Identifier
Mp00038: Integer compositions reverseInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00034: Dyck paths to binary tree: up step, left tree, down step, right tree Binary trees
Images
=>
Cc0005;cc-rep-2Cc0010;cc-rep-3
[1]=>[1]=>[1,0]=>[.,.] [1,1]=>[1,1]=>[1,0,1,0]=>[.,[.,.]] [2]=>[2]=>[1,1,0,0]=>[[.,.],.] [1,1,1]=>[1,1,1]=>[1,0,1,0,1,0]=>[.,[.,[.,.]]] [1,2]=>[2,1]=>[1,1,0,0,1,0]=>[[.,.],[.,.]] [2,1]=>[1,2]=>[1,0,1,1,0,0]=>[.,[[.,.],.]] [3]=>[3]=>[1,1,1,0,0,0]=>[[[.,.],.],.] [1,1,1,1]=>[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,.]]]] [1,1,2]=>[2,1,1]=>[1,1,0,0,1,0,1,0]=>[[.,.],[.,[.,.]]] [1,2,1]=>[1,2,1]=>[1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,.]]] [1,3]=>[3,1]=>[1,1,1,0,0,0,1,0]=>[[[.,.],.],[.,.]] [2,1,1]=>[1,1,2]=>[1,0,1,0,1,1,0,0]=>[.,[.,[[.,.],.]]] [2,2]=>[2,2]=>[1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],.]] [3,1]=>[1,3]=>[1,0,1,1,1,0,0,0]=>[.,[[[.,.],.],.]] [4]=>[4]=>[1,1,1,1,0,0,0,0]=>[[[[.,.],.],.],.] [1,1,1,1,1]=>[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,.]]]]] [1,1,1,2]=>[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,.]]]] [1,1,2,1]=>[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[.,[.,.]]]] [1,1,3]=>[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[[[.,.],.],[.,[.,.]]] [1,2,1,1]=>[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[.,.]]]] [1,2,2]=>[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[.,.]]] [1,3,1]=>[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[.,[[[.,.],.],[.,.]]] [1,4]=>[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[[[[.,.],.],.],[.,.]] [2,1,1,1]=>[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[[.,.],.]]]] [2,1,2]=>[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[[.,.],[.,[[.,.],.]]] [2,2,1]=>[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,.],.]]] [2,3]=>[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[[[.,.],.],[[.,.],.]] [3,1,1]=>[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[.,[.,[[[.,.],.],.]]] [3,2]=>[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[[.,.],[[[.,.],.],.]] [4,1]=>[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[.,[[[[.,.],.],.],.]] [5]=>[5]=>[1,1,1,1,1,0,0,0,0,0]=>[[[[[.,.],.],.],.],.] [1,1,1,1,1,1]=>[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,.]]]]]] [1,1,1,1,2]=>[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,.]]]]] [1,1,1,2,1]=>[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,.]]]]] [1,1,1,3]=>[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[[[.,.],.],[.,[.,[.,.]]]] [1,1,2,1,1]=>[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,.]]]]] [1,1,2,2]=>[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,.]]]] [1,1,3,1]=>[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[.,[[[.,.],.],[.,[.,.]]]] [1,1,4]=>[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[.,.],.],.],[.,[.,.]]] [1,2,1,1,1]=>[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[[.,.],[.,.]]]]] [1,2,1,2]=>[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[[.,.],[.,.]]]] [1,2,2,1]=>[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[.,.],[[.,.],[.,.]]]] [1,2,3]=>[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[[[.,.],.],[[.,.],[.,.]]] [1,3,1,1]=>[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[.,[.,[[[.,.],.],[.,.]]]] [1,3,2]=>[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[[.,.],[[[.,.],.],[.,.]]] [1,4,1]=>[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[.,[[[[.,.],.],.],[.,.]]] [1,5]=>[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[[[[[.,.],.],.],.],[.,.]] [2,1,1,1,1]=>[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[[.,.],.]]]]] [2,1,1,2]=>[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[[.,.],.]]]] [2,1,2,1]=>[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[[.,.],.]]]] [2,1,3]=>[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[[[.,.],.],[.,[[.,.],.]]] [2,2,1,1]=>[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[[.,.],[[.,.],.]]]] [2,2,2]=>[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],[[.,.],.]]] [2,3,1]=>[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[.,[[[.,.],.],[[.,.],.]]] [2,4]=>[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[[[[.,.],.],.],[[.,.],.]] [3,1,1,1]=>[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[[[.,.],.],.]]]] [3,1,2]=>[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[[.,.],[.,[[[.,.],.],.]]] [3,2,1]=>[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[[.,.],[[[.,.],.],.]]] [3,3]=>[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[[[.,.],.],[[[.,.],.],.]] [4,1,1]=>[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[[[[.,.],.],.],.]]] [4,2]=>[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[[.,.],[[[[.,.],.],.],.]] [5,1]=>[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[[[[[.,.],.],.],.],.]] [6]=>[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[[[[[[.,.],.],.],.],.],.] [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]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,2]=>[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [1,1,1,1,2,1]=>[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,.]]]]]] [1,1,1,1,3]=>[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [1,1,1,2,1,1]=>[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,.]]]]]] [1,1,1,2,2]=>[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,.]]]]] [1,1,1,3,1]=>[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[.,[[[.,.],.],[.,[.,[.,.]]]]] [1,1,1,4]=>[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [1,1,2,1,1,1]=>[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,.]]]]]] [1,1,2,1,2]=>[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[.,.],[.,[[.,.],[.,[.,.]]]]] [1,1,2,2,1]=>[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[[.,.],[.,[.,.]]]]] [1,1,2,3]=>[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[.,.],.],[[.,.],[.,[.,.]]]] [1,1,3,1,1]=>[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[.,[.,[[[.,.],.],[.,[.,.]]]]] [1,1,3,2]=>[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[.,.],[[[.,.],.],[.,[.,.]]]] [1,1,4,1]=>[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[.,[[[[.,.],.],.],[.,[.,.]]]] [1,1,5]=>[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[[[[.,.],.],.],.],[.,[.,.]]] [1,2,1,1,1,1]=>[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,.]]]]]] [1,2,1,1,2]=>[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[.,[[.,.],[.,.]]]]] [1,2,1,2,1]=>[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,[[.,.],[.,.]]]]] [1,2,1,3]=>[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[[.,.],.],[.,[[.,.],[.,.]]]] [1,2,2,1,1]=>[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[[.,.],[.,.]]]]] [1,2,2,2]=>[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[[.,.],[.,.]]]] [1,2,3,1]=>[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[.,[[[.,.],.],[[.,.],[.,.]]]] [1,2,4]=>[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[[[.,.],.],.],[[.,.],[.,.]]] [1,3,1,1,1]=>[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[.,[.,[.,[[[.,.],.],[.,.]]]]] [1,3,1,2]=>[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[.,.],[.,[[[.,.],.],[.,.]]]] [1,3,2,1]=>[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[.,[[.,.],[[[.,.],.],[.,.]]]] [1,3,3]=>[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[[.,.],.],[[[.,.],.],[.,.]]] [1,4,1,1]=>[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[.,[.,[[[[.,.],.],.],[.,.]]]] [1,4,2]=>[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[.,.],[[[[.,.],.],.],[.,.]]] [1,5,1]=>[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[.,[[[[[.,.],.],.],.],[.,.]]] [1,6]=>[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[[[[[.,.],.],.],.],.],[.,.]] [2,1,1,1,1,1]=>[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[[.,.],.]]]]]] [2,1,1,1,2]=>[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[[.,.],.]]]]] [2,1,1,2,1]=>[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[.,[[.,.],.]]]]] [2,1,1,3]=>[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[[.,.],.],[.,[.,[[.,.],.]]]] [2,1,2,1,1]=>[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[.,[[.,.],[.,[[.,.],.]]]]] [2,1,2,2]=>[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[.,.],[[.,.],[.,[[.,.],.]]]] [2,1,3,1]=>[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[.,[[[.,.],.],[.,[[.,.],.]]]] [2,1,4]=>[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[[[.,.],.],.],[.,[[.,.],.]]] [2,2,1,1,1]=>[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[[.,.],[[.,.],.]]]]] [2,2,1,2]=>[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[.,.],[.,[[.,.],[[.,.],.]]]] [2,2,2,1]=>[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,.],[[.,.],.]]]] [2,2,3]=>[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[.,.],.],[[.,.],[[.,.],.]]] [2,3,1,1]=>[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[.,[.,[[[.,.],.],[[.,.],.]]]] [2,3,2]=>[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[.,.],[[[.,.],.],[[.,.],.]]] [2,4,1]=>[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[.,[[[[.,.],.],.],[[.,.],.]]] [2,5]=>[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[[[[.,.],.],.],.],[[.,.],.]] [3,1,1,1,1]=>[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[[[.,.],.],.]]]]] [3,1,1,2]=>[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[.,.],[.,[.,[[[.,.],.],.]]]] [3,1,2,1]=>[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[.,[[.,.],[.,[[[.,.],.],.]]]] [3,1,3]=>[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[[.,.],.],[.,[[[.,.],.],.]]] [3,2,1,1]=>[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[.,[[.,.],[[[.,.],.],.]]]] [3,2,2]=>[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[.,.],[[.,.],[[[.,.],.],.]]] [3,3,1]=>[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[.,[[[.,.],.],[[[.,.],.],.]]] [3,4]=>[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[[[.,.],.],.],[[[.,.],.],.]] [4,1,1,1]=>[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[[[[.,.],.],.],.]]]] [4,1,2]=>[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[.,.],[.,[[[[.,.],.],.],.]]] [4,2,1]=>[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[.,[[.,.],[[[[.,.],.],.],.]]] [4,3]=>[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[[.,.],.],[[[[.,.],.],.],.]] [5,1,1]=>[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[[[[[.,.],.],.],.],.]]] [5,2]=>[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,.],[[[[[.,.],.],.],.],.]] [6,1]=>[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[[[[[[.,.],.],.],.],.],.]] [7]=>[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[[[[[[.,.],.],.],.],.],.],.] [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]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,2,1]=>[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,3]=>[3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]] [1,1,1,1,2,1,1]=>[1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]] [1,1,1,1,2,2]=>[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]] [1,1,1,1,3,1]=>[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]] [1,1,1,1,4]=>[4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[[[[.,.],.],.],[.,[.,[.,[.,.]]]]] [1,1,1,2,1,1,1]=>[1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]] [1,1,1,2,1,2]=>[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]] [1,1,1,2,2,1]=>[1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]] [1,1,1,2,3]=>[3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[.,.],.],[[.,.],[.,[.,[.,.]]]]] [1,1,1,3,1,1]=>[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]] [1,1,1,3,2]=>[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[.,.],[[[.,.],.],[.,[.,[.,.]]]]] [1,1,1,4,1]=>[1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[.,[[[[.,.],.],.],[.,[.,[.,.]]]]] [1,1,1,5]=>[5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[[[[[.,.],.],.],.],[.,[.,[.,.]]]] [1,1,2,1,1,1,1]=>[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]] [1,1,2,1,1,2]=>[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] [1,1,2,1,2,1]=>[1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]] [1,1,2,1,3]=>[3,1,2,1,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[.,.],.],[.,[[.,.],[.,[.,.]]]]] [1,1,2,2,1,1]=>[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] [1,1,2,2,2]=>[2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[.,.],[[.,.],[[.,.],[.,[.,.]]]]] [1,1,2,3,1]=>[1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[.,[[[.,.],.],[[.,.],[.,[.,.]]]]] [1,1,2,4]=>[4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[[[[.,.],.],.],[[.,.],[.,[.,.]]]] [1,1,3,1,1,1]=>[1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]] [1,1,3,1,2]=>[2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[.,.],[.,[[[.,.],.],[.,[.,.]]]]] [1,1,3,2,1]=>[1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[.,[[.,.],[[[.,.],.],[.,[.,.]]]]] [1,1,3,3]=>[3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[.,.],.],[[[.,.],.],[.,[.,.]]]] [1,1,4,1,1]=>[1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[.,[.,[[[[.,.],.],.],[.,[.,.]]]]] [1,1,4,2]=>[2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[.,.],[[[[.,.],.],.],[.,[.,.]]]] [1,1,5,1]=>[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[.,[[[[[.,.],.],.],.],[.,[.,.]]]] [1,1,6]=>[6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[[[[[[.,.],.],.],.],.],[.,[.,.]]] [1,2,1,1,1,1,1]=>[1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]] [1,2,1,1,1,2]=>[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] [1,2,1,1,2,1]=>[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] [1,2,1,1,3]=>[3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[.,.],.],[.,[.,[[.,.],[.,.]]]]] [1,2,1,2,1,1]=>[1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]] [1,2,1,2,2]=>[2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[.,[[.,.],[.,.]]]]] [1,2,1,3,1]=>[1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[.,[[[.,.],.],[.,[[.,.],[.,.]]]]] [1,2,1,4]=>[4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[[[[.,.],.],.],[.,[[.,.],[.,.]]]] [1,2,2,1,1,1]=>[1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]] [1,2,2,1,2]=>[2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[.,.],[.,[[.,.],[[.,.],[.,.]]]]] [1,2,2,2,1]=>[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[.,.],[[.,.],[[.,.],[.,.]]]]] [1,2,2,3]=>[3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[.,.],.],[[.,.],[[.,.],[.,.]]]] [1,2,3,1,1]=>[1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[.,[.,[[[.,.],.],[[.,.],[.,.]]]]] [1,2,3,2]=>[2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[.,.],[[[.,.],.],[[.,.],[.,.]]]] [1,2,4,1]=>[1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[.,[[[[.,.],.],.],[[.,.],[.,.]]]] [1,2,5]=>[5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[[[[[.,.],.],.],.],[[.,.],[.,.]]] [1,3,1,1,1,1]=>[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]] [1,3,1,1,2]=>[2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[.,.],[.,[.,[[[.,.],.],[.,.]]]]] [1,3,1,2,1]=>[1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[.,[[.,.],[.,[[[.,.],.],[.,.]]]]] [1,3,1,3]=>[3,1,3,1]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[.,.],.],[.,[[[.,.],.],[.,.]]]] [1,3,2,1,1]=>[1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[.,[.,[[.,.],[[[.,.],.],[.,.]]]]] [1,3,2,2]=>[2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[.,.],[[.,.],[[[.,.],.],[.,.]]]] [1,3,3,1]=>[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[.,[[[.,.],.],[[[.,.],.],[.,.]]]] [1,3,4]=>[4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[[[[.,.],.],.],[[[.,.],.],[.,.]]] [1,4,1,1,1]=>[1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[.,[.,[.,[[[[.,.],.],.],[.,.]]]]] [1,4,1,2]=>[2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[.,.],[.,[[[[.,.],.],.],[.,.]]]] [1,4,2,1]=>[1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[.,[[.,.],[[[[.,.],.],.],[.,.]]]] [1,4,3]=>[3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,.],.],[[[[.,.],.],.],[.,.]]] [1,5,1,1]=>[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[.,[.,[[[[[.,.],.],.],.],[.,.]]]] [1,5,2]=>[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[.,.],[[[[[.,.],.],.],.],[.,.]]] [1,6,1]=>[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[.,[[[[[[.,.],.],.],.],.],[.,.]]] [1,7]=>[7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[[[[[[[.,.],.],.],.],.],.],[.,.]] [2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]] [2,1,1,1,1,2]=>[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] [2,1,1,1,2,1]=>[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]] [2,1,1,1,3]=>[3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[.,.],.],[.,[.,[.,[[.,.],.]]]]] [2,1,1,2,1,1]=>[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]] [2,1,1,2,2]=>[2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[.,.],[[.,.],[.,[.,[[.,.],.]]]]] [2,1,1,3,1]=>[1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[.,[[[.,.],.],[.,[.,[[.,.],.]]]]] [2,1,1,4]=>[4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[[[[.,.],.],.],[.,[.,[[.,.],.]]]] [2,1,2,1,1,1]=>[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]] [2,1,2,1,2]=>[2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[.,.],[.,[[.,.],[.,[[.,.],.]]]]] [2,1,2,2,1]=>[1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[.,[[.,.],[[.,.],[.,[[.,.],.]]]]] [2,1,2,3]=>[3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[.,.],.],[[.,.],[.,[[.,.],.]]]] [2,1,3,1,1]=>[1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[.,[.,[[[.,.],.],[.,[[.,.],.]]]]] [2,1,3,2]=>[2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[.,.],[[[.,.],.],[.,[[.,.],.]]]] [2,1,4,1]=>[1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[.,[[[[.,.],.],.],[.,[[.,.],.]]]] [2,1,5]=>[5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[[[[[.,.],.],.],.],[.,[[.,.],.]]] [2,2,1,1,1,1]=>[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]] [2,2,1,1,2]=>[2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[.,.],[.,[.,[[.,.],[[.,.],.]]]]] [2,2,1,2,1]=>[1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[.,[[.,.],[[.,.],.]]]]] [2,2,1,3]=>[3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[.,.],.],[.,[[.,.],[[.,.],.]]]] [2,2,2,1,1]=>[1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[.,[.,[[.,.],[[.,.],[[.,.],.]]]]] [2,2,2,2]=>[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [2,2,3,1]=>[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[.,[[[.,.],.],[[.,.],[[.,.],.]]]] [2,2,4]=>[4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[[[[.,.],.],.],[[.,.],[[.,.],.]]] [2,3,1,1,1]=>[1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[.,[.,[.,[[[.,.],.],[[.,.],.]]]]] [2,3,1,2]=>[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[.,.],[.,[[[.,.],.],[[.,.],.]]]] [2,3,2,1]=>[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[.,[[.,.],[[[.,.],.],[[.,.],.]]]] [2,3,3]=>[3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,.],.],[[[.,.],.],[[.,.],.]]] [2,4,1,1]=>[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[.,[.,[[[[.,.],.],.],[[.,.],.]]]] [2,4,2]=>[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[.,.],[[[[.,.],.],.],[[.,.],.]]] [2,5,1]=>[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[.,[[[[[.,.],.],.],.],[[.,.],.]]] [2,6]=>[6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[[[[[[.,.],.],.],.],.],[[.,.],.]] [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]=>[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]] [3,1,1,1,2]=>[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[.,.],[.,[.,[.,[[[.,.],.],.]]]]] [3,1,1,2,1]=>[1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[.,[[.,.],[.,[.,[[[.,.],.],.]]]]] [3,1,1,3]=>[3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[.,.],.],[.,[.,[[[.,.],.],.]]]] [3,1,2,1,1]=>[1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[.,[.,[[.,.],[.,[[[.,.],.],.]]]]] [3,1,2,2]=>[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[.,.],[[.,.],[.,[[[.,.],.],.]]]] [3,1,3,1]=>[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[.,[[[.,.],.],[.,[[[.,.],.],.]]]] [3,1,4]=>[4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[[[[.,.],.],.],[.,[[[.,.],.],.]]] [3,2,1,1,1]=>[1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[.,[.,[[.,.],[[[.,.],.],.]]]]] [3,2,1,2]=>[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[.,.],[.,[[.,.],[[[.,.],.],.]]]] [3,2,2,1]=>[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[.,[[.,.],[[.,.],[[[.,.],.],.]]]] [3,2,3]=>[3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,.],.],[[.,.],[[[.,.],.],.]]] [3,3,1,1]=>[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[.,[.,[[[.,.],.],[[[.,.],.],.]]]] [3,3,2]=>[2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[.,.],[[[.,.],.],[[[.,.],.],.]]] [3,4,1]=>[1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[.,[[[[.,.],.],.],[[[.,.],.],.]]] [3,5]=>[5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[[[[[.,.],.],.],.],[[[.,.],.],.]] [4,1,1,1,1]=>[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[[[[.,.],.],.],.]]]]] [4,1,1,2]=>[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[.,.],[.,[.,[[[[.,.],.],.],.]]]] [4,1,2,1]=>[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[.,[[.,.],[.,[[[[.,.],.],.],.]]]] [4,1,3]=>[3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,.],.],[.,[[[[.,.],.],.],.]]] [4,2,1,1]=>[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[.,[.,[[.,.],[[[[.,.],.],.],.]]]] [4,2,2]=>[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[.,.],[[.,.],[[[[.,.],.],.],.]]] [4,3,1]=>[1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[.,[[[.,.],.],[[[[.,.],.],.],.]]] [4,4]=>[4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [5,1,1,1]=>[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[[[[[.,.],.],.],.],.]]]] [5,1,2]=>[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[.,.],[.,[[[[[.,.],.],.],.],.]]] [5,2,1]=>[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[.,[[.,.],[[[[[.,.],.],.],.],.]]] [5,3]=>[3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,.],.],[[[[[.,.],.],.],.],.]] [6,1,1]=>[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[[[[[[.,.],.],.],.],.],.]]] [6,2]=>[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,.],[[[[[[.,.],.],.],.],.],.]] [7,1]=>[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[[[[[[[.,.],.],.],.],.],.],.]] [8]=>[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]=>[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,1,1,2,1]=>[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]=>[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,1,1,3]=>[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]=>[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,2,1,1]=>[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]=>[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,1,2,2]=>[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]=>[[.,.],[[.,.],[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,2,1,1,1]=>[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]=>[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,2,1,2]=>[2,1,2,1,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[[.,.],[.,[.,[.,[.,.]]]]]]] [1,1,1,2,1,1,1,1]=>[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]=>[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]] [1,1,1,2,1,1,2]=>[2,1,1,2,1,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[[.,.],[.,[.,[.,.]]]]]]] [1,1,2,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]] [1,1,2,1,1,1,2]=>[2,1,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[.,.],[.,[.,[.,[[.,.],[.,[.,.]]]]]]] [1,1,7]=>[7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]] [1,2,1,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]] [1,2,1,1,1,1,2]=>[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]=>[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]] [1,3,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]] [1,6,1,1]=>[1,1,6,1]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[.,[.,[[[[[[.,.],.],.],.],.],[.,.]]]] [1,7,1]=>[1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[.,[[[[[[[.,.],.],.],.],.],.],[.,.]]] [1,8]=>[8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,.]] [2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] [2,1,1,1,1,1,2]=>[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]=>[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] [2,2,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]] [2,2,1,1,1,2]=>[2,1,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[.,.],[.,[.,[.,[[.,.],[[.,.],.]]]]]] [2,7]=>[7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[[[[[[[.,.],.],.],.],.],.],[[.,.],.]] [3,1,1,1,1,1,1]=>[1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]] [3,2,1,1,1,1]=>[1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]] [3,6]=>[6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[[[[[[.,.],.],.],.],.],[[[.,.],.],.]] [4,1,1,1,1,1]=>[1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]] [4,2,1,1,1]=>[1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[[.,.],[[[[.,.],.],.],.]]]]] [4,5]=>[5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[[[[[.,.],.],.],.],[[[[.,.],.],.],.]] [5,1,1,1,1]=>[1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]] [5,1,3]=>[3,1,5]=>[1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,.],.],[.,[[[[[.,.],.],.],.],.]]] [5,2,1,1]=>[1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[[.,.],[[[[[.,.],.],.],.],.]]]] [5,4]=>[4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[[.,.],.],.],[[[[[.,.],.],.],.],.]] [6,1,1,1]=>[1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]] [6,2,1]=>[1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[[.,.],[[[[[[.,.],.],.],.],.],.]]] [6,3]=>[3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],.],[[[[[[.,.],.],.],.],.],.]] [7,1,1]=>[1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]] [7,2]=>[2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,.],[[[[[[[.,.],.],.],.],.],.],.]] [8,1]=>[1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[[[[[[[[.,.],.],.],.],.],.],.],.]] [9]=>[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]=>[1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] [1,1,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,1,1,1,2,1]=>[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]=>[.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,1,1,2,1,1]=>[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]=>[.,[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,1,2,1,1,1]=>[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]=>[.,[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,2,1,1,1,1]=>[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]=>[.,[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]] [1,1,1,2,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]] [1,1,2,1,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]] [1,2,1,1,1,1,1,1,1]=>[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]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]] [1,8,1]=>[1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]] [1,9]=>[9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] [2,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]] [2,1,1,1,1,1,1,2]=>[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]=>[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] [2,8]=>[8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]] [3,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]] [3,7]=>[7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]] [4,1,1,1,1,1,1]=>[1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]] [4,6]=>[6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]] [5,1,1,1,1,1]=>[1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]] [5,5]=>[5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]] [6,1,1,1,1]=>[1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]] [6,4]=>[4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[[.,.],.],.],[[[[[[.,.],.],.],.],.],.]] [7,1,1,1]=>[1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]] [7,3]=>[3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[[.,.],.],[[[[[[[.,.],.],.],.],.],.],.]] [8,1,1]=>[1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]] [8,2]=>[2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]] [9,1]=>[1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]] [10]=>[10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary tree: up step, left tree, down step, right tree
Description
Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path $D$ of semilength $n$ with $ n > 1$ may be uniquely decomposed into $1L0R$ for Dyck paths L,R of respective semilengths $n_1, n_2$ with $n_1 + n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.