Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Images
=>
Cc0005;cc-rep-1Cc0021;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]=>[[[]],[[[]]],[],[]]=>[[.,[[[.,[.,[.,.]]],.],.]],.] [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[]],[[[]]],[[]]]=>[[.,[[[.,[[.,.],.]],.],.]],.] [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[]],[[[[]]]],[]]=>[[.,[[[[.,[.,.]],.],.],.]],.] [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[]],[[[[[]]]]]]=>[[.,[[[[[.,.],.],.],.],.]],.] [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[[[]]],[],[],[],[]]=>[[[.,[.,[.,[.,[.,.]]]]],.],.] [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[[[]]],[],[],[[]]]=>[[[.,[.,[.,[[.,.],.]]]],.],.] [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[[[]]],[],[[]],[]]=>[[[.,[.,[[.,[.,.]],.]]],.],.] [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[[[]]],[],[[[]]]]=>[[[.,[.,[[[.,.],.],.]]],.],.] [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[[]]],[[]],[],[]]=>[[[.,[[.,[.,[.,.]]],.]],.],.] [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[[]]],[[]],[[]]]=>[[[.,[[.,[[.,.],.]],.]],.],.] [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[[[]]],[[[]]],[]]=>[[[.,[[[.,[.,.]],.],.]],.],.] [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[[[]]],[[[[]]]]]=>[[[.,[[[[.,.],.],.],.]],.],.] [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[[[[]]]],[],[],[]]=>[[[[.,[.,[.,[.,.]]]],.],.],.] [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[[[[]]]],[],[[]]]=>[[[[.,[.,[[.,.],.]]],.],.],.] [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[[[[]]]],[[]],[]]=>[[[[.,[[.,[.,.]],.]],.],.],.] [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[[[[]]]],[[[]]]]=>[[[[.,[[[.,.],.],.]],.],.],.] [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[[[[[]]]]],[],[]]=>[[[[[.,[.,[.,.]]],.],.],.],.] [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[[[[[]]]]],[[]]]=>[[[[[.,[[.,.],.]],.],.],.],.] [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[[[[[[]]]]]],[]]=>[[[[[[.,[.,.]],.],.],.],.],.] [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[[[[[[[]]]]]]]]=>[[[[[[[.,.],.],.],.],.],.],.] 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[1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[],[],[[]],[],[[]],[]]=>[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]] [1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[],[],[[]],[],[[[]]]]=>[.,[.,[[.,[.,[[[.,.],.],.]]],.]]] [1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[],[],[[]],[[]],[],[]]=>[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]] [1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[],[],[[]],[[]],[[]]]=>[.,[.,[[.,[[.,[[.,.],.]],.]],.]]] [1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[],[],[[]],[[[]]],[]]=>[.,[.,[[.,[[[.,[.,.]],.],.]],.]]] [1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[],[],[[]],[[[[]]]]]=>[.,[.,[[.,[[[[.,.],.],.],.]],.]]] [1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[],[],[[[]]],[],[],[]]=>[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]] [1,1,3,1,2]=>[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]=>[[],[],[[[[[[]]]]]]]=>[.,[.,[[[[[[.,.],.],.],.],.],.]]] [1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[],[[]],[],[],[],[],[]]=>[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]] [1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[],[[]],[],[],[],[[]]]=>[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]] [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]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[],[[]],[],[],[[[]]]]=>[.,[[.,[.,[.,[[[.,.],.],.]]]],.]] 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[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[],[[[[[[[]]]]]]]]=>[.,[[[[[[[.,.],.],.],.],.],.],.]] [2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[]],[],[],[],[],[],[]]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.] [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]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[]],[],[],[],[[]],[]]=>[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.] [2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[]],[],[],[],[[[]]]]=>[[.,[.,[.,[.,[[[.,.],.],.]]]]],.] [2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[]],[],[],[[]],[],[]]=>[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.] [2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[]],[],[],[[]],[[]]]=>[[.,[.,[.,[[.,[[.,.],.]],.]]]],.] [2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[]],[],[],[[[]]],[]]=>[[.,[.,[.,[[[.,[.,.]],.],.]]]],.] [2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[]],[],[],[[[[]]]]]=>[[.,[.,[.,[[[[.,.],.],.],.]]]],.] [2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[]],[],[[]],[],[],[]]=>[[.,[.,[[.,[.,[.,[.,.]]]],.]]],.] [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]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[]],[],[[]],[[]],[]]=>[[.,[.,[[.,[[.,[.,.]],.]],.]]],.] [2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[]],[],[[]],[[[]]]]=>[[.,[.,[[.,[[[.,.],.],.]],.]]],.] [2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[]],[],[[[]]],[],[]]=>[[.,[.,[[[.,[.,[.,.]]],.],.]]],.] [2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[]],[],[[[]]],[[]]]=>[[.,[.,[[[.,[[.,.],.]],.],.]]],.] [2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[]],[],[[[[]]]],[]]=>[[.,[.,[[[[.,[.,.]],.],.],.]]],.] [2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[]],[],[[[[[]]]]]]=>[[.,[.,[[[[[.,.],.],.],.],.]]],.] [2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[]],[[]],[],[],[],[]]=>[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.] [2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[]],[[]],[],[],[[]]]=>[[.,[[.,[.,[.,[[.,.],.]]]],.]],.] [2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[]],[[]],[],[[]],[]]=>[[.,[[.,[.,[[.,[.,.]],.]]],.]],.] [2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[]],[[]],[],[[[]]]]=>[[.,[[.,[.,[[[.,.],.],.]]],.]],.] [2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[]],[[]],[[]],[],[]]=>[[.,[[.,[[.,[.,[.,.]]],.]],.]],.] [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[]],[[]],[[]],[[]]]=>[[.,[[.,[[.,[[.,.],.]],.]],.]],.] [2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[]],[[]],[[[]]],[]]=>[[.,[[.,[[[.,[.,.]],.],.]],.]],.] [2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[]],[[]],[[[[]]]]]=>[[.,[[.,[[[[.,.],.],.],.]],.]],.] [2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[]],[[[]]],[],[],[]]=>[[.,[[[.,[.,[.,[.,.]]]],.],.]],.] [2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[]],[[[]]],[],[[]]]=>[[.,[[[.,[.,[[.,.],.]]],.],.]],.] [2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[]],[[[]]],[[]],[]]=>[[.,[[[.,[[.,[.,.]],.]],.],.]],.] [2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[]],[[[]]],[[[]]]]=>[[.,[[[.,[[[.,.],.],.]],.],.]],.] [2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[]],[[[[]]]],[],[]]=>[[.,[[[[.,[.,[.,.]]],.],.],.]],.] [2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[]],[[[[]]]],[[]]]=>[[.,[[[[.,[[.,.],.]],.],.],.]],.] [2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[]],[[[[[]]]]],[]]=>[[.,[[[[[.,[.,.]],.],.],.],.]],.] [2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[]],[[[[[[]]]]]]]=>[[.,[[[[[[.,.],.],.],.],.],.]],.] [3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[]]],[],[],[],[],[]]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.] [3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[]]],[],[],[],[[]]]=>[[[.,[.,[.,[.,[[.,.],.]]]]],.],.] [3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[]]],[],[],[[]],[]]=>[[[.,[.,[.,[[.,[.,.]],.]]]],.],.] [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]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[]]],[],[[]],[],[]]=>[[[.,[.,[[.,[.,[.,.]]],.]]],.],.] [3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[]]],[],[[]],[[]]]=>[[[.,[.,[[.,[[.,.],.]],.]]],.],.] [3,1,3,1]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[]]],[],[[[]]],[]]=>[[[.,[.,[[[.,[.,.]],.],.]]],.],.] [3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[[]]],[],[[[[]]]]]=>[[[.,[.,[[[[.,.],.],.],.]]],.],.] [3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[]]],[[]],[],[],[]]=>[[[.,[[.,[.,[.,[.,.]]]],.]],.],.] [3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[]]],[[]],[],[[]]]=>[[[.,[[.,[.,[[.,.],.]]],.]],.],.] [3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[]]],[[]],[[]],[]]=>[[[.,[[.,[[.,[.,.]],.]],.]],.],.] [3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[[]]],[[]],[[[]]]]=>[[[.,[[.,[[[.,.],.],.]],.]],.],.] [3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[]]],[[[]]],[],[]]=>[[[.,[[[.,[.,[.,.]]],.],.]],.],.] [3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[[]]],[[[]]],[[]]]=>[[[.,[[[.,[[.,.],.]],.],.]],.],.] [3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[[]]],[[[[]]]],[]]=>[[[.,[[[[.,[.,.]],.],.],.]],.],.] [3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[[]]],[[[[[]]]]]]=>[[[.,[[[[[.,.],.],.],.],.]],.],.] [4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[]]]],[],[],[],[]]=>[[[[.,[.,[.,[.,[.,.]]]]],.],.],.] [4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[[[[[]]]],[],[],[[]]]=>[[[[.,[.,[.,[[.,.],.]]]],.],.],.] [4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[[[[[]]]],[],[[]],[]]=>[[[[.,[.,[[.,[.,.]],.]]],.],.],.] [4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[[[[[]]]],[],[[[]]]]=>[[[[.,[.,[[[.,.],.],.]]],.],.],.] [4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[[[[[]]]],[[]],[],[]]=>[[[[.,[[.,[.,[.,.]]],.]],.],.],.] [4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[[[[[]]]],[[]],[[]]]=>[[[[.,[[.,[[.,.],.]],.]],.],.],.] [4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[[[[[]]]],[[[]]],[]]=>[[[[.,[[[.,[.,.]],.],.]],.],.],.] [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[[[[[]]]],[[[[]]]]]=>[[[[.,[[[[.,.],.],.],.]],.],.],.] [5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[[[[[[]]]]],[],[],[]]=>[[[[[.,[.,[.,[.,.]]]],.],.],.],.] [5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[[[[[[]]]]],[],[[]]]=>[[[[[.,[.,[[.,.],.]]],.],.],.],.] [5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[[[[[[]]]]],[[]],[]]=>[[[[[.,[[.,[.,.]],.]],.],.],.],.] [5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[[[[[[]]]]],[[[]]]]=>[[[[[.,[[[.,.],.],.]],.],.],.],.] [6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[[[[[[[]]]]]],[],[]]=>[[[[[[.,[.,[.,.]]],.],.],.],.],.] [6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[[[[[[[]]]]]],[[]]]=>[[[[[[.,[[.,.],.]],.],.],.],.],.] [7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[[[[[[[[]]]]]]],[]]=>[[[[[[[.,[.,.]],.],.],.],.],.],.] [8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[[[[[[[[]]]]]]]]]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to ordered tree
Description
Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.
Map
Zeilberger's Strahler bijection
Description
Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.