>[8] [5,4,4,2]=>[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[1,7]=>[7,1] [5,4,3,3]=>[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[1,7]=>[7,1] [5,4,3,2,1]=>[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,8]=>[8,1] [4,4,4,3]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[7]=>[7] [4,4,4,2,1]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[8]=>[8] [4,4,3,3,1]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]=>[8]=>[8] [4,4,3,2,2]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[8]=>[8] [4,3,3,3,2]=>[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[1,7]=>[7,1] [4,3,3,3,1,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]=>[1,8]=>[8,1] [3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[7]=>[7] [3,3,3,3,2,1]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]=>[8]=>[8] [3,3,3,3,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]=>[9]=>[9] [3,3,3,2,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[8]=>[8] [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]=>[1,8]=>[8,1] [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]=>[9]=>[9] [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]=>[10]=>[10] [9,7]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,1,8]=>[8,1,1] [8,8]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[9]=>[9] [8,6,2]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[1,1,8]=>[8,1,1] [8,4,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,1,1,1,6]=>[6,1,1,1,1] [7,7,2]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[9]=>[9] [7,7,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[10]=>[10] [7,3,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,1,1,6]=>[6,1,1,1,1] [6,6,4]=>[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]=>[8]=>[8] [6,6,3,1]=>[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,0]=>[9]=>[9] [6,6,2,2]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]=>[9]=>[9] [6,6,1,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[11]=>[11] [6,5,5]=>[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[1,7]=>[7,1] [6,4,3,2,1]=>[1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,1,8]=>[8,1,1] [5,5,5,1]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]=>[8]=>[8] [5,5,4,2]=>[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]=>[8]=>[8] [5,5,3,3]=>[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]=>[8]=>[8] [5,5,2,2,2]=>[1,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[9]=>[9] [5,4,4,3]=>[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[1,7]=>[7,1] [5,3,3,3,2]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[1,1,7]=>[7,1,1] [4,4,4,4]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7]=>[7] [4,4,4,3,1]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]=>[8]=>[8] [4,4,4,2,2]=>[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]=>[8]=>[8] [4,4,3,3,2]=>[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]=>[8]=>[8] [4,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[1,7]=>[7,1] [4,3,3,2,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[1,8]=>[8,1] [3,3,3,3,3,1]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]=>[8]=>[8] [3,3,3,3,2,2]=>[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]=>[8]=>[8] [3,3,3,3,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]=>[10]=>[10] [3,3,2,2,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[9]=>[9] [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]=>[9]=>[9] [9,8]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,9]=>[9,1] [8,8,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[10]=>[10] [8,7,2]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[1,9]=>[9,1] [7,7,3]=>[1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]=>[9]=>[9] [7,5,5]=>[1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[1,1,7]=>[7,1,1] [6,6,5]=>[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]=>[8]=>[8] [6,5,3,3]=>[1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]=>[1,8]=>[8,1] [6,4,4,3]=>[1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[1,1,7]=>[7,1,1] [5,5,5,2]=>[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]=>[8]=>[8] [5,5,4,3]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]=>[8]=>[8] [5,4,4,4]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,7]=>[7,1] [5,3,3,3,3]=>[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[1,1,7]=>[7,1,1] [4,4,4,4,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]=>[8]=>[8] [4,4,4,3,2]=>[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]=>[8]=>[8] [4,4,4,3,1,1]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]=>[9]=>[9] [4,4,3,3,3]=>[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]=>[8]=>[8] [4,3,3,3,3,1]=>[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]=>[1,8]=>[8,1] [4,3,3,3,2,2]=>[1,0,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]=>[1,8]=>[8,1] [3,3,3,3,3,2]=>[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]=>[8]=>[8] [3,3,3,3,3,1,1]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,0]=>[9]=>[9] [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]=>[9]=>[9] [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]=>[1,9]=>[9,1] [5,5,4,3,2,1]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]=>[10]=>[10] [5,5,4,2,1,1]=>[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]=>[10]=>[10] [5,5,4,3,2]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]=>[9]=>[9] [5,5,4,3,1]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]=>[9]=>[9] [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]=>[1,9]=>[9,1] [6,6,5,4,3,2,1]=>[1,1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0,0]=>[12]=>[12] [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]=>[1,11]=>[11,1] [4,3,3,3,3,2]=>[1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]=>[1,8]=>[8,1] [4,4,4,4,3,1]=>[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]=>[9]=>[9] [6,6,5,4,3]=>[1,1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0]=>[10]=>[10] [5,4,4,4,3]=>[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]=>[1,8]=>[8,1] [4,4,4,4,2]=>[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]=>[8]=>[8] [6,5,5,4]=>[1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]=>[1,8]=>[8,1] [7,6,5]=>[1,0,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]=>[1,8]=>[8,1] [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]=>[10]=>[10] [3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[8]=>[8] [3,3,3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[10]=>[10] [3,3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[9]=>[9] [3,3,3,3,3,2,2]=>[1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0]=>[9]=>[9] [4,3,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[1,8]=>[8,1] [4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[8]=>[8] [4,4,4,3,3,3]=>[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]=>[9]=>[9] [4,4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[9]=>[9] [5,5,5,5]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[8]=>[8] [6,6,6,6]=>[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]=>[9]=>[9] [6,6,6]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[8]=>[8] [7,6,6]=>[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,8]=>[8,1] [10,10]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[11]=>[11] [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]=>[10]=>[10] [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]=>[10]=>[10] [7,7,7,7]=>[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]=>[10]=>[10] [7,7,7]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[9]=>[9] [5,5,5,5,5]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[9]=>[9] [9,9]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[10]=>[10] [8,8,8]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[10]=>[10] [4,4,4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]=>[10]=>[10] [6,5,5,5]=>[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[1,8]=>[8,1] [5,4,4,4,4]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,8]=>[8,1] [4,4,4,4,4,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[9]=>[9] [5,5,5,5,1]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]=>[9]=>[9]
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
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
Map
rotate back to front
Description
The back to front rotation of an integer composition.