Identifier
Mp00231: Integer compositions bounce pathDyck paths
Mp00034: to binary tree: up step, left tree, down step, right treeBinary trees
Mp00018: left border symmetry
Images
=>
Cc0005;cc-rep-1Cc0010;cc-rep-2Cc0010;cc-rep-3
[1]=>[1,0]=>[.,.]=>[.,.] [1,1]=>[1,0,1,0]=>[.,[.,.]]=>[.,[.,.]] [2]=>[1,1,0,0]=>[[.,.],.]=>[[.,.],.] [1,1,1]=>[1,0,1,0,1,0]=>[.,[.,[.,.]]]=>[.,[.,[.,.]]] [1,2]=>[1,0,1,1,0,0]=>[.,[[.,.],.]]=>[.,[[.,.],.]] [2,1]=>[1,1,0,0,1,0]=>[[.,.],[.,.]]=>[[.,[.,.]],.] [3]=>[1,1,1,0,0,0]=>[[[.,.],.],.]=>[[[.,.],.],.] [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,.]]]]=>[.,[.,[.,[.,.]]]] [1,1,2]=>[1,0,1,0,1,1,0,0]=>[.,[.,[[.,.],.]]]=>[.,[.,[[.,.],.]]] [1,2,1]=>[1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,.]]]=>[.,[[.,[.,.]],.]] [1,3]=>[1,0,1,1,1,0,0,0]=>[.,[[[.,.],.],.]]=>[.,[[[.,.],.],.]] [2,1,1]=>[1,1,0,0,1,0,1,0]=>[[.,.],[.,[.,.]]]=>[[.,[.,[.,.]]],.] [2,2]=>[1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],.]]=>[[.,[[.,.],.]],.] [3,1]=>[1,1,1,0,0,0,1,0]=>[[[.,.],.],[.,.]]=>[[[.,[.,.]],.],.] [4]=>[1,1,1,1,0,0,0,0]=>[[[[.,.],.],.],.]=>[[[[.,.],.],.],.] [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[.,[.,[.,[.,[.,.]]]]]=>[.,[.,[.,[.,[.,.]]]]] [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[[.,.],.]]]]=>[.,[.,[.,[[.,.],.]]]] [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[.,.]]]]=>[.,[.,[[.,[.,.]],.]]] [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[.,[.,[[[.,.],.],.]]]=>[.,[.,[[[.,.],.],.]]] [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[.,[.,.]]]]=>[.,[[.,[.,[.,.]]],.]] [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,.],.]]]=>[.,[[.,[[.,.],.]],.]] [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[.,[[[.,.],.],[.,.]]]=>[.,[[[.,[.,.]],.],.]] [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[.,[[[[.,.],.],.],.]]=>[.,[[[[.,.],.],.],.]] [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],.] [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[[.,.],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],.] [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],.] [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[[.,.],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],.] [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[[[.,.],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],.] [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[[[.,.],.],[[.,.],.]]=>[[[.,[[.,.],.]],.],.] [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[[[[.,.],.],.],[.,.]]=>[[[[.,[.,.]],.],.],.] [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]=>[.,[.,[.,[.,[.,[.,.]]]]]]=>[.,[.,[.,[.,[.,[.,.]]]]]] [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[[.,.],.]]]]]=>[.,[.,[.,[.,[[.,.],.]]]]] [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[[.,.],[.,.]]]]]=>[.,[.,[.,[[.,[.,.]],.]]]] [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[[[.,.],.],.]]]]=>[.,[.,[.,[[[.,.],.],.]]]] [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,.]]]]]=>[.,[.,[[.,[.,[.,.]]],.]]] [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[[.,.],[[.,.],.]]]]=>[.,[.,[[.,[[.,.],.]],.]]] [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[.,[.,[[[.,.],.],[.,.]]]]=>[.,[.,[[[.,[.,.]],.],.]]] [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[[[[.,.],.],.],.]]]=>[.,[.,[[[[.,.],.],.],.]]] [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,.]]]]]=>[.,[[.,[.,[.,[.,.]]]],.]] [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[[.,.],.]]]]=>[.,[[.,[.,[[.,.],.]]],.]] [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[[.,.],[[.,.],[.,.]]]]=>[.,[[.,[[.,[.,.]],.]],.]] [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[[.,.],[[[.,.],.],.]]]=>[.,[[.,[[[.,.],.],.]],.]] [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[.,[[[.,.],.],[.,[.,.]]]]=>[.,[[[.,[.,[.,.]]],.],.]] [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[.,[[[.,.],.],[[.,.],.]]]=>[.,[[[.,[[.,.],.]],.],.]] [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[.,[[[[.,.],.],.],[.,.]]]=>[.,[[[[.,[.,.]],.],.],.]] [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[[[[[.,.],.],.],.],.]]=>[.,[[[[[.,.],.],.],.],.]] [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,.]]]]]=>[[.,[.,[.,[.,[.,.]]]]],.] [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[[.,.],.]]]]=>[[.,[.,[.,[[.,.],.]]]],.] [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[[.,.],[.,.]]]]=>[[.,[.,[[.,[.,.]],.]]],.] [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[[.,.],[.,[[[.,.],.],.]]]=>[[.,[.,[[[.,.],.],.]]],.] [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,.]]]]=>[[.,[[.,[.,[.,.]]],.]],.] [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[.,.],[[.,.],[[.,.],.]]]=>[[.,[[.,[[.,.],.]],.]],.] [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[[.,.],[[[.,.],.],[.,.]]]=>[[.,[[[.,[.,.]],.],.]],.] [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[[.,.],[[[[.,.],.],.],.]]=>[[.,[[[[.,.],.],.],.]],.] [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[[[.,.],.],[.,[.,[.,.]]]]=>[[[.,[.,[.,[.,.]]]],.],.] [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[[[.,.],.],[.,[[.,.],.]]]=>[[[.,[.,[[.,.],.]]],.],.] [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[[[.,.],.],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],.],.] [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[[[.,.],.],[[[.,.],.],.]]=>[[[.,[[[.,.],.],.]],.],.] [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[.,.],.],.],[.,[.,.]]]=>[[[[.,[.,[.,.]]],.],.],.] [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[[[[.,.],.],.],[[.,.],.]]=>[[[[.,[[.,.],.]],.],.],.] [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[[[[[.,.],.],.],.],[.,.]]=>[[[[[.,[.,.]],.],.],.],.] [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]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[[.,.],.]]]]]]=>[.,[.,[.,[.,[.,[[.,.],.]]]]]] [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[.,[.,[.,[.,[[.,.],[.,.]]]]]]=>[.,[.,[.,[.,[[.,[.,.]],.]]]]] [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[[[.,.],.],.]]]]]=>[.,[.,[.,[.,[[[.,.],.],.]]]]] [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[.,[.,[.,[[.,.],[.,[.,.]]]]]]=>[.,[.,[.,[[.,[.,[.,.]]],.]]]] [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[[.,.],[[.,.],.]]]]]=>[.,[.,[.,[[.,[[.,.],.]],.]]]] [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[.,[.,[.,[[[.,.],.],[.,.]]]]]=>[.,[.,[.,[[[.,[.,.]],.],.]]]] [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[[[[.,.],.],.],.]]]]=>[.,[.,[.,[[[[.,.],.],.],.]]]] [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[.,[.,[[.,.],[.,[.,[.,.]]]]]]=>[.,[.,[[.,[.,[.,[.,.]]]],.]]] [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[.,[.,[[.,.],[.,[[.,.],.]]]]]=>[.,[.,[[.,[.,[[.,.],.]]],.]]] [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[.,[.,[[.,.],[[.,.],[.,.]]]]]=>[.,[.,[[.,[[.,[.,.]],.]],.]]] [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[.,[[.,.],[[[.,.],.],.]]]]=>[.,[.,[[.,[[[.,.],.],.]],.]]] [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[.,[.,[[[.,.],.],[.,[.,.]]]]]=>[.,[.,[[[.,[.,[.,.]]],.],.]]] [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[.,[.,[[[.,.],.],[[.,.],.]]]]=>[.,[.,[[[.,[[.,.],.]],.],.]]] [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[.,[.,[[[[.,.],.],.],[.,.]]]]=>[.,[.,[[[[.,[.,.]],.],.],.]]] [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[[[[[.,.],.],.],.],.]]]=>[.,[.,[[[[[.,.],.],.],.],.]]] [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[.,[[.,.],[.,[.,[.,[.,.]]]]]]=>[.,[[.,[.,[.,[.,[.,.]]]]],.]] [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[.,[[.,.],[.,[.,[[.,.],.]]]]]=>[.,[[.,[.,[.,[[.,.],.]]]],.]] [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[.,[[.,.],[.,[[.,.],[.,.]]]]]=>[.,[[.,[.,[[.,[.,.]],.]]],.]] [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[.,[[.,.],[.,[[[.,.],.],.]]]]=>[.,[[.,[.,[[[.,.],.],.]]],.]] [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[.,[[.,.],[[.,.],[.,[.,.]]]]]=>[.,[[.,[[.,[.,[.,.]]],.]],.]] [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[.,[[.,.],[[.,.],[[.,.],.]]]]=>[.,[[.,[[.,[[.,.],.]],.]],.]] [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[.,[[.,.],[[[.,.],.],[.,.]]]]=>[.,[[.,[[[.,[.,.]],.],.]],.]] [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[.,[[.,.],[[[[.,.],.],.],.]]]=>[.,[[.,[[[[.,.],.],.],.]],.]] [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[.,[[[.,.],.],[.,[.,[.,.]]]]]=>[.,[[[.,[.,[.,[.,.]]]],.],.]] [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[.,[[[.,.],.],[.,[[.,.],.]]]]=>[.,[[[.,[.,[[.,.],.]]],.],.]] [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[.,[[[.,.],.],[[.,.],[.,.]]]]=>[.,[[[.,[[.,[.,.]],.]],.],.]] [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[.,[[[.,.],.],[[[.,.],.],.]]]=>[.,[[[.,[[[.,.],.],.]],.],.]] [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[.,[[[[.,.],.],.],[.,[.,.]]]]=>[.,[[[[.,[.,[.,.]]],.],.],.]] [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[.,[[[[.,.],.],.],[[.,.],.]]]=>[.,[[[[.,[[.,.],.]],.],.],.]] [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[.,[[[[[.,.],.],.],.],[.,.]]]=>[.,[[[[[.,[.,.]],.],.],.],.]] [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[[[[[[.,.],.],.],.],.],.]]=>[.,[[[[[[.,.],.],.],.],.],.]] [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]=>[[.,[.,[.,[.,[.,[.,.]]]]]],.] [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[.,.],[.,[.,[.,[[.,.],.]]]]]=>[[.,[.,[.,[.,[[.,.],.]]]]],.] [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[.,.],[.,[.,[[.,.],[.,.]]]]]=>[[.,[.,[.,[[.,[.,.]],.]]]],.] [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[.,.],[.,[.,[[[.,.],.],.]]]]=>[[.,[.,[.,[[[.,.],.],.]]]],.] [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[.,.],[.,[[.,.],[.,[.,.]]]]]=>[[.,[.,[[.,[.,[.,.]]],.]]],.] [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[.,.],[.,[[.,.],[[.,.],.]]]]=>[[.,[.,[[.,[[.,.],.]],.]]],.] [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[.,.],[.,[[[.,.],.],[.,.]]]]=>[[.,[.,[[[.,[.,.]],.],.]]],.] [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[.,.],[.,[[[[.,.],.],.],.]]]=>[[.,[.,[[[[.,.],.],.],.]]],.] [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[[.,[[.,[.,[.,[.,.]]]],.]],.] [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[.,.],[[.,.],[.,[[.,.],.]]]]=>[[.,[[.,[.,[[.,.],.]]],.]],.] [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[.,.],[[.,.],[[.,.],[.,.]]]]=>[[.,[[.,[[.,[.,.]],.]],.]],.] [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[.,.],[[.,.],[[[.,.],.],.]]]=>[[.,[[.,[[[.,.],.],.]],.]],.] [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[.,.],[[[.,.],.],[.,[.,.]]]]=>[[.,[[[.,[.,[.,.]]],.],.]],.] 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[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]=>[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]] [1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[.,[.,[[[.,.],.],[.,[[.,.],.]]]]]=>[.,[.,[[[.,[.,[[.,.],.]]],.],.]]] [1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[.,[.,[[[.,.],.],[[.,.],[.,.]]]]]=>[.,[.,[[[.,[[.,[.,.]],.]],.],.]]] [1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[.,[.,[[[.,.],.],[[[.,.],.],.]]]]=>[.,[.,[[[.,[[[.,.],.],.]],.],.]]] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[.,[.,[[[[.,.],.],.],[[.,.],.]]]]=>[.,[.,[[[[.,[[.,.],.]],.],.],.]]] [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[.,[.,[[[[[.,.],.],.],.],[.,.]]]]=>[.,[.,[[[[[.,[.,.]],.],.],.],.]]] [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[[[[[[.,.],.],.],.],.],.]]]=>[.,[.,[[[[[[.,.],.],.],.],.],.]]] 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[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]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]] [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]=>[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]=>[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]] [1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]=>[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]] [1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]=>[.,[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]] [1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]=>[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]] [1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]]=>[.,[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]] [1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]=>[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]] [1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[[.,.],[[[[.,.],.],.],.]]]]]=>[.,[.,[.,[[.,[[[[.,.],.],.],.]],.]]]] [1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]=>[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]] [1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[[.,.],[[[[[.,.],.],.],.],.]]]]=>[.,[.,[[.,[[[[[.,.],.],.],.],.]],.]]] [1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]=>[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]] [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]=>[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]] [1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[[.,.],[[[[[[.,.],.],.],.],.],.]]]=>[.,[[.,[[[[[[.,.],.],.],.],.],.]],.]] [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[.,[[[[[[[[.,.],.],.],.],.],.],.],.]] [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]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] [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]=>[[.,.],[[.,.],[.,[.,[.,[.,[.,.]]]]]]]=>[[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],.] [2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,.],[[[[[[[.,.],.],.],.],.],.],.]]=>[[.,[[[[[[[.,.],.],.],.],.],.],.]],.] [3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],.],[[[[[[.,.],.],.],.],.],.]]=>[[[.,[[[[[[.,.],.],.],.],.],.]],.],.] [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]=>[[[[[[[[.,[.,.]],.],.],.],.],.],.],.] [9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],.]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],.] [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]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] [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]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]] [1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]] [1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]=>[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]] [1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]=>[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]] [1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]=>[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]] [1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]=>[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]] [1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]=>[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]] [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]=>[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]] [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]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.] [2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.] [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]=>[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.] [10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
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.
Map
left border symmetry
Description
Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.