Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Images
=>
Cc0005;cc-rep-1Cc0021;cc-rep-2Cc0021;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]=>[[[[[[[[]]]]]]]]=>[[[[[[[[]]]]]]]] [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]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[],[],[],[],[],[],[[]]]=>[[[]],[],[],[],[],[],[]] [1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[],[],[],[],[],[[]],[]]=>[[],[[]],[],[],[],[],[]] [1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[],[],[],[],[],[[[]]]]=>[[[[]]],[],[],[],[],[]] [1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[],[],[],[],[[]],[],[]]=>[[],[],[[]],[],[],[],[]] [1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[],[],[],[],[[]],[[]]]=>[[[]],[[]],[],[],[],[]] [1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[],[],[],[],[[[]]],[]]=>[[],[[[]]],[],[],[],[]] [1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[],[],[],[],[[[[]]]]]=>[[[[[]]]],[],[],[],[]] [1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[],[],[],[[]],[],[],[]]=>[[],[],[],[[]],[],[],[]] [1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[],[],[],[[]],[],[[]]]=>[[[]],[],[[]],[],[],[]] [1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[],[],[],[[]],[[]],[]]=>[[],[[]],[[]],[],[],[]] [1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[],[],[],[[]],[[[]]]]=>[[[[]]],[[]],[],[],[]] [1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[],[],[],[[[]]],[],[]]=>[[],[],[[[]]],[],[],[]] [1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[],[],[],[[[]]],[[]]]=>[[[]],[[[]]],[],[],[]] [1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[],[],[],[[[[]]]],[]]=>[[],[[[[]]]],[],[],[]] [1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[],[],[],[[[[[]]]]]]=>[[[[[[]]]]],[],[],[]] [1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[],[],[[]],[],[],[],[]]=>[[],[],[],[],[[]],[],[]] [1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[],[],[[]],[],[],[[]]]=>[[[]],[],[],[[]],[],[]] [1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[],[],[[]],[],[[]],[]]=>[[],[[]],[],[[]],[],[]] [1,1,2,1,3]=>[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]=>[[],[[]],[],[],[[[]]]]=>[[[[]]],[],[],[[]],[]] [1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[],[[]],[],[[]],[],[]]=>[[],[],[[]],[],[[]],[]] [1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[],[[]],[],[[]],[[]]]=>[[[]],[[]],[],[[]],[]] [1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[],[[]],[],[[[]]],[]]=>[[],[[[]]],[],[[]],[]] [1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[],[[]],[],[[[[]]]]]=>[[[[[]]]],[],[[]],[]] [1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[],[[]],[[]],[],[],[]]=>[[],[],[],[[]],[[]],[]] [1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[],[[]],[[]],[],[[]]]=>[[[]],[],[[]],[[]],[]] [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]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[],[[]],[[]],[[[]]]]=>[[[[]]],[[]],[[]],[]] [1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[],[[]],[[[]]],[],[]]=>[[],[],[[[]]],[[]],[]] [1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[],[[]],[[[]]],[[]]]=>[[[]],[[[]]],[[]],[]] [1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[],[[]],[[[[]]]],[]]=>[[],[[[[]]]],[[]],[]] [1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[],[[]],[[[[[]]]]]]=>[[[[[[]]]]],[[]],[]] [1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[],[[[]]],[],[],[],[]]=>[[],[],[],[],[[[]]],[]] [1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[],[[[]]],[],[],[[]]]=>[[[]],[],[],[[[]]],[]] [1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[],[[[]]],[],[[]],[]]=>[[],[[]],[],[[[]]],[]] [1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[],[[[]]],[],[[[]]]]=>[[[[]]],[],[[[]]],[]] [1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[],[[[]]],[[]],[],[]]=>[[],[],[[]],[[[]]],[]] [1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[],[[[]]],[[]],[[]]]=>[[[]],[[]],[[[]]],[]] [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[],[[[]]],[[[]]],[]]=>[[],[[[]]],[[[]]],[]] [1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[],[[[]]],[[[[]]]]]=>[[[[[]]]],[[[]]],[]] [1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[],[[[[]]]],[],[],[]]=>[[],[],[],[[[[]]]],[]] [1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[],[[[[]]]],[],[[]]]=>[[[]],[],[[[[]]]],[]] [1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[],[[[[]]]],[[]],[]]=>[[],[[]],[[[[]]]],[]] [1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[],[[[[]]]],[[[]]]]=>[[[[]]],[[[[]]]],[]] [1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[],[[[[[]]]]],[],[]]=>[[],[],[[[[[]]]]],[]] [1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[],[[[[[]]]]],[[]]]=>[[[]],[[[[[]]]]],[]] [1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[],[[[[[[]]]]]],[]]=>[[],[[[[[[]]]]]],[]] [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]=>[[[[[[[[[]]]]]]]]]=>[[[[[[[[[]]]]]]]]] [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]=>[[],[],[],[],[],[],[],[[]]]=>[[[]],[],[],[],[],[],[],[]] [1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[],[],[],[],[],[],[[[]]]]=>[[[[]]],[],[],[],[],[],[]] [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[],[[[[[[[[]]]]]]]]]=>[[[[[[[[[]]]]]]]],[]] [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[[]]]]]]]],[]]=>[[],[[[[[[[[]]]]]]]]] [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]=>[[],[],[],[],[],[],[],[],[[]]]=>[[[]],[],[],[],[],[],[],[],[]] [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[[],[[[[[[[[[]]]]]]]]]]=>[[[[[[[[[[]]]]]]]]],[]] [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[[[[[[[[[[]]]]]]]]],[]]=>[[],[[[[[[[[[]]]]]]]]]]
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
left-right symmetry
Description
Return the left-right symmetrized tree of an ordered tree.