Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Images
=>
Cc0005;cc-rep-1Cc0010;cc-rep-2
[1]=>[1,0]=>[.,.]=>[1] [1,1]=>[1,0,1,0]=>[[.,.],.]=>[1,2] [2]=>[1,1,0,0]=>[.,[.,.]]=>[2,1] [1,1,1]=>[1,0,1,0,1,0]=>[[[.,.],.],.]=>[1,2,3] [1,2]=>[1,0,1,1,0,0]=>[[.,.],[.,.]]=>[3,1,2] [2,1]=>[1,1,0,0,1,0]=>[[.,[.,.]],.]=>[2,1,3] [3]=>[1,1,1,0,0,0]=>[.,[.,[.,.]]]=>[3,2,1] [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[[[[.,.],.],.],.]=>[1,2,3,4] [1,1,2]=>[1,0,1,0,1,1,0,0]=>[[[.,.],.],[.,.]]=>[4,1,2,3] [1,2,1]=>[1,0,1,1,0,0,1,0]=>[[[.,.],[.,.]],.]=>[3,1,2,4] [1,3]=>[1,0,1,1,1,0,0,0]=>[[.,.],[.,[.,.]]]=>[4,3,1,2] [2,1,1]=>[1,1,0,0,1,0,1,0]=>[[[.,[.,.]],.],.]=>[2,1,3,4] [2,2]=>[1,1,0,0,1,1,0,0]=>[[.,[.,.]],[.,.]]=>[4,2,1,3] [3,1]=>[1,1,1,0,0,0,1,0]=>[[.,[.,[.,.]]],.]=>[3,2,1,4] [4]=>[1,1,1,1,0,0,0,0]=>[.,[.,[.,[.,.]]]]=>[4,3,2,1] [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[[[[[.,.],.],.],.],.]=>[1,2,3,4,5] [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[[[[.,.],.],.],[.,.]]=>[5,1,2,3,4] [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[[[[.,.],.],[.,.]],.]=>[4,1,2,3,5] [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3] [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[[[[.,.],[.,.]],.],.]=>[3,1,2,4,5] [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4] [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5] [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2] [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5] [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4] [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5] [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3] [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5] [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4] [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5] [5]=>[1,1,1,1,1,0,0,0,0,0]=>[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1] [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6] [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],.],.],.],[.,.]]=>[6,1,2,3,4,5] [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],.],.],[.,.]],.]=>[5,1,2,3,4,6] [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4] [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],.],[.,.]],.],.]=>[4,1,2,3,5,6] [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,.],.],[.,.]],[.,.]]=>[6,4,1,2,3,5] [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,.],.],[.,[.,.]]],.]=>[5,4,1,2,3,6] [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3] [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,.]],.],.],.]=>[3,1,2,4,5,6] [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,.]],.],[.,.]]=>[6,3,1,2,4,5] [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,.]],[.,.]],.]=>[5,3,1,2,4,6] [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4] [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,.]]],.],.]=>[4,3,1,2,5,6] [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5] [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,.]]]],.]=>[5,4,3,1,2,6] [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2] [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6] [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,.]],.],.],[.,.]]=>[6,2,1,3,4,5] [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,.]],.],[.,.]],.]=>[5,2,1,3,4,6] [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4] [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,.]],.],.]=>[4,2,1,3,5,6] [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5] [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,.]]],.]=>[5,4,2,1,3,6] [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3] [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6] [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,.]]],.],[.,.]]=>[6,3,2,1,4,5] [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,.]]],[.,.]],.]=>[5,3,2,1,4,6] [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4] [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6] [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5] [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6] [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1] [1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7] [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[.,.],.],.],.],.],[.,.]]=>[7,1,2,3,4,5,6] [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[.,.],.],.],.],[.,.]],.]=>[6,1,2,3,4,5,7] [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[.,.],.],.],.],[.,[.,.]]]=>[7,6,1,2,3,4,5] [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[.,.],.],.],[.,.]],.],.]=>[5,1,2,3,4,6,7] [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[.,.],.],.],[.,.]],[.,.]]=>[7,5,1,2,3,4,6] [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[.,.],.],.],[.,[.,.]]],.]=>[6,5,1,2,3,4,7] [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[.,.],.],.],[.,[.,[.,.]]]]=>[7,6,5,1,2,3,4] [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[.,.],.],[.,.]],.],.],.]=>[4,1,2,3,5,6,7] [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[.,.],.],[.,.]],.],[.,.]]=>[7,4,1,2,3,5,6] [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,.],.],[.,.]],[.,.]],.]=>[6,4,1,2,3,5,7] [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[.,.],.],[.,.]],[.,[.,.]]]=>[7,6,4,1,2,3,5] [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[.,.],.],[.,[.,.]]],.],.]=>[5,4,1,2,3,6,7] [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[[.,.],.],[.,[.,.]]],[.,.]]=>[7,5,4,1,2,3,6] [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[[.,.],.],[.,[.,[.,.]]]],.]=>[6,5,4,1,2,3,7] [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]=>[7,6,5,4,1,2,3] [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],[.,.]],.],.],.],.]=>[3,1,2,4,5,6,7] [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],[.,.]],.],.],[.,.]]=>[7,3,1,2,4,5,6] [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],[.,.]],.],[.,.]],.]=>[6,3,1,2,4,5,7] [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[[.,.],[.,.]],.],[.,[.,.]]]=>[7,6,3,1,2,4,5] [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],[.,.]],[.,.]],.],.]=>[5,3,1,2,4,6,7] [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[[.,.],[.,.]],[.,.]],[.,.]]=>[7,5,3,1,2,4,6] [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[[.,.],[.,.]],[.,[.,.]]],.]=>[6,5,3,1,2,4,7] [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[7,6,5,3,1,2,4] [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,[.,.]]],.],.],.]=>[4,3,1,2,5,6,7] [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,[.,.]]],.],[.,.]]=>[7,4,3,1,2,5,6] [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,[.,.]]],[.,.]],.]=>[6,4,3,1,2,5,7] [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[7,6,4,3,1,2,5] [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,[.,.]]]],.],.]=>[5,4,3,1,2,6,7] [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]=>[7,5,4,3,1,2,6] [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,[.,.]]]]],.]=>[6,5,4,3,1,2,7] [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]=>[7,6,5,4,3,1,2] [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,[.,.]],.],.],.],.],.]=>[2,1,3,4,5,6,7] [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,.]],.],.],.],[.,.]]=>[7,2,1,3,4,5,6] [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,[.,.]],.],.],[.,.]],.]=>[6,2,1,3,4,5,7] [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,[.,.]],.],.],[.,[.,.]]]=>[7,6,2,1,3,4,5] [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,[.,.]],.],[.,.]],.],.]=>[5,2,1,3,4,6,7] [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,[.,.]],.],[.,.]],[.,.]]=>[7,5,2,1,3,4,6] [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,[.,.]],.],[.,[.,.]]],.]=>[6,5,2,1,3,4,7] [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,[.,.]],.],[.,[.,[.,.]]]]=>[7,6,5,2,1,3,4] [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],[.,.]],.],.],.]=>[4,2,1,3,5,6,7] [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,[.,.]],[.,.]],.],[.,.]]=>[7,4,2,1,3,5,6] [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,[.,.]],[.,.]],[.,.]],.]=>[6,4,2,1,3,5,7] [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,[.,.]],[.,.]],[.,[.,.]]]=>[7,6,4,2,1,3,5] [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,[.,.]]],.],.]=>[5,4,2,1,3,6,7] [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,[.,.]],[.,[.,.]]],[.,.]]=>[7,5,4,2,1,3,6] [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,[.,.]]]],.]=>[6,5,4,2,1,3,7] [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]]=>[7,6,5,4,2,1,3] [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,[.,.]]],.],.],.],.]=>[3,2,1,4,5,6,7] [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,[.,.]]],.],.],[.,.]]=>[7,3,2,1,4,5,6] [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,[.,.]]],.],[.,.]],.]=>[6,3,2,1,4,5,7] [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,[.,.]]],.],[.,[.,.]]]=>[7,6,3,2,1,4,5] [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,[.,.]]],[.,.]],.],.]=>[5,3,2,1,4,6,7] [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,[.,.]]],[.,.]],[.,.]]=>[7,5,3,2,1,4,6] [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,[.,.]]],[.,[.,.]]],.]=>[6,5,3,2,1,4,7] [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]]=>[7,6,5,3,2,1,4] [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,[.,.]]]],.],.],.]=>[4,3,2,1,5,6,7] [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,[.,.]]]],.],[.,.]]=>[7,4,3,2,1,5,6] [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,[.,.]]]],[.,.]],.]=>[6,4,3,2,1,5,7] [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]]=>[7,6,4,3,2,1,5] [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,[.,.]]]]],.],.]=>[5,4,3,2,1,6,7] [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]]=>[7,5,4,3,2,1,6] [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],.]=>[6,5,4,3,2,1,7] [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]]=>[7,6,5,4,3,2,1] [1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[[.,.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8] [1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[[.,.],.],.],.],.],.],[.,.]]=>[8,1,2,3,4,5,6,7] [1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[[.,.],.],.],.],.],[.,.]],.]=>[7,1,2,3,4,5,6,8] [1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[[.,.],.],.],.],.],[.,[.,.]]]=>[8,7,1,2,3,4,5,6] [1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[[.,.],.],.],.],[.,.]],.],.]=>[6,1,2,3,4,5,7,8] [1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[[.,.],.],.],.],[.,.]],[.,.]]=>[8,6,1,2,3,4,5,7] [1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[[.,.],.],.],.],[.,[.,.]]],.]=>[7,6,1,2,3,4,5,8] [1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[[.,.],.],.],.],[.,[.,[.,.]]]]=>[8,7,6,1,2,3,4,5] [1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[[.,.],.],.],[.,.]],.],.],.]=>[5,1,2,3,4,6,7,8] [1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[[.,.],.],.],[.,.]],.],[.,.]]=>[8,5,1,2,3,4,6,7] [1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[[.,.],.],.],[.,.]],[.,.]],.]=>[7,5,1,2,3,4,6,8] [1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[[.,.],.],.],[.,.]],[.,[.,.]]]=>[8,7,5,1,2,3,4,6] [1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[[.,.],.],.],[.,[.,.]]],.],.]=>[6,5,1,2,3,4,7,8] [1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[[[.,.],.],.],[.,[.,.]]],[.,.]]=>[8,6,5,1,2,3,4,7] [1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[[[.,.],.],.],[.,[.,[.,.]]]],.]=>[7,6,5,1,2,3,4,8] [1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]=>[8,7,6,5,1,2,3,4] [1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,.],.],[.,.]],.],.],.],.]=>[4,1,2,3,5,6,7,8] [1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[[.,.],.],[.,.]],.],.],[.,.]]=>[8,4,1,2,3,5,6,7] [1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[[[.,.],.],[.,.]],.],[.,.]],.]=>[7,4,1,2,3,5,6,8] [1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[[[.,.],.],[.,.]],.],[.,[.,.]]]=>[8,7,4,1,2,3,5,6] [1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[[[.,.],.],[.,.]],[.,.]],.],.]=>[6,4,1,2,3,5,7,8] [1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[[[.,.],.],[.,.]],[.,.]],[.,.]]=>[8,6,4,1,2,3,5,7] [1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[[[.,.],.],[.,.]],[.,[.,.]]],.]=>[7,6,4,1,2,3,5,8] [1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]=>[8,7,6,4,1,2,3,5] [1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[[[.,.],.],[.,[.,.]]],.],.],.]=>[5,4,1,2,3,6,7,8] [1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[[.,.],.],[.,[.,.]]],.],[.,.]]=>[8,5,4,1,2,3,6,7] [1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[[.,.],.],[.,[.,.]]],[.,.]],.]=>[7,5,4,1,2,3,6,8] [1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[[.,.],.],[.,[.,.]]],[.,[.,.]]]=>[8,7,5,4,1,2,3,6] [1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[[.,.],.],[.,[.,[.,.]]]],.],.]=>[6,5,4,1,2,3,7,8] [1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[[.,.],.],[.,[.,[.,.]]]],[.,.]]=>[8,6,5,4,1,2,3,7] [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[[.,.],.],[.,[.,[.,[.,.]]]]],.]=>[7,6,5,4,1,2,3,8] [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]=>[8,7,6,5,4,1,2,3] [1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,.],[.,.]],.],.],.],.],.]=>[3,1,2,4,5,6,7,8] [1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[.,.],[.,.]],.],.],.],[.,.]]=>[8,3,1,2,4,5,6,7] [1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[.,.],[.,.]],.],.],[.,.]],.]=>[7,3,1,2,4,5,6,8] [1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[.,.],[.,.]],.],.],[.,[.,.]]]=>[8,7,3,1,2,4,5,6] [1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[.,.],[.,.]],.],[.,.]],.],.]=>[6,3,1,2,4,5,7,8] [1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[.,.],[.,.]],.],[.,.]],[.,.]]=>[8,6,3,1,2,4,5,7] [1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[.,.],[.,.]],.],[.,[.,.]]],.]=>[7,6,3,1,2,4,5,8] [1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]=>[8,7,6,3,1,2,4,5] [1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[.,.],[.,.]],[.,.]],.],.],.]=>[5,3,1,2,4,6,7,8] [1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[.,.],[.,.]],[.,.]],.],[.,.]]=>[8,5,3,1,2,4,6,7] [1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,.],[.,.]],[.,.]],[.,.]],.]=>[7,5,3,1,2,4,6,8] [1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[[.,.],[.,.]],[.,.]],[.,[.,.]]]=>[8,7,5,3,1,2,4,6] [1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[.,.],[.,.]],[.,[.,.]]],.],.]=>[6,5,3,1,2,4,7,8] [1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[[.,.],[.,.]],[.,[.,.]]],[.,.]]=>[8,6,5,3,1,2,4,7] [1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[[.,.],[.,.]],[.,[.,[.,.]]]],.]=>[7,6,5,3,1,2,4,8] [1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]=>[8,7,6,5,3,1,2,4] [1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[[.,.],[.,[.,.]]],.],.],.],.]=>[4,3,1,2,5,6,7,8] [1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[[[[[.,.],[.,[.,.]]],.],.],[.,.]]=>[8,4,3,1,2,5,6,7] [1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[[[[[.,.],[.,[.,.]]],.],[.,.]],.]=>[7,4,3,1,2,5,6,8] [1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]=>[8,7,4,3,1,2,5,6] [1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[[[[[.,.],[.,[.,.]]],[.,.]],.],.]=>[6,4,3,1,2,5,7,8] [1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]=>[8,6,4,3,1,2,5,7] [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]=>[7,6,4,3,1,2,5,8] [1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]=>[8,7,6,4,3,1,2,5] [1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[[[[[.,.],[.,[.,[.,.]]]],.],.],.]=>[5,4,3,1,2,6,7,8] [1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[[[[.,.],[.,[.,[.,.]]]],.],[.,.]]=>[8,5,4,3,1,2,6,7] [1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[[[[.,.],[.,[.,[.,.]]]],[.,.]],.]=>[7,5,4,3,1,2,6,8] [1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]=>[8,7,5,4,3,1,2,6] [1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[[[[.,.],[.,[.,[.,[.,.]]]]],.],.]=>[6,5,4,3,1,2,7,8] [1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]=>[8,6,5,4,3,1,2,7] [1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]=>[7,6,5,4,3,1,2,8] [1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[8,7,6,5,4,3,1,2] [2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[[.,[.,.]],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8] [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[[.,[.,.]],.],.],.],.],[.,.]]=>[8,2,1,3,4,5,6,7] [2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[[.,[.,.]],.],.],.],[.,.]],.]=>[7,2,1,3,4,5,6,8] [2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[[[[.,[.,.]],.],.],.],[.,[.,.]]]=>[8,7,2,1,3,4,5,6] [2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[[.,[.,.]],.],.],[.,.]],.],.]=>[6,2,1,3,4,5,7,8] [2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[[[[.,[.,.]],.],.],[.,.]],[.,.]]=>[8,6,2,1,3,4,5,7] [2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[[[[[.,[.,.]],.],.],[.,[.,.]]],.]=>[7,6,2,1,3,4,5,8] [2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[[[.,[.,.]],.],.],[.,[.,[.,.]]]]=>[8,7,6,2,1,3,4,5] [2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[[.,[.,.]],.],[.,.]],.],.],.]=>[5,2,1,3,4,6,7,8] [2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[[[[.,[.,.]],.],[.,.]],.],[.,.]]=>[8,5,2,1,3,4,6,7] [2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[[[[.,[.,.]],.],[.,.]],[.,.]],.]=>[7,5,2,1,3,4,6,8] [2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]=>[8,7,5,2,1,3,4,6] [2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[[[[[.,[.,.]],.],[.,[.,.]]],.],.]=>[6,5,2,1,3,4,7,8] [2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[[[[.,[.,.]],.],[.,[.,.]]],[.,.]]=>[8,6,5,2,1,3,4,7] [2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[[[[.,[.,.]],.],[.,[.,[.,.]]]],.]=>[7,6,5,2,1,3,4,8] [2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]=>[8,7,6,5,2,1,3,4] [2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[[[[[.,[.,.]],[.,.]],.],.],.],.]=>[4,2,1,3,5,6,7,8] [2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,.]],[.,.]],.],.],[.,.]]=>[8,4,2,1,3,5,6,7] [2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[[[[.,[.,.]],[.,.]],.],[.,.]],.]=>[7,4,2,1,3,5,6,8] [2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[[[[.,[.,.]],[.,.]],.],[.,[.,.]]]=>[8,7,4,2,1,3,5,6] [2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[[[[.,[.,.]],[.,.]],[.,.]],.],.]=>[6,4,2,1,3,5,7,8] [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]=>[8,6,4,2,1,3,5,7] [2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[[[[.,[.,.]],[.,.]],[.,[.,.]]],.]=>[7,6,4,2,1,3,5,8] [2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]=>[8,7,6,4,2,1,3,5] [2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[[[[[.,[.,.]],[.,[.,.]]],.],.],.]=>[5,4,2,1,3,6,7,8] [2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[.,[.,.]],[.,[.,.]]],.],[.,.]]=>[8,5,4,2,1,3,6,7] [2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[.,[.,.]],[.,[.,.]]],[.,.]],.]=>[7,5,4,2,1,3,6,8] [2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]=>[8,7,5,4,2,1,3,6] [2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[[[[.,[.,.]],[.,[.,[.,.]]]],.],.]=>[6,5,4,2,1,3,7,8] [2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]=>[8,6,5,4,2,1,3,7] [2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]=>[7,6,5,4,2,1,3,8] [2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[8,7,6,5,4,2,1,3] [3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[[[[[[.,[.,[.,.]]],.],.],.],.],.]=>[3,2,1,4,5,6,7,8] [3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[[[[[.,[.,[.,.]]],.],.],.],[.,.]]=>[8,3,2,1,4,5,6,7] [3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[[[[[.,[.,[.,.]]],.],.],[.,.]],.]=>[7,3,2,1,4,5,6,8] [3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]=>[8,7,3,2,1,4,5,6] [3,1,2,1,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[[[[[.,[.,[.,.]]],.],[.,.]],.],.]=>[6,3,2,1,4,5,7,8] [3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]=>[8,6,3,2,1,4,5,7] [3,1,3,1]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[[[[.,[.,[.,.]]],.],[.,[.,.]]],.]=>[7,6,3,2,1,4,5,8] [3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]=>[8,7,6,3,2,1,4,5] [3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[[[[[.,[.,[.,.]]],[.,.]],.],.],.]=>[5,3,2,1,4,6,7,8] [3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[[[[.,[.,[.,.]]],[.,.]],.],[.,.]]=>[8,5,3,2,1,4,6,7] [3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]=>[7,5,3,2,1,4,6,8] [3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]=>[8,7,5,3,2,1,4,6] [3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[[[[.,[.,[.,.]]],[.,[.,.]]],.],.]=>[6,5,3,2,1,4,7,8] [3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]=>[8,6,5,3,2,1,4,7] [3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]=>[7,6,5,3,2,1,4,8] [3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]=>[8,7,6,5,3,2,1,4] [4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[[[[[.,[.,[.,[.,.]]]],.],.],.],.]=>[4,3,2,1,5,6,7,8] [4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]=>[8,4,3,2,1,5,6,7] [4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[[[[.,[.,[.,[.,.]]]],.],[.,.]],.]=>[7,4,3,2,1,5,6,8] [4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]=>[8,7,4,3,2,1,5,6] [4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[[[[.,[.,[.,[.,.]]]],[.,.]],.],.]=>[6,4,3,2,1,5,7,8] [4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]=>[8,6,4,3,2,1,5,7] [4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]=>[7,6,4,3,2,1,5,8] [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]=>[8,7,6,4,3,2,1,5] [5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]=>[5,4,3,2,1,6,7,8] [5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]=>[8,5,4,3,2,1,6,7] [5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]=>[7,5,4,3,2,1,6,8] [5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]=>[8,7,5,4,3,2,1,6] [6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]=>[6,5,4,3,2,1,7,8] [6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]=>[8,6,5,4,3,2,1,7] [7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]=>[7,6,5,4,3,2,1,8] [8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[8,7,6,5,4,3,2,1]
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.
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.