Identifier
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
Mp00034: Dyck paths to binary tree: up step, left tree, down step, right tree Binary trees
Images
=>
Cc0005;cc-rep-1Cc0005;cc-rep-2Cc0010;cc-rep-3
[1]=>[1,0]=>[1,0]=>[.,.] [1,1]=>[1,0,1,0]=>[1,1,0,0]=>[[.,.],.] [2]=>[1,1,0,0]=>[1,0,1,0]=>[.,[.,.]] [1,1,1]=>[1,0,1,0,1,0]=>[1,1,0,1,0,0]=>[[.,[.,.]],.] [1,2]=>[1,0,1,1,0,0]=>[1,1,0,0,1,0]=>[[.,.],[.,.]] [2,1]=>[1,1,0,0,1,0]=>[1,0,1,1,0,0]=>[.,[[.,.],.]] [3]=>[1,1,1,0,0,0]=>[1,0,1,0,1,0]=>[.,[.,[.,.]]] [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,0]=>[[.,[.,[.,.]]],.] [1,1,2]=>[1,0,1,0,1,1,0,0]=>[1,1,0,1,0,0,1,0]=>[[.,[.,.]],[.,.]] [1,2,1]=>[1,0,1,1,0,0,1,0]=>[1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],.]] [1,3]=>[1,0,1,1,1,0,0,0]=>[1,1,0,0,1,0,1,0]=>[[.,.],[.,[.,.]]] [2,1,1]=>[1,1,0,0,1,0,1,0]=>[1,0,1,1,0,1,0,0]=>[.,[[.,[.,.]],.]] [2,2]=>[1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,.]]] [3,1]=>[1,1,1,0,0,0,1,0]=>[1,0,1,0,1,1,0,0]=>[.,[.,[[.,.],.]]] [4]=>[1,1,1,1,0,0,0,0]=>[1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,.]]]] [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,.]]]],.] [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,.]]],[.,.]] [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,0,1,1,0,0]=>[[.,[.,.]],[[.,.],.]] [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,0,1,0,1,0]=>[[.,[.,.]],[.,[.,.]]] [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[1,1,0,0,1,1,0,1,0,0]=>[[.,.],[[.,[.,.]],.]] [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[.,.]]] [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[1,1,0,0,1,0,1,1,0,0]=>[[.,.],[.,[[.,.],.]]] [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[1,1,0,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,.]]]] [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[1,0,1,1,0,1,0,1,0,0]=>[.,[[.,[.,[.,.]]],.]] [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[1,0,1,1,0,1,0,0,1,0]=>[.,[[.,[.,.]],[.,.]]] [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[1,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,.],.]]] [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[1,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[.,[.,.]]]] [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[1,0,1,0,1,1,0,1,0,0]=>[.,[.,[[.,[.,.]],.]]] [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[1,0,1,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[.,.]]]] [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[[.,.],.]]]] [5]=>[1,1,1,1,1,0,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,.]]]]] [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[.,.]]]]],.] [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,.]]]],[.,.]] [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,1,0,0,1,1,0,0]=>[[.,[.,[.,.]]],[[.,.],.]] [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,0]=>[[.,[.,[.,.]]],[.,[.,.]]] [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,1,0,1,0,0,1,1,0,1,0,0]=>[[.,[.,.]],[[.,[.,.]],.]] [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,0,1,1,0,0,1,0]=>[[.,[.,.]],[[.,.],[.,.]]] [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,1,0,1,0,0,1,0,1,1,0,0]=>[[.,[.,.]],[.,[[.,.],.]]] [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,0]=>[[.,[.,.]],[.,[.,[.,.]]]] [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,1,0,0,1,1,0,1,0,1,0,0]=>[[.,.],[[.,[.,[.,.]]],.]] [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,1,0,0,1,1,0,1,0,0,1,0]=>[[.,.],[[.,[.,.]],[.,.]]] [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],[[.,.],.]]] [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,.]]]] [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,1,0,0,1,0,1,1,0,1,0,0]=>[[.,.],[.,[[.,[.,.]],.]]] [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[[.,.],[.,.]]]] [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[[.,.],.]]]] [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,.]]]]] [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>[.,[[.,[.,[.,[.,.]]]],.]] [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[1,0,1,1,0,1,0,1,0,0,1,0]=>[.,[[.,[.,[.,.]]],[.,.]]] [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[1,0,1,1,0,1,0,0,1,1,0,0]=>[.,[[.,[.,.]],[[.,.],.]]] [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[1,0,1,1,0,1,0,0,1,0,1,0]=>[.,[[.,[.,.]],[.,[.,.]]]] [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[1,0,1,1,0,0,1,1,0,1,0,0]=>[.,[[.,.],[[.,[.,.]],.]]] [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[.,.],[[.,.],[.,.]]]] [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[[.,.],.]]]] [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,.]]]]] [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>[.,[.,[[.,[.,[.,.]]],.]]] [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[1,0,1,0,1,1,0,1,0,0,1,0]=>[.,[.,[[.,[.,.]],[.,.]]]] [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[[.,.],[[.,.],.]]]] [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,.]]]]] [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>[.,[.,[.,[[.,[.,.]],.]]]] [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[[.,.],[.,.]]]]] [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[[.,.],.]]]]] [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,.]]]]]] [1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],.] [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]] [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[[.,[.,[.,[.,.]]]],[[.,.],.]] [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[[.,[.,[.,.]]],[[.,[.,.]],.]] [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>[[.,[.,[.,.]]],[.,[[.,.],.]]] [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[[.,[.,.]],[[.,[.,[.,.]]],.]] [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>[[.,[.,.]],[[.,[.,.]],[.,.]]] [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[[.,[.,.]],[[.,.],[[.,.],.]]] [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[[.,[.,.]],[[.,.],[.,[.,.]]]] [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[[.,[.,.]],[.,[[.,[.,.]],.]]] [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>[[.,[.,.]],[.,[[.,.],[.,.]]]] [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[[.,[.,.]],[.,[.,[[.,.],.]]]] [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[[.,.],[[.,[.,[.,[.,.]]]],.]] [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[[.,.],[[.,[.,[.,.]]],[.,.]]] [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>[[.,.],[[.,[.,.]],[[.,.],.]]] [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>[[.,.],[[.,[.,.]],[.,[.,.]]]] [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[[.,.],[[.,.],[[.,[.,.]],.]]] [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[[.,.],[.,.]]]] [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[.,.],[[.,.],[.,[[.,.],.]]]] [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,.]]]]] [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>[[.,.],[.,[[.,[.,[.,.]]],.]]] [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>[[.,.],[.,[[.,[.,.]],[.,.]]]] [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[.,.],[.,[[.,.],[[.,.],.]]]] [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[.,.],[.,[[.,.],[.,[.,.]]]]] [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>[[.,.],[.,[.,[[.,[.,.]],.]]]] [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[.,[[.,.],[.,.]]]]] [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[[.,.],.]]]]] [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[.,[[.,[.,[.,[.,[.,.]]]]],.]] [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[.,[[.,[.,[.,[.,.]]]],[.,.]]] [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>[.,[[.,[.,[.,.]]],[[.,.],.]]] [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>[.,[[.,[.,[.,.]]],[.,[.,.]]]] [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>[.,[[.,[.,.]],[[.,[.,.]],.]]] [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[.,[[.,[.,.]],[[.,.],[.,.]]]] [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[.,[[.,[.,.]],[.,[[.,.],.]]]] [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[.,[[.,[.,.]],[.,[.,[.,.]]]]] [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[.,[[.,.],[[.,[.,[.,.]]],.]]] [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[.,[[.,.],[[.,[.,.]],[.,.]]]] [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,.],[[.,.],.]]]] [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[[.,.],[.,[.,.]]]]] [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>[.,[[.,.],[.,[[.,[.,.]],.]]]] [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,[[.,.],[.,.]]]]] [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[.,[[.,.],.]]]]] [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,.]]]]]] [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[.,[.,[[.,[.,[.,[.,.]]]],.]]] [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[.,[.,[[.,[.,[.,.]]],[.,.]]]] [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>[.,[.,[[.,[.,.]],[[.,.],.]]]] [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[.,[.,[[.,[.,.]],[.,[.,.]]]]] [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[.,[.,[[.,.],[[.,[.,.]],.]]]] [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[[.,.],[.,.]]]]] [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[.,[[.,.],[.,[[.,.],.]]]]] [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,.]]]]]] [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[.,[.,[.,[[.,[.,[.,.]]],.]]]] [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[.,[.,[.,[[.,[.,.]],[.,.]]]]] [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[[.,.],[[.,.],.]]]]] [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,.]]]]]] [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[.,[.,[.,[.,[[.,[.,.]],.]]]]] [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,.]]]]]] [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[[.,.],.]]]]]] [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.] [1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]] [1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]] [1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]] [1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]] [1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[[.,[.,[.,[.,.]]]],[[.,.],[.,.]]] [1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]=>[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]] [1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] [1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]] [1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]=>[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]] [1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[[.,[.,[.,.]]],[[.,.],[[.,.],.]]] [1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]] [1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]] [1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]=>[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]] [1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]] [1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]] [1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[[.,[.,.]],[[.,[.,[.,[.,.]]]],.]] [1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[[.,[.,.]],[[.,[.,[.,.]]],[.,.]]] [1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]=>[[.,[.,.]],[[.,[.,.]],[[.,.],.]]] [1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]=>[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]] [1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[[.,[.,.]],[[.,.],[[.,[.,.]],.]]] [1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[.,[.,.]],[[.,.],[[.,.],[.,.]]]] [1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[.,[.,.]],[[.,.],[.,[[.,.],.]]]] [1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[.,[.,.]],[[.,.],[.,[.,[.,.]]]]] [1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]=>[[.,[.,.]],[.,[[.,[.,[.,.]]],.]]] [1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]=>[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]] [1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[.,[.,.]],[.,[[.,.],[[.,.],.]]]] [1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[.,[.,.]],[.,[[.,.],[.,[.,.]]]]] [1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]=>[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]] [1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[.,[.,.]],[.,[.,[[.,.],[.,.]]]]] [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[.,[.,.]],[.,[.,[.,[[.,.],.]]]]] [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]] [1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]] [1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]] [1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]=>[[.,.],[[.,[.,[.,.]]],[[.,.],.]]] [1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]=>[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]] [1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]=>[[.,.],[[.,[.,.]],[[.,[.,.]],.]]] [1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,[.,.]],[[.,.],[.,.]]]] [1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[[.,.],[[.,[.,.]],[.,[[.,.],.]]]] [1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[[.,.],[[.,[.,.]],[.,[.,[.,.]]]]] [1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[[.,.],[[.,.],[[.,[.,[.,.]]],.]]] [1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[[.,.],[[.,.],[[.,[.,.]],[.,.]]]] [1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[.,.],[[.,.],[[.,.],[.,[.,.]]]]] [1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]=>[[.,.],[[.,.],[.,[[.,[.,.]],.]]]] [1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[.,[[.,.],[.,.]]]]] [1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[.,.],[[.,.],[.,[.,[[.,.],.]]]]] [1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]] [1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]] [1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]] [1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]=>[[.,.],[.,[[.,[.,.]],[[.,.],.]]]] [1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]] [1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[[.,.],[.,[[.,.],[[.,[.,.]],.]]]] [1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[.,.],[.,[[.,.],[[.,.],[.,.]]]]] [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[.,.],[.,[[.,.],[.,[[.,.],.]]]]] [1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]] [1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]] [1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]] [1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[.,.],[.,[.,[[.,.],[[.,.],.]]]]] [1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] [1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]] [1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] [1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] [1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] [2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]] [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]] [2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]] [2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]] [2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[.,[[.,[.,[.,.]]],[[.,[.,.]],.]]] [2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[.,[[.,[.,[.,.]]],[[.,.],[.,.]]]] [2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]] [2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]] [2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[.,[[.,[.,.]],[[.,[.,[.,.]]],.]]] [2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]] [2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[.,[[.,[.,.]],[[.,.],[[.,.],.]]]] [2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]] [2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[.,[[.,[.,.]],[.,[[.,[.,.]],.]]]] [2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]] [2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[.,[[.,[.,.]],[.,[.,[[.,.],.]]]]] [2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]] [2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]] [2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[.,[[.,.],[[.,[.,[.,.]]],[.,.]]]] [2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,[.,.]],[[.,.],.]]]] [2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]] [2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[.,[[.,.],[[.,.],[[.,[.,.]],.]]]] [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[.,.],[[.,.],[[.,.],[.,.]]]]] [2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[.,[[.,.],[[.,.],[.,[[.,.],.]]]]] [2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]] [2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>[.,[[.,.],[.,[[.,[.,[.,.]]],.]]]] [2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]] [2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[.,[[.,.],[[.,.],.]]]]] [2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]] [2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]] [2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]] [2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]] [2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]] [3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]] [3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]] [3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]] [3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]] [3,1,2,1,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>[.,[.,[[.,[.,.]],[[.,[.,.]],.]]]] [3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]] [3,1,3,1]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]] [3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]] [3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]] [3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]] [3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[.,[.,[[.,.],[[.,.],[[.,.],.]]]]] [3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]] [3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]] [3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]] [3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]] [3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]] [4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]] [4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]] [4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]] [4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]] [4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]] [4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]] [4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]] [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]] [5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]] [5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]] [5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]] [5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]] [6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]] [6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]] [7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]] [8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] [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]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]] [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]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[[.,.],.]] [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]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]] [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]=>[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],[[.,.],[.,.]]] [1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]] [1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[[.,[.,[.,[.,.]]]],[[.,.],[.,[.,.]]]] [1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]] [1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[.,[.,[.,.]]],[[.,.],[.,[.,[.,.]]]]] [1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]] [1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,[.,.]],[[.,.],[.,[.,[.,[.,.]]]]]] [1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]] [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]=>[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]] [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]=>[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,.],[[.,[.,[.,[.,[.,.]]]]],[.,.]]] [1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,[.,[.,.]]]]]]] [1,6,2]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]] [1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]] [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [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]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]] [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]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]] [2,1,6]=>[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]] [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]=>[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[.,[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]] [2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]] [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]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]] [3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]] [4,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]] [4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]] [5,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]] [5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]] [6,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]] [6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]] [7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]] [7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]] [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] [9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[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]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.] [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]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]] [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]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]] [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]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]] [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]=>[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]] [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]=>[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]] [1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [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]=>[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[.,.],[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]] [1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [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]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]] [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]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]] [2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [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]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]] [3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]] [4,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]] [4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]] [5,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]] [5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]] [6,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]] [6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]] [7,1,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]] [7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]] [8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]] [8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]] [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]] [10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
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.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.
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.