Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Images
=>
Cc0005;cc-rep-1Cc0002;cc-rep-2Cc0002;cc-rep-3
[1]=>[1,0]=>[]=>[] [1,1]=>[1,0,1,0]=>[1]=>[1] [2]=>[1,1,0,0]=>[]=>[] [1,1,1]=>[1,0,1,0,1,0]=>[2,1]=>[2,1] [1,2]=>[1,0,1,1,0,0]=>[1,1]=>[2] [2,1]=>[1,1,0,0,1,0]=>[2]=>[1,1] [3]=>[1,1,1,0,0,0]=>[]=>[] [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[3,2,1]=>[3,2,1] [1,1,2]=>[1,0,1,0,1,1,0,0]=>[2,2,1]=>[3,2] [1,2,1]=>[1,0,1,1,0,0,1,0]=>[3,1,1]=>[3,1,1] [1,3]=>[1,0,1,1,1,0,0,0]=>[1,1,1]=>[3] [2,1,1]=>[1,1,0,0,1,0,1,0]=>[3,2]=>[2,2,1] [2,2]=>[1,1,0,0,1,1,0,0]=>[2,2]=>[2,2] [3,1]=>[1,1,1,0,0,0,1,0]=>[3]=>[1,1,1] [4]=>[1,1,1,1,0,0,0,0]=>[]=>[] [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[4,3,2,1]=>[4,3,2,1] [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[3,3,2,1]=>[4,3,2] [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[4,2,2,1]=>[4,3,1,1] [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[2,2,2,1]=>[4,3] [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[4,3,1,1]=>[4,2,2,1] [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[3,3,1,1]=>[4,2,2] [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[4,1,1,1]=>[4,1,1,1] [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[1,1,1,1]=>[4] [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[4,3,2]=>[3,3,2,1] [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[3,3,2]=>[3,3,2] [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[4,2,2]=>[3,3,1,1] [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,2,2]=>[3,3] [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[4,3]=>[2,2,2,1] [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[3,3]=>[2,2,2] [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[4]=>[1,1,1,1] [5]=>[1,1,1,1,1,0,0,0,0,0]=>[]=>[] [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[5,4,3,2,1]=>[5,4,3,2,1] [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[4,4,3,2,1]=>[5,4,3,2] [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[5,3,3,2,1]=>[5,4,3,1,1] [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[3,3,3,2,1]=>[5,4,3] [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[5,4,2,2,1]=>[5,4,2,2,1] [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[4,4,2,2,1]=>[5,4,2,2] [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[5,2,2,2,1]=>[5,4,1,1,1] [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[2,2,2,2,1]=>[5,4] [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[5,4,3,1,1]=>[5,3,3,2,1] [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[4,4,3,1,1]=>[5,3,3,2] [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[5,3,3,1,1]=>[5,3,3,1,1] [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[3,3,3,1,1]=>[5,3,3] [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[5,4,1,1,1]=>[5,2,2,2,1] [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[4,4,1,1,1]=>[5,2,2,2] [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[5,1,1,1,1]=>[5,1,1,1,1] [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,1,1,1]=>[5] [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[5,4,3,2]=>[4,4,3,2,1] [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[4,4,3,2]=>[4,4,3,2] [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[5,3,3,2]=>[4,4,3,1,1] [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[3,3,3,2]=>[4,4,3] [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[5,4,2,2]=>[4,4,2,2,1] [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[4,4,2,2]=>[4,4,2,2] [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[5,2,2,2]=>[4,4,1,1,1] [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,2,2,2]=>[4,4] [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[5,4,3]=>[3,3,3,2,1] [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[4,4,3]=>[3,3,3,2] [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[5,3,3]=>[3,3,3,1,1] [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,3,3]=>[3,3,3] [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[5,4]=>[2,2,2,2,1] [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,4]=>[2,2,2,2] [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[5]=>[1,1,1,1,1] [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[]=>[] [1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1] [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[5,5,4,3,2,1]=>[6,5,4,3,2] [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[6,4,4,3,2,1]=>[6,5,4,3,1,1] [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[4,4,4,3,2,1]=>[6,5,4,3] [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[6,5,3,3,2,1]=>[6,5,4,2,2,1] [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[5,5,3,3,2,1]=>[6,5,4,2,2] [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[6,3,3,3,2,1]=>[6,5,4,1,1,1] [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[3,3,3,3,2,1]=>[6,5,4] [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[6,5,4,2,2,1]=>[6,5,3,3,2,1] [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[5,5,4,2,2,1]=>[6,5,3,3,2] [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[6,4,4,2,2,1]=>[6,5,3,3,1,1] [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[4,4,4,2,2,1]=>[6,5,3,3] [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[6,5,2,2,2,1]=>[6,5,2,2,2,1] [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[5,5,2,2,2,1]=>[6,5,2,2,2] [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[6,2,2,2,2,1]=>[6,5,1,1,1,1] [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,2,2,2,2,1]=>[6,5] [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[6,5,4,3,1,1]=>[6,4,4,3,2,1] [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[5,5,4,3,1,1]=>[6,4,4,3,2] [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[6,4,4,3,1,1]=>[6,4,4,3,1,1] [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[4,4,4,3,1,1]=>[6,4,4,3] [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[6,5,3,3,1,1]=>[6,4,4,2,2,1] [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[5,5,3,3,1,1]=>[6,4,4,2,2] [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[6,3,3,3,1,1]=>[6,4,4,1,1,1] [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[3,3,3,3,1,1]=>[6,4,4] [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[6,5,4,1,1,1]=>[6,3,3,3,2,1] [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[5,5,4,1,1,1]=>[6,3,3,3,2] [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[6,4,4,1,1,1]=>[6,3,3,3,1,1] [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[4,4,4,1,1,1]=>[6,3,3,3] [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[6,5,1,1,1,1]=>[6,2,2,2,2,1] [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[5,5,1,1,1,1]=>[6,2,2,2,2] [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[6,1,1,1,1,1]=>[6,1,1,1,1,1] [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,1,1,1,1]=>[6] [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[6,5,4,3,2]=>[5,5,4,3,2,1] [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[5,5,4,3,2]=>[5,5,4,3,2] [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[6,4,4,3,2]=>[5,5,4,3,1,1] [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[4,4,4,3,2]=>[5,5,4,3] [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[6,5,3,3,2]=>[5,5,4,2,2,1] [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[5,5,3,3,2]=>[5,5,4,2,2] [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[6,3,3,3,2]=>[5,5,4,1,1,1] [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[3,3,3,3,2]=>[5,5,4] [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[6,5,4,2,2]=>[5,5,3,3,2,1] [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[5,5,4,2,2]=>[5,5,3,3,2] [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[6,4,4,2,2]=>[5,5,3,3,1,1] [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[4,4,4,2,2]=>[5,5,3,3] [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[6,5,2,2,2]=>[5,5,2,2,2,1] [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[5,5,2,2,2]=>[5,5,2,2,2] [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[6,2,2,2,2]=>[5,5,1,1,1,1] [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,2,2,2,2]=>[5,5] [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[6,5,4,3]=>[4,4,4,3,2,1] [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[5,5,4,3]=>[4,4,4,3,2] [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[6,4,4,3]=>[4,4,4,3,1,1] [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[4,4,4,3]=>[4,4,4,3] [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[6,5,3,3]=>[4,4,4,2,2,1] [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[5,5,3,3]=>[4,4,4,2,2] [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[6,3,3,3]=>[4,4,4,1,1,1] [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,3,3,3]=>[4,4,4] [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[6,5,4]=>[3,3,3,3,2,1] [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[5,5,4]=>[3,3,3,3,2] [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[6,4,4]=>[3,3,3,3,1,1] [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[4,4,4]=>[3,3,3,3] [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[6,5]=>[2,2,2,2,2,1] [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[5,5]=>[2,2,2,2,2] [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[6]=>[1,1,1,1,1,1] [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[]=>[] [1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1] [1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[6,6,5,4,3,2,1]=>[7,6,5,4,3,2] [1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[7,5,5,4,3,2,1]=>[7,6,5,4,3,1,1] [1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[5,5,5,4,3,2,1]=>[7,6,5,4,3] [1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[7,6,4,4,3,2,1]=>[7,6,5,4,2,2,1] [1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[6,6,4,4,3,2,1]=>[7,6,5,4,2,2] [1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[7,4,4,4,3,2,1]=>[7,6,5,4,1,1,1] [1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,3,2,1]=>[7,6,5,4] [1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[7,6,5,3,3,2,1]=>[7,6,5,3,3,2,1] [1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[6,6,5,3,3,2,1]=>[7,6,5,3,3,2] [1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[5,5,5,3,3,2,1]=>[7,6,5,3,3] [1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[7,6,3,3,3,2,1]=>[7,6,5,2,2,2,1] [1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[6,6,3,3,3,2,1]=>[7,6,5,2,2,2] [1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[7,3,3,3,3,2,1]=>[7,6,5,1,1,1,1] [1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[3,3,3,3,3,2,1]=>[7,6,5] [1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,2,2,1]=>[7,6,4,4,3,2,1] [1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[6,6,5,4,2,2,1]=>[7,6,4,4,3,2] [1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[7,6,4,4,2,2,1]=>[7,6,4,4,2,2,1] [1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[6,6,4,4,2,2,1]=>[7,6,4,4,2,2] [1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,2,2,1]=>[7,6,4,4] [1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[7,6,5,2,2,2,1]=>[7,6,3,3,3,2,1] [1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[5,5,5,2,2,2,1]=>[7,6,3,3,3] [1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[7,6,2,2,2,2,1]=>[7,6,2,2,2,2,1] [1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[6,6,2,2,2,2,1]=>[7,6,2,2,2,2] [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[7,2,2,2,2,2,1]=>[7,6,1,1,1,1,1] [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,2,2,2,2,2,1]=>[7,6] [1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,3,1,1]=>[7,5,5,4,3,2,1] [1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[6,6,5,4,3,1,1]=>[7,5,5,4,3,2] [1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[7,5,5,4,3,1,1]=>[7,5,5,4,3,1,1] [1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,3,1,1]=>[7,5,5,4] [1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[7,6,5,3,3,1,1]=>[7,5,5,3,3,2,1] [1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[7,5,5,3,3,1,1]=>[7,5,5,3,3,1,1] [1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[5,5,5,3,3,1,1]=>[7,5,5,3,3] [1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[7,6,3,3,3,1,1]=>[7,5,5,2,2,2,1] [1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[7,3,3,3,3,1,1]=>[7,5,5,1,1,1,1] [1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[3,3,3,3,3,1,1]=>[7,5,5] [1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,1,1,1]=>[7,4,4,4,3,2,1] [1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[6,6,4,4,1,1,1]=>[7,4,4,4,2,2] [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[7,4,4,4,1,1,1]=>[7,4,4,4,1,1,1] [1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,1,1,1]=>[7,4,4,4] [1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[7,6,5,1,1,1,1]=>[7,3,3,3,3,2,1] [1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[5,5,5,1,1,1,1]=>[7,3,3,3,3] [1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[7,6,1,1,1,1,1]=>[7,2,2,2,2,2,1] [1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[6,6,1,1,1,1,1]=>[7,2,2,2,2,2] [1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[7,1,1,1,1,1,1]=>[7,1,1,1,1,1,1] [1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,1,1,1,1,1,1]=>[7] [2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,3,2]=>[6,6,5,4,3,2,1] [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[6,6,5,4,3,2]=>[6,6,5,4,3,2] [2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[7,5,5,4,3,2]=>[6,6,5,4,3,1,1] [2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[5,5,5,4,3,2]=>[6,6,5,4,3] [2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[7,6,4,4,3,2]=>[6,6,5,4,2,2,1] [2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,3,2]=>[6,6,5,4] [2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[3,3,3,3,3,2]=>[6,6,5] [2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,2,2]=>[6,6,4,4,3,2,1] [2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[7,6,4,4,2,2]=>[6,6,4,4,2,2,1] [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[6,6,4,4,2,2]=>[6,6,4,4,2,2] [2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[7,4,4,4,2,2]=>[6,6,4,4,1,1,1] [2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,2,2]=>[6,6,4,4] [2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[7,6,5,2,2,2]=>[6,6,3,3,3,2,1] [2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[7,6,2,2,2,2]=>[6,6,2,2,2,2,1] [2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[6,6,2,2,2,2]=>[6,6,2,2,2,2] [2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[7,2,2,2,2,2]=>[6,6,1,1,1,1,1] [2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,2,2,2,2,2]=>[6,6] [3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,3]=>[5,5,5,4,3,2,1] [3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[6,6,5,4,3]=>[5,5,5,4,3,2] [3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[5,5,5,4,3]=>[5,5,5,4,3] [3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[4,4,4,4,3]=>[5,5,5,4] [3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[7,3,3,3,3]=>[5,5,5,1,1,1,1] [3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[3,3,3,3,3]=>[5,5,5] [4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[7,6,5,4]=>[4,4,4,4,3,2,1] [4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[6,6,5,4]=>[4,4,4,4,3,2] [4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[5,5,5,4]=>[4,4,4,4,3] [4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[6,6,4,4]=>[4,4,4,4,2,2] [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[4,4,4,4]=>[4,4,4,4] [5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[7,6,5]=>[3,3,3,3,3,2,1] [5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[6,6,5]=>[3,3,3,3,3,2] [5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[7,5,5]=>[3,3,3,3,3,1,1] [5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[5,5,5]=>[3,3,3,3,3] [6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[7,6]=>[2,2,2,2,2,2,1] [6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[6,6]=>[2,2,2,2,2,2] [7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[7]=>[1,1,1,1,1,1,1] [8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[]=>[] [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]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1] [1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2] [1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[8,6,6,5,4,3,2,1]=>[8,7,6,5,4,3,1,1] [1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[6,6,6,5,4,3,2,1]=>[8,7,6,5,4,3] [1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[8,7,5,5,4,3,2,1]=>[8,7,6,5,4,2,2,1] [1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[7,7,5,5,4,3,2,1]=>[8,7,6,5,4,2,2] [1,1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[8,5,5,5,4,3,2,1]=>[8,7,6,5,4,1,1,1] [1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[5,5,5,5,4,3,2,1]=>[8,7,6,5,4] [1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[8,7,6,4,4,3,2,1]=>[8,7,6,5,3,3,2,1] [1,1,1,1,3,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[8,7,4,4,4,3,2,1]=>[8,7,6,5,2,2,2,1] [1,1,1,1,3,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[7,7,4,4,4,3,2,1]=>[8,7,6,5,2,2,2] [1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[4,4,4,4,4,3,2,1]=>[8,7,6,5] [1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,3,3,2,1]=>[8,7,6,4,4,3,2,1] [1,1,1,2,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[7,7,5,5,3,3,2,1]=>[8,7,6,4,4,2,2] [1,1,1,3,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[8,7,6,3,3,3,2,1]=>[8,7,6,3,3,3,2,1] [1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,3,3,3,3,3,2,1]=>[8,7,6] [1,1,2,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,2,2,1]=>[8,7,5,5,4,3,2,1] [1,1,3,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,2,2,2,1]=>[8,7,4,4,4,3,2,1] [1,1,3,2,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[7,7,5,5,2,2,2,1]=>[8,7,4,4,4,2,2] [1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,2,2,2,2,2,2,1]=>[8,7] [1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,3,1,1]=>[8,6,6,5,4,3,2,1] [1,2,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,7,6,5,4,3,1,1]=>[8,6,6,5,4,3,2] [1,2,2,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,3,3,1,1]=>[8,6,6,4,4,3,2,1] [1,2,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[8,7,5,5,3,3,1,1]=>[8,6,6,4,4,2,2,1] [1,2,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[7,7,5,5,3,3,1,1]=>[8,6,6,4,4,2,2] [1,2,2,3,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[8,5,5,5,3,3,1,1]=>[8,6,6,4,4,1,1,1] [1,2,3,1,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[8,7,6,3,3,3,1,1]=>[8,6,6,3,3,3,2,1] [1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,3,3,3,3,3,1,1]=>[8,6,6] [1,3,1,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,1,1,1]=>[8,5,5,5,4,3,2,1] [1,4,4]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[5,5,5,5,1,1,1,1]=>[8,4,4,4,4] [1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[8,1,1,1,1,1,1,1]=>[8,1,1,1,1,1,1,1] [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,1,1,1,1,1,1,1]=>[8] [2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,3,2]=>[7,7,6,5,4,3,2,1] [2,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,7,6,5,4,3,2]=>[7,7,6,5,4,3,2] [2,1,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[8,6,6,5,4,3,2]=>[7,7,6,5,4,3,1,1] [2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,2,2]=>[7,7,5,5,4,3,2,1] [2,2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[8,7,6,4,4,2,2]=>[7,7,5,5,3,3,2,1] [2,2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[8,6,6,4,4,2,2]=>[7,7,5,5,3,3,1,1] [2,2,2,3]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[6,6,6,4,4,2,2]=>[7,7,5,5,3,3] [2,2,3,1,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[8,7,4,4,4,2,2]=>[7,7,5,5,2,2,2,1] [2,3,1,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,2,2,2]=>[7,7,4,4,4,3,2,1] [2,6,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[8,2,2,2,2,2,2]=>[7,7,1,1,1,1,1,1] [2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,2,2,2,2,2,2]=>[7,7] [3,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,3]=>[6,6,6,5,4,3,2,1] [3,2,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[7,7,5,5,3,3]=>[6,6,6,4,4,2,2] [3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,3,3,3,3,3]=>[6,6,6] [4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[4,4,4,4,4]=>[5,5,5,5] [5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[5,5,5,5]=>[4,4,4,4,4] [6,2,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]=>[8,6,6]=>[3,3,3,3,3,3,1,1] [6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[6,6,6]=>[3,3,3,3,3,3] [7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[8,7]=>[2,2,2,2,2,2,2,1] [7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[7,7]=>[2,2,2,2,2,2,2] [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[8]=>[1,1,1,1,1,1,1,1] [9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[]=>[] [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]=>[9,8,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,2,1] [1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[8,8,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,2] [1,1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[9,7,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,1,1] [1,1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[9,8,6,6,5,4,3,2,1]=>[9,8,7,6,5,4,2,2,1] [1,1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[8,8,6,6,5,4,3,2,1]=>[9,8,7,6,5,4,2,2] [1,1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[9,8,7,5,5,4,3,2,1]=>[9,8,7,6,5,3,3,2,1] [1,1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,4,4,3,2,1]=>[9,8,7,6,4,4,3,2,1] [1,1,1,1,2,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[8,8,6,6,4,4,3,2,1]=>[9,8,7,6,4,4,2,2] [1,1,1,2,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,3,3,2,1]=>[9,8,7,5,5,4,3,2,1] [1,1,2,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,4,2,2,1]=>[9,8,6,6,5,4,3,2,1] [1,1,2,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[8,8,6,6,4,4,2,2,1]=>[9,8,6,6,4,4,2,2] [1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,2,2,2,2,2,2,2,1]=>[9,8] [1,2,1,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,4,3,1,1]=>[9,7,7,6,5,4,3,2,1] [1,2,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,3,3,1,1]=>[9,7,7,5,5,4,3,2,1] [1,2,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[9,8,7,5,5,3,3,1,1]=>[9,7,7,5,5,3,3,2,1] [1,2,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[9,7,7,5,5,3,3,1,1]=>[9,7,7,5,5,3,3,1,1] [1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[9,1,1,1,1,1,1,1,1]=>[9,1,1,1,1,1,1,1,1] [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[1,1,1,1,1,1,1,1,1]=>[9] [2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,4,3,2]=>[8,8,7,6,5,4,3,2,1] [2,1,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[8,8,7,6,5,4,3,2]=>[8,8,7,6,5,4,3,2] [2,2,1,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,4,2,2]=>[8,8,6,6,5,4,3,2,1] [2,2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,4,4,2,2]=>[8,8,6,6,4,4,3,2,1] [2,2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[9,8,6,6,4,4,2,2]=>[8,8,6,6,4,4,2,2,1] [2,2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[8,8,6,6,4,4,2,2]=>[8,8,6,6,4,4,2,2] [2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,2,2,2,2,2,2,2]=>[8,8] [3,1,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,4,3]=>[7,7,7,6,5,4,3,2,1] [3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[3,3,3,3,3,3,3]=>[7,7,7] [4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[4,4,4,4,4,4]=>[6,6,6,6] [5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[5,5,5,5,5]=>[5,5,5,5,5] [6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[6,6,6,6]=>[4,4,4,4,4,4] [7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[7,7,7]=>[3,3,3,3,3,3,3] [8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>[9,8]=>[2,2,2,2,2,2,2,2,1] [8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[8,8]=>[2,2,2,2,2,2,2,2] [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[9]=>[1,1,1,1,1,1,1,1,1] [10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[]=>[] [1,10]=>[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[1,1,1,1,1,1,1,1,1,1]=>[10] [10,1]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]=>[10]=>[1,1,1,1,1,1,1,1,1,1] [1,1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[9,9,8,7,6,5,4,3,2,1]=>[10,9,8,7,6,5,4,3,2]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.