Identifier
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Images
=>
Cc0005;cc-rep-2Cc0010;cc-rep-3
[1]=>[1]=>[1,0]=>[.,.] [1,1]=>[1,1]=>[1,0,1,0]=>[[.,.],.] [2]=>[2]=>[1,1,0,0]=>[.,[.,.]] [1,1,1]=>[1,1,1]=>[1,0,1,0,1,0]=>[[[.,.],.],.] [1,2]=>[2,1]=>[1,1,0,0,1,0]=>[[.,[.,.]],.] [2,1]=>[1,2]=>[1,0,1,1,0,0]=>[[.,.],[.,.]] [3]=>[3]=>[1,1,1,0,0,0]=>[.,[.,[.,.]]] [1,1,1,1]=>[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[[[[.,.],.],.],.] [1,1,2]=>[2,1,1]=>[1,1,0,0,1,0,1,0]=>[[[.,[.,.]],.],.] [1,2,1]=>[1,2,1]=>[1,0,1,1,0,0,1,0]=>[[[.,.],[.,.]],.] [1,3]=>[3,1]=>[1,1,1,0,0,0,1,0]=>[[.,[.,[.,.]]],.] [2,1,1]=>[1,1,2]=>[1,0,1,0,1,1,0,0]=>[[[.,.],.],[.,.]] [2,2]=>[2,2]=>[1,1,0,0,1,1,0,0]=>[[.,[.,.]],[.,.]] [3,1]=>[1,3]=>[1,0,1,1,1,0,0,0]=>[[.,.],[.,[.,.]]] [4]=>[4]=>[1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,.]]]] [1,1,1,1,1]=>[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[[[[[.,.],.],.],.],.] [1,1,1,2]=>[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[[[[.,[.,.]],.],.],.] [1,1,2,1]=>[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[[[[.,.],[.,.]],.],.] [1,1,3]=>[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[[[.,[.,[.,.]]],.],.] [1,2,1,1]=>[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[[[[.,.],.],[.,.]],.] [1,2,2]=>[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[[[.,[.,.]],[.,.]],.] [1,3,1]=>[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[[[.,.],[.,[.,.]]],.] [1,4]=>[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[[.,[.,[.,[.,.]]]],.] [2,1,1,1]=>[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[[[[.,.],.],.],[.,.]] [2,1,2]=>[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[[[.,[.,.]],.],[.,.]] [2,2,1]=>[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[[[.,.],[.,.]],[.,.]] [2,3]=>[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[[.,[.,[.,.]]],[.,.]] [3,1,1]=>[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[[[.,.],.],[.,[.,.]]] [3,2]=>[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[[.,[.,.]],[.,[.,.]]] [4,1]=>[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[[.,.],[.,[.,[.,.]]]] [5]=>[5]=>[1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[.,.]]]]] [1,1,1,1,1,1]=>[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],.],.],.],.],.] [1,1,1,1,2]=>[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],.],.],.],.] [1,1,1,2,1]=>[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,.]],.],.],.] [1,1,1,3]=>[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,.]]],.],.],.] [1,1,2,1,1]=>[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],.],[.,.]],.],.] [1,1,2,2]=>[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,.]],.],.] [1,1,3,1]=>[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,.]]],.],.] [1,1,4]=>[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,.]]]],.],.] [1,2,1,1,1]=>[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],.],.],[.,.]],.] [1,2,1,2]=>[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,.]],.],[.,.]],.] [1,2,2,1]=>[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,.]],[.,.]],.] [1,2,3]=>[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,.]]],[.,.]],.] [1,3,1,1]=>[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,.],.],[.,[.,.]]],.] [1,3,2]=>[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,.]]],.] [1,4,1]=>[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,.]]]],.] [1,5]=>[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],.] [2,1,1,1,1]=>[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],.],.],.],[.,.]] [2,1,1,2]=>[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,.]],.],.],[.,.]] [2,1,2,1]=>[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,.]],.],[.,.]] [2,1,3]=>[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,.]]],.],[.,.]] [2,2,1,1]=>[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,.],.],[.,.]],[.,.]] [2,2,2]=>[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,.]],[.,.]],[.,.]] [2,3,1]=>[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,.]]],[.,.]] [2,4]=>[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,.]]]],[.,.]] [3,1,1,1]=>[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,.],.],.],[.,[.,.]]] [3,1,2]=>[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,.]],.],[.,[.,.]]] [3,2,1]=>[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,.],[.,.]],[.,[.,.]]] [3,3]=>[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,.]]],[.,[.,.]]] [4,1,1]=>[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,.],.],[.,[.,[.,.]]]] [4,2]=>[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,.]]]] [5,1]=>[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,.]]]]] [6]=>[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,.]]]]]] [1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,.],.],.],.],.],.],.] [1,1,1,1,1,2]=>[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,[.,.]],.],.],.],.],.] [1,1,1,1,2,1]=>[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],[.,.]],.],.],.],.] [1,1,1,1,3]=>[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,[.,.]]],.],.],.],.] [1,1,1,2,1,1]=>[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[.,.],.],[.,.]],.],.],.] [1,1,1,2,2]=>[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],[.,.]],.],.],.] [1,1,1,3,1]=>[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,[.,.]]],.],.],.] [1,1,1,4]=>[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,[.,.]]]],.],.],.] [1,1,2,1,1,1]=>[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[.,.],.],.],[.,.]],.],.] [1,1,2,1,2]=>[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,[.,.]],.],[.,.]],.],.] [1,1,2,2,1]=>[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],[.,.]],[.,.]],.],.] [1,1,2,3]=>[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,[.,.]]],[.,.]],.],.] [1,1,3,1,1]=>[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[.,.],.],[.,[.,.]]],.],.] [1,1,3,2]=>[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,[.,.]]],.],.] [1,1,4,1]=>[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,[.,.]]]],.],.] [1,1,5]=>[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,[.,.]]]]],.],.] [1,2,1,1,1,1]=>[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[.,.],.],.],.],[.,.]],.] [1,2,1,1,2]=>[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,[.,.]],.],.],[.,.]],.] [1,2,1,2,1]=>[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],[.,.]],.],[.,.]],.] [1,2,1,3]=>[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,[.,.]]],.],[.,.]],.] [1,2,2,1,1]=>[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,.],.],[.,.]],[.,.]],.] [1,2,2,2]=>[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,[.,.]],[.,.]],[.,.]],.] [1,2,3,1]=>[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,[.,.]]],[.,.]],.] [1,2,4]=>[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,[.,.]]]],[.,.]],.] [1,3,1,1,1]=>[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[.,.],.],.],[.,[.,.]]],.] [1,3,1,2]=>[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,[.,.]],.],[.,[.,.]]],.] [1,3,2,1]=>[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[[.,.],[.,.]],[.,[.,.]]],.] [1,3,3]=>[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,[.,.]]],[.,[.,.]]],.] [1,4,1,1]=>[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[[.,.],.],[.,[.,[.,.]]]],.] [1,4,2]=>[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,[.,.]]]],.] [1,5,1]=>[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,[.,.]]]]],.] [1,6]=>[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],.] [2,1,1,1,1,1]=>[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[.,.],.],.],.],.],[.,.]] [2,1,1,1,2]=>[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,.]],.],.],.],[.,.]] [2,1,1,2,1]=>[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],[.,.]],.],.],[.,.]] [2,1,1,3]=>[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,[.,.]]],.],.],[.,.]] [2,1,2,1,1]=>[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[.,.],.],[.,.]],.],[.,.]] [2,1,2,2]=>[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,[.,.]],[.,.]],.],[.,.]] [2,1,3,1]=>[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,[.,.]]],.],[.,.]] [2,1,4]=>[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,[.,.]]]],.],[.,.]] [2,2,1,1,1]=>[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[.,.],.],.],[.,.]],[.,.]] [2,2,1,2]=>[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,[.,.]],.],[.,.]],[.,.]] [2,2,2,1]=>[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [2,2,3]=>[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,[.,.]]],[.,.]],[.,.]] [2,3,1,1]=>[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[[.,.],.],[.,[.,.]]],[.,.]] [2,3,2]=>[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,[.,.]],[.,[.,.]]],[.,.]] [2,4,1]=>[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,[.,.]]]],[.,.]] [2,5]=>[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]] [3,1,1,1,1]=>[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[.,.],.],.],.],[.,[.,.]]] [3,1,1,2]=>[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,[.,.]],.],.],[.,[.,.]]] [3,1,2,1]=>[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[[.,.],[.,.]],.],[.,[.,.]]] [3,1,3]=>[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,[.,.]]],.],[.,[.,.]]] [3,2,1,1]=>[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [3,2,2]=>[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [3,3,1]=>[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [3,4]=>[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [4,1,1,1]=>[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [4,1,2]=>[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,[.,.]],.],[.,[.,[.,.]]]] [4,2,1]=>[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [4,3]=>[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [5,1,1]=>[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [5,2]=>[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [6,1]=>[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [7]=>[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]] [1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[.,.],.],.],.],.],.],.],.] [1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,[.,.]],.],.],.],.],.],.] [1,1,1,1,1,2,1]=>[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,.],[.,.]],.],.],.],.],.] [1,1,1,1,1,3]=>[3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,[.,[.,.]]],.],.],.],.],.] [1,1,1,1,2,1,1]=>[1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,.],.],[.,.]],.],.],.],.] [1,1,1,1,2,2]=>[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[.,[.,.]],[.,.]],.],.],.],.] [1,1,1,1,3,1]=>[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],[.,[.,.]]],.],.],.],.] [1,1,1,1,4]=>[4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,[.,[.,.]]]],.],.],.],.] [1,1,1,2,1,1,1]=>[1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[[.,.],.],.],[.,.]],.],.],.] [1,1,1,2,1,2]=>[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[.,[.,.]],.],[.,.]],.],.],.] [1,1,1,2,2,1]=>[1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[.,.],[.,.]],[.,.]],.],.],.] [1,1,1,2,3]=>[3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,[.,[.,.]]],[.,.]],.],.],.] [1,1,1,3,1,1]=>[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[[[.,.],.],[.,[.,.]]],.],.],.] [1,1,1,3,2]=>[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],[.,[.,.]]],.],.],.] [1,1,1,4,1]=>[1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,[.,[.,.]]]],.],.],.] [1,1,1,5]=>[5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,[.,[.,.]]]]],.],.],.] [1,1,2,1,1,1,1]=>[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[[.,.],.],.],.],[.,.]],.],.] [1,1,2,1,1,2]=>[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[.,[.,.]],.],.],[.,.]],.],.] [1,1,2,1,2,1]=>[1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[.,.],[.,.]],.],[.,.]],.],.] [1,1,2,1,3]=>[3,1,2,1,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,[.,[.,.]]],.],[.,.]],.],.] [1,1,2,2,1,1]=>[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[[[.,.],.],[.,.]],[.,.]],.],.] [1,1,2,2,2]=>[2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[[.,[.,.]],[.,.]],[.,.]],.],.] [1,1,2,3,1]=>[1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],[.,[.,.]]],[.,.]],.],.] [1,1,2,4]=>[4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,[.,[.,.]]]],[.,.]],.],.] [1,1,3,1,1,1]=>[1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[[.,.],.],.],[.,[.,.]]],.],.] [1,1,3,1,2]=>[2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[.,[.,.]],.],[.,[.,.]]],.],.] [1,1,3,2,1]=>[1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[.,.],[.,.]],[.,[.,.]]],.],.] [1,1,3,3]=>[3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,[.,[.,.]]],[.,[.,.]]],.],.] [1,1,4,1,1]=>[1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[[.,.],.],[.,[.,[.,.]]]],.],.] [1,1,4,2]=>[2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,[.,[.,.]]]],.],.] [1,1,5,1]=>[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,[.,[.,.]]]]],.],.] [1,1,6]=>[6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.] [1,2,1,1,1,1,1]=>[1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[[.,.],.],.],.],.],[.,.]],.] [1,2,1,1,1,2]=>[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[.,[.,.]],.],.],.],[.,.]],.] [1,2,1,1,2,1]=>[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[.,.],[.,.]],.],.],[.,.]],.] [1,2,1,1,3]=>[3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,[.,[.,.]]],.],.],[.,.]],.] [1,2,1,2,1,1]=>[1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[[[.,.],.],[.,.]],.],[.,.]],.] [1,2,1,2,2]=>[2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[[.,[.,.]],[.,.]],.],[.,.]],.] [1,2,1,3,1]=>[1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],[.,[.,.]]],.],[.,.]],.] [1,2,1,4]=>[4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,[.,[.,.]]]],.],[.,.]],.] [1,2,2,1,1,1]=>[1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[[.,.],.],.],[.,.]],[.,.]],.] [1,2,2,1,2]=>[2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,[.,.]],.],[.,.]],[.,.]],.] [1,2,2,2,1]=>[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,.],[.,.]],[.,.]],[.,.]],.] [1,2,2,3]=>[3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,[.,[.,.]]],[.,.]],[.,.]],.] [1,2,3,1,1]=>[1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[[.,.],.],[.,[.,.]]],[.,.]],.] [1,2,3,2]=>[2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[.,[.,.]],[.,[.,.]]],[.,.]],.] [1,2,4,1]=>[1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,[.,[.,.]]]],[.,.]],.] [1,2,5]=>[5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.] [1,3,1,1,1,1]=>[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[[.,.],.],.],.],[.,[.,.]]],.] [1,3,1,1,2]=>[2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[.,[.,.]],.],.],[.,[.,.]]],.] [1,3,1,2,1]=>[1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[.,.],[.,.]],.],[.,[.,.]]],.] [1,3,1,3]=>[3,1,3,1]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,[.,[.,.]]],.],[.,[.,.]]],.] [1,3,2,1,1]=>[1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[[[.,.],.],[.,.]],[.,[.,.]]],.] [1,3,2,2]=>[2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[[.,[.,.]],[.,.]],[.,[.,.]]],.] [1,3,3,1]=>[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[[[.,.],[.,[.,.]]],[.,[.,.]]],.] [1,3,4]=>[4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.] [1,4,1,1,1]=>[1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[[[.,.],.],.],[.,[.,[.,.]]]],.] [1,4,1,2]=>[2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[[.,[.,.]],.],[.,[.,[.,.]]]],.] [1,4,2,1]=>[1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[[.,.],[.,.]],[.,[.,[.,.]]]],.] [1,4,3]=>[3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,[.,[.,.]]],[.,[.,[.,.]]]],.] [1,5,1,1]=>[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[[.,.],.],[.,[.,[.,[.,.]]]]],.] [1,5,2]=>[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.] [1,6,1]=>[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.] [1,7]=>[7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.] [2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[[.,.],.],.],.],.],.],[.,.]] [2,1,1,1,1,2]=>[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[.,[.,.]],.],.],.],.],[.,.]] [2,1,1,1,2,1]=>[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[.,.],[.,.]],.],.],.],[.,.]] [2,1,1,1,3]=>[3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,[.,.]]],.],.],.],[.,.]] [2,1,1,2,1,1]=>[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[[.,.],.],[.,.]],.],.],[.,.]] [2,1,1,2,2]=>[2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,.]],[.,.]],.],.],[.,.]] [2,1,1,3,1]=>[1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],[.,[.,.]]],.],.],[.,.]] [2,1,1,4]=>[4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,[.,[.,.]]]],.],.],[.,.]] [2,1,2,1,1,1]=>[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[[.,.],.],.],[.,.]],.],[.,.]] [2,1,2,1,2]=>[2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[.,[.,.]],.],[.,.]],.],[.,.]] [2,1,2,2,1]=>[1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[.,.],[.,.]],[.,.]],.],[.,.]] [2,1,2,3]=>[3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,[.,[.,.]]],[.,.]],.],[.,.]] [2,1,3,1,1]=>[1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[[.,.],.],[.,[.,.]]],.],[.,.]] [2,1,3,2]=>[2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[.,[.,.]],[.,[.,.]]],.],[.,.]] [2,1,4,1]=>[1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,[.,[.,.]]]],.],[.,.]] [2,1,5]=>[5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]] [2,2,1,1,1,1]=>[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[[.,.],.],.],.],[.,.]],[.,.]] [2,2,1,1,2]=>[2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[.,[.,.]],.],.],[.,.]],[.,.]] [2,2,1,2,1]=>[1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[.,.],[.,.]],.],[.,.]],[.,.]] [2,2,1,3]=>[3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,[.,[.,.]]],.],[.,.]],[.,.]] [2,2,2,1,1]=>[1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[[[.,.],.],[.,.]],[.,.]],[.,.]] [2,2,2,2]=>[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]] [2,2,3,1]=>[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[[.,.],[.,[.,.]]],[.,.]],[.,.]] [2,2,4]=>[4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]] [2,3,1,1,1]=>[1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[[[.,.],.],.],[.,[.,.]]],[.,.]] [2,3,1,2]=>[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[[.,[.,.]],.],[.,[.,.]]],[.,.]] [2,3,2,1]=>[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[[.,.],[.,.]],[.,[.,.]]],[.,.]] [2,3,3]=>[3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]] [2,4,1,1]=>[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[[.,.],.],[.,[.,[.,.]]]],[.,.]] [2,4,2]=>[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]] [2,5,1]=>[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,[.,[.,.]]]]],[.,.]] [2,6]=>[6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]] [3,1,1,1,1,1]=>[1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[[.,.],.],.],.],.],[.,[.,.]]] [3,1,1,1,2]=>[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[.,[.,.]],.],.],.],[.,[.,.]]] [3,1,1,2,1]=>[1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[.,.],[.,.]],.],.],[.,[.,.]]] [3,1,1,3]=>[3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,[.,[.,.]]],.],.],[.,[.,.]]] [3,1,2,1,1]=>[1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[[[.,.],.],[.,.]],.],[.,[.,.]]] [3,1,2,2]=>[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[[.,[.,.]],[.,.]],.],[.,[.,.]]] [3,1,3,1]=>[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[[[.,.],[.,[.,.]]],.],[.,[.,.]]] [3,1,4]=>[4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,[.,[.,.]]]],.],[.,[.,.]]] [3,2,1,1,1]=>[1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[[.,.],.],.],[.,.]],[.,[.,.]]] [3,2,1,2]=>[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[.,[.,.]],.],[.,.]],[.,[.,.]]] [3,2,2,1]=>[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[[.,.],[.,.]],[.,.]],[.,[.,.]]] [3,2,3]=>[3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,[.,[.,.]]],[.,.]],[.,[.,.]]] [3,3,1,1]=>[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[[.,.],.],[.,[.,.]]],[.,[.,.]]] [3,3,2]=>[2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]] [3,4,1]=>[1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]] [3,5]=>[5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]] [4,1,1,1,1]=>[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[[.,.],.],.],.],[.,[.,[.,.]]]] [4,1,1,2]=>[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[.,[.,.]],.],.],[.,[.,[.,.]]]] [4,1,2,1]=>[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[[.,.],[.,.]],.],[.,[.,[.,.]]]] [4,1,3]=>[3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,[.,[.,.]]],.],[.,[.,[.,.]]]] [4,2,1,1]=>[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[[.,.],.],[.,.]],[.,[.,[.,.]]]] [4,2,2]=>[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] [4,3,1]=>[1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]] [4,4]=>[4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] [5,1,1,1]=>[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[[.,.],.],.],[.,[.,[.,[.,.]]]]] [5,1,2]=>[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,[.,.]],.],[.,[.,[.,[.,.]]]]] [5,2,1]=>[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]] [5,3]=>[3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]] [6,1,1]=>[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]] [6,2]=>[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]] [7,1]=>[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] [8]=>[8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],.] [1,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[.,[.,.]],.],.],.],.],.],.],.] [1,1,1,1,1,1,2,1]=>[1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[.,.],[.,.]],.],.],.],.],.],.] [1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.] [1,1,1,1,1,2,1,1]=>[1,1,2,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[.,.],.],[.,.]],.],.],.],.],.] [1,1,1,1,1,2,2]=>[2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,[.,.]],[.,.]],.],.],.],.],.] [1,1,1,1,2,1,1,1]=>[1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[[[.,.],.],.],[.,.]],.],.],.],.] [1,1,1,1,2,1,2]=>[2,1,2,1,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,[.,.]],.],[.,.]],.],.],.],.] [1,1,1,2,1,1,1,1]=>[1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[[[.,.],.],.],.],[.,.]],.],.],.] [1,1,1,2,1,1,2]=>[2,1,1,2,1,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[[.,[.,.]],.],.],[.,.]],.],.],.] [1,1,2,1,1,1,1,1]=>[1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[[[.,.],.],.],.],.],[.,.]],.],.] [1,1,2,1,1,1,2]=>[2,1,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[[.,[.,.]],.],.],.],[.,.]],.],.] [1,1,7]=>[7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.] [1,2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[[[.,.],.],.],.],.],.],[.,.]],.] [1,2,1,1,1,1,2]=>[2,1,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[[.,[.,.]],.],.],.],.],[.,.]],.] [1,3,1,1,1,1,1]=>[1,1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[[[.,.],.],.],.],.],[.,[.,.]]],.] [1,6,1,1]=>[1,1,6,1]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],.] [1,7,1]=>[1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.] [1,8]=>[8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] [2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,.]] [2,1,1,1,1,1,2]=>[2,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]] [2,2,1,1,1,1,1]=>[1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[[[.,.],.],.],.],.],[.,.]],[.,.]] [2,2,1,1,1,2]=>[2,1,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[[.,[.,.]],.],.],.],[.,.]],[.,.]] [2,7]=>[7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]] [3,1,1,1,1,1,1]=>[1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]] [3,2,1,1,1,1]=>[1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[[[.,.],.],.],.],[.,.]],[.,[.,.]]] [3,6]=>[6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]] [4,1,1,1,1,1]=>[1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]] [4,2,1,1,1]=>[1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[[[.,.],.],.],[.,.]],[.,[.,[.,.]]]] [4,5]=>[5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]] [5,1,1,1,1]=>[1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]] [5,1,3]=>[3,1,5]=>[1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,[.,[.,.]]],.],[.,[.,[.,[.,.]]]]] [5,2,1,1]=>[1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[[.,.],.],[.,.]],[.,[.,[.,[.,.]]]]] [5,4]=>[4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]] [6,1,1,1]=>[1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]] [6,2,1]=>[1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]] [6,3]=>[3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]] [7,1,1]=>[1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]] [7,2]=>[2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]] [8,1]=>[1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [9]=>[9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [1,1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.] [1,1,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.] [1,1,1,1,1,1,1,2,1]=>[1,2,1,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[.,.],[.,.]],.],.],.],.],.],.],.] [1,1,1,1,1,1,2,1,1]=>[1,1,2,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[.,.],.],[.,.]],.],.],.],.],.],.] [1,1,1,1,1,2,1,1,1]=>[1,1,1,2,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[.,.],.],.],[.,.]],.],.],.],.],.] [1,1,1,1,2,1,1,1,1]=>[1,1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[[[[.,.],.],.],.],[.,.]],.],.],.],.] [1,1,1,2,1,1,1,1,1]=>[1,1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],[.,.]],.],.],.] [1,1,2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],.],[.,.]],.],.] [1,2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],[.,.]],.] [1,8,1]=>[1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.] [1,9]=>[9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.] [2,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]] [2,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]] [2,8]=>[8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]] [3,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]] [3,7]=>[7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]] [4,1,1,1,1,1,1]=>[1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]] [4,6]=>[6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]] [5,1,1,1,1,1]=>[1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]] [5,5]=>[5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]] [6,1,1,1,1]=>[1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]] [6,4]=>[4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]] [7,1,1,1]=>[1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]] [7,3]=>[3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]] [8,1,1]=>[1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [8,2]=>[2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [9,1]=>[1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]] [10]=>[10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary tree: left tree, up step, right tree, down step
Description
Return the binary tree corresponding to the Dyck path under the transformation left tree - up step - right tree - down step.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ 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.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.