Identifier
Mp00038: Integer compositions reverseInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00029: Dyck paths to binary tree: left tree, up step, right tree, down step Binary trees
Images
[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] => [[[[.,[.,.]],[.,.]],.],[.,.]]
>>> Load all 548 entries. <<<
[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] => [[[[[[[.,[.,.]],[.,.]],.],.],.],.],.]
[1,1,1,1,1,3,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],[.,[.,.]]],.],.],.],.],.]
[1,1,1,1,1,4] => [4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]
[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] => [[[[[[[.,[.,.]],.],[.,.]],.],.],.],.]
[1,1,1,1,2,2,1] => [1,2,2,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],[.,.]],[.,.]],.],.],.],.]
[1,1,1,1,2,3] => [3,2,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[[.,[.,[.,.]]],[.,.]],.],.],.],.]
[1,1,1,1,3,1,1] => [1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],.],[.,[.,.]]],.],.],.],.]
[1,1,1,1,3,2] => [2,3,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[.,[.,.]],[.,[.,.]]],.],.],.],.]
[1,1,1,1,4,1] => [1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],[.,[.,[.,.]]]],.],.],.],.]
[1,1,1,1,5] => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
[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] => [[[[[[[.,[.,.]],.],.],[.,.]],.],.],.]
[1,1,1,2,1,2,1] => [1,2,1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[[[.,.],[.,.]],.],[.,.]],.],.],.]
[1,1,1,2,2,1,1] => [1,1,2,2,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[[[[[[.,.],.],[.,.]],[.,.]],.],.],.]
[1,1,1,2,2,2] => [2,2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[[[[[.,[.,.]],[.,.]],[.,.]],.],.],.]
[1,1,1,2,4] => [4,2,1,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0] => [[[[[.,[.,[.,[.,.]]]],[.,.]],.],.],.]
[1,1,1,3,1,1,1] => [1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[[[[[.,.],.],.],[.,[.,.]]],.],.],.]
[1,1,1,3,2,1] => [1,2,3,1,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[[[[.,.],[.,.]],[.,[.,.]]],.],.],.]
[1,1,1,3,3] => [3,3,1,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[[[.,[.,[.,.]]],[.,[.,.]]],.],.],.]
[1,1,1,4,1,1] => [1,1,4,1,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[[[.,.],.],[.,[.,[.,.]]]],.],.],.]
[1,1,1,4,2] => [2,4,1,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[[.,[.,.]],[.,[.,[.,.]]]],.],.],.]
[1,1,1,5,1] => [1,5,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,[.,[.,[.,.]]]]],.],.],.]
[1,1,1,6] => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => [[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
[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] => [[[[[[[.,[.,.]],.],.],.],[.,.]],.],.]
[1,1,2,2,1,1,1] => [1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[[[[[[.,.],.],.],[.,.]],[.,.]],.],.]
[1,1,2,2,2,1] => [1,2,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[[[[[.,.],[.,.]],[.,.]],[.,.]],.],.]
[1,1,2,4,1] => [1,4,2,1,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0] => [[[[[.,.],[.,[.,[.,.]]]],[.,.]],.],.]
[1,1,2,5] => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0] => [[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.],.]
[1,1,3,1,1,1,1] => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[[[[[[.,.],.],.],.],[.,[.,.]]],.],.]
[1,1,3,2,2] => [2,2,3,1,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[[[[.,[.,.]],[.,.]],[.,[.,.]]],.],.]
[1,1,3,3,1] => [1,3,3,1,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => [[[[[.,.],[.,[.,.]]],[.,[.,.]]],.],.]
[1,1,3,4] => [4,3,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.],.]
[1,1,5,1,1] => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[[[.,.],.],[.,[.,[.,[.,.]]]]],.],.]
[1,1,5,2] => [2,5,1,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.],.]
[1,1,6,1] => [1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.],.]
[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] => [[[[[[[.,[.,.]],.],.],.],.],[.,.]],.]
[1,2,1,1,1,2,1] => [1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[[.,.],[.,.]],.],.],.],[.,.]],.]
[1,2,1,1,1,3] => [3,1,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[.,[.,[.,.]]],.],.],.],[.,.]],.]
[1,2,1,1,2,1,1] => [1,1,2,1,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[[.,.],.],[.,.]],.],.],[.,.]],.]
[1,2,1,2,1,1,1] => [1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[[[[[[.,.],.],.],[.,.]],.],[.,.]],.]
[1,2,1,5] => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0] => [[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]],.]
[1,2,2,1,1,1,1] => [1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[[[[[[.,.],.],.],.],[.,.]],[.,.]],.]
[1,2,2,1,3] => [3,1,2,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[[[[.,[.,[.,.]]],.],[.,.]],[.,.]],.]
[1,2,2,2,1,1] => [1,1,2,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[[.,.],.],[.,.]],[.,.]],[.,.]],.]
[1,2,2,2,2] => [2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]
[1,2,2,3,1] => [1,3,2,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[.,.],[.,[.,.]]],[.,.]],[.,.]],.]
[1,2,2,4] => [4,2,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]],.]
[1,2,3,1,1,1] => [1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[[[[[.,.],.],.],[.,[.,.]]],[.,.]],.]
[1,2,6] => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0] => [[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],.]
[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] => [[[[[[[.,.],.],.],.],.],[.,[.,.]]],.]
[1,3,1,1,1,2] => [2,1,1,1,3,1] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[[[[[.,[.,.]],.],.],.],[.,[.,.]]],.]
[1,3,1,4] => [4,1,3,1] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,[.,[.,[.,.]]]],.],[.,[.,.]]],.]
[1,3,2,2,1] => [1,2,2,3,1] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[[[[.,.],[.,.]],[.,.]],[.,[.,.]]],.]
[1,3,3,2] => [2,3,3,1] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]],.]
[1,3,5] => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0] => [[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]
[1,4,1,1,1,1] => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[[[[.,.],.],.],.],[.,[.,[.,.]]]],.]
[1,4,1,1,2] => [2,1,1,4,1] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[[[.,[.,.]],.],.],[.,[.,[.,.]]]],.]
[1,4,1,3] => [3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[[.,[.,[.,.]]],.],[.,[.,[.,.]]]],.]
[1,4,4] => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0] => [[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]],.]
[1,5,1,1,1] => [1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[.,.],.],.],[.,[.,[.,[.,.]]]]],.]
[1,5,1,2] => [2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[[.,[.,.]],.],[.,[.,[.,[.,.]]]]],.]
[1,5,2,1] => [1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]],.]
[1,5,3] => [3,5,1] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]],.]
[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] => [[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],.]
[1,6,2] => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
[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] => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
[2,1,1,1,1,2,1] => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
[2,1,1,1,1,3] => [3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[.,[.,[.,.]]],.],.],.],.],[.,.]]
[2,1,1,1,2,1,1] => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[[.,.],.],[.,.]],.],.],.],[.,.]]
[2,1,1,1,2,2] => [2,2,1,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[.,[.,.]],[.,.]],.],.],.],[.,.]]
[2,1,1,1,4] => [4,1,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,[.,[.,[.,.]]]],.],.],.],[.,.]]
[2,1,1,2,2,1] => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[[[[[.,.],[.,.]],[.,.]],.],.],[.,.]]
[2,1,1,5] => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],[.,.]]
[2,1,2,1,1,1,1] => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[[[[[[.,.],.],.],.],[.,.]],.],[.,.]]
[2,1,2,1,3] => [3,1,2,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[[[[.,[.,[.,.]]],.],[.,.]],.],[.,.]]
[2,1,2,2,2] => [2,2,2,1,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[[[[.,[.,.]],[.,.]],[.,.]],.],[.,.]]
[2,1,6] => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,.]]
[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] => [[[[[[.,[.,.]],.],.],.],[.,.]],[.,.]]
[2,2,1,2,1,1] => [1,1,2,1,2,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[[[[[.,.],.],[.,.]],.],[.,.]],[.,.]]
[2,2,2,1,1,1] => [1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[[.,.],.],.],[.,.]],[.,.]],[.,.]]
[2,2,2,1,2] => [2,1,2,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[.,[.,.]],.],[.,.]],[.,.]],[.,.]]
[2,2,2,2,1] => [1,2,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]]
[2,2,2,3] => [3,2,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[.,[.,[.,.]]],[.,.]],[.,.]],[.,.]]
[2,2,3,1,1] => [1,1,3,2,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[[[.,.],.],[.,[.,.]]],[.,.]],[.,.]]
[2,3,1,1,1,1] => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[[[[[.,.],.],.],.],[.,[.,.]]],[.,.]]
[2,3,2,1,1] => [1,1,2,3,2] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[[[[.,.],.],[.,.]],[.,[.,.]]],[.,.]]
[2,3,2,2] => [2,2,3,2] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],[.,.]]
[2,3,3,1] => [1,3,3,2] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],[.,.]]
[2,3,4] => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],[.,.]]
[2,4,3] => [3,4,2] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],[.,.]]
[2,5,1,1] => [1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],[.,.]]
[2,5,2] => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],[.,.]]
[2,6,1] => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],[.,.]]
[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] => [[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
[3,1,1,1,1,2] => [2,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[[[[[.,[.,.]],.],.],.],.],[.,[.,.]]]
[3,1,1,1,2,1] => [1,2,1,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[[[[[.,.],[.,.]],.],.],.],[.,[.,.]]]
[3,1,1,1,3] => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[[[[.,[.,[.,.]]],.],.],.],[.,[.,.]]]
[3,1,1,2,1,1] => [1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[[[[[.,.],.],[.,.]],.],.],[.,[.,.]]]
[3,1,1,4] => [4,1,1,3] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,[.,[.,[.,.]]]],.],.],[.,[.,.]]]
[3,1,4,1] => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0] => [[[[.,.],[.,[.,[.,.]]]],.],[.,[.,.]]]
[3,1,5] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,[.,.]]]
[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] => [[[[[[.,.],.],.],.],[.,.]],[.,[.,.]]]
[3,2,1,1,2] => [2,1,1,2,3] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[[[[.,[.,.]],.],.],[.,.]],[.,[.,.]]]
[3,2,2,1,1] => [1,1,2,2,3] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[[[[.,.],.],[.,.]],[.,.]],[.,[.,.]]]
[3,2,2,2] => [2,2,2,3] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[[[.,[.,.]],[.,.]],[.,.]],[.,[.,.]]]
[3,2,4] => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,[.,.]]]
[3,3,1,1,1] => [1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[[[.,.],.],.],[.,[.,.]]],[.,[.,.]]]
[3,3,1,2] => [2,1,3,3] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[[.,[.,.]],.],[.,[.,.]]],[.,[.,.]]]
[3,3,2,1] => [1,2,3,3] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,.]]]
[3,3,3] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,[.,.]]]
[3,4,1,1] => [1,1,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[[.,.],.],[.,[.,[.,.]]]],[.,[.,.]]]
[3,4,2] => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,[.,.]]]
[3,5,1] => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,[.,.]]]
[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] => [[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]
[4,1,1,1,2] => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[[[[.,[.,.]],.],.],.],[.,[.,[.,.]]]]
[4,1,2,2] => [2,2,1,4] => [1,1,0,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[[[.,[.,.]],[.,.]],.],[.,[.,[.,.]]]]
[4,1,4] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0] => [[[.,[.,[.,[.,.]]]],.],[.,[.,[.,.]]]]
[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] => [[[[[.,.],.],.],[.,.]],[.,[.,[.,.]]]]
[4,2,2,1] => [1,2,2,4] => [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[[[.,.],[.,.]],[.,.]],[.,[.,[.,.]]]]
[4,3,2] => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[4,4,1] => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [[[.,.],[.,[.,[.,.]]]],[.,[.,[.,.]]]]
[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] => [[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
[5,1,1,2] => [2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[[[.,[.,.]],.],.],[.,[.,[.,[.,.]]]]]
[5,1,2,1] => [1,2,1,5] => [1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[[[.,.],[.,.]],.],[.,[.,[.,[.,.]]]]]
[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] => [[[[.,.],.],[.,.]],[.,[.,[.,[.,.]]]]]
[5,2,2] => [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[[.,[.,.]],[.,.]],[.,[.,[.,[.,.]]]]]
[5,3,1] => [1,3,5] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,[.,.]]],[.,[.,[.,[.,.]]]]]
[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] => [[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]
[6,1,2] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[[.,[.,.]],.],[.,[.,[.,[.,[.,.]]]]]]
[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] => [[[[[[[[[.,.],[.,.]],.],.],.],.],.],.],.]
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]
[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] => [[[[[[[[[.,.],.],[.,.]],.],.],.],.],.],.]
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,[.,.]],[.,.]],.],.],.],.],.],.]
[1,1,1,1,1,1,3,1] => [1,3,1,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,.],[.,[.,.]]],.],.],.],.],.],.]
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
[1,1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[[.,.],.],.],[.,.]],.],.],.],.],.]
[1,1,1,1,1,2,1,2] => [2,1,2,1,1,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,[.,.]],.],[.,.]],.],.],.],.],.]
[1,1,1,1,1,2,2,1] => [1,2,2,1,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,.],[.,.]],[.,.]],.],.],.],.],.]
[1,1,1,1,1,2,3] => [3,2,1,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,[.,[.,.]]],[.,.]],.],.],.],.],.]
[1,1,1,1,1,3,1,1] => [1,1,3,1,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,.],.],[.,[.,.]]],.],.],.],.],.]
[1,1,1,1,1,3,2] => [2,3,1,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,[.,.]],[.,[.,.]]],.],.],.],.],.]
[1,1,1,1,1,4,1] => [1,4,1,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],[.,[.,[.,.]]]],.],.],.],.],.]
[1,1,1,1,1,5] => [5,1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]
[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] => [[[[[[[[[.,.],.],.],.],[.,.]],.],.],.],.]
[1,1,1,1,2,2,1,1] => [1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,.],.],[.,.]],[.,.]],.],.],.],.]
[1,1,1,1,2,2,2] => [2,2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[[[.,[.,.]],[.,.]],[.,.]],.],.],.],.]
[1,1,1,1,3,1,1,1] => [1,1,1,3,1,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[[[.,.],.],.],[.,[.,.]]],.],.],.],.]
[1,1,1,1,4,1,1] => [1,1,4,1,1,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],.],[.,[.,[.,.]]]],.],.],.],.]
[1,1,1,1,5,1] => [1,5,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],[.,[.,[.,[.,.]]]]],.],.],.],.]
[1,1,1,1,6] => [6,1,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
[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] => [[[[[[[[[.,.],.],.],.],.],[.,.]],.],.],.]
[1,1,1,2,2,1,1,1] => [1,1,1,2,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[[[[[[[.,.],.],.],[.,.]],[.,.]],.],.],.]
[1,1,1,2,2,2,1] => [1,2,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[[[[[[.,.],[.,.]],[.,.]],[.,.]],.],.],.]
[1,1,1,3,1,1,1,1] => [1,1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[[[[[[.,.],.],.],.],[.,[.,.]]],.],.],.]
[1,1,1,3,2,2] => [2,2,3,1,1,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[[[[.,[.,.]],[.,.]],[.,[.,.]]],.],.],.]
[1,1,1,4,2,1] => [1,2,4,1,1,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[[[.,.],[.,.]],[.,[.,[.,.]]]],.],.],.]
[1,1,1,5,1,1] => [1,1,5,1,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [[[[[[.,.],.],[.,[.,[.,[.,.]]]]],.],.],.]
[1,1,1,6,1] => [1,6,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.],.],.]
[1,1,1,7] => [7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0] => [[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
[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] => [[[[[[[[[.,.],.],.],.],.],.],[.,.]],.],.]
[1,1,2,1,1,1,1,2] => [2,1,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[[[[.,[.,.]],.],.],.],.],[.,.]],.],.]
[1,1,2,2,2,2] => [2,2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.],.]
[1,1,4,4] => [4,4,1,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]],.],.]
[1,1,6,2] => [2,6,1,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [[[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.],.]
[1,1,7,1] => [1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
[1,1,8] => [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
[1,2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[[[[.,.],.],.],.],.],.],.],[.,.]],.]
[1,2,1,1,1,1,1,2] => [2,1,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]],.]
[1,2,1,1,1,1,2,1] => [1,2,1,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]],.]
[1,2,1,1,1,1,3] => [3,1,1,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[[.,[.,[.,.]]],.],.],.],.],[.,.]],.]
[1,2,1,2,2,2] => [2,2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[[[[[.,[.,.]],[.,.]],[.,.]],.],[.,.]],.]
[1,2,1,6] => [6,1,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0,1,0] => [[[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,.]],.]
[1,2,2,1,1,1,1,1] => [1,1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[[[[[[[.,.],.],.],.],.],[.,.]],[.,.]],.]
[1,2,2,2,1,1,1] => [1,1,1,2,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[[[.,.],.],.],[.,.]],[.,.]],[.,.]],.]
[1,2,2,2,2,1] => [1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]],.]
[1,2,2,2,3] => [3,2,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[.,[.,[.,.]]],[.,.]],[.,.]],[.,.]],.]
[1,2,2,5] => [5,2,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,[.,[.,[.,[.,.]]]]],[.,.]],[.,.]],.]
[1,2,4,1,1,1] => [1,1,1,4,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[[[[.,.],.],.],[.,[.,[.,.]]]],[.,.]],.]
[1,2,7] => [7,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]],.]
[1,3,1,1,1,1,1,1] => [1,1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]],.]
[1,3,1,1,1,1,2] => [2,1,1,1,1,3,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[[[[[[.,[.,.]],.],.],.],.],[.,[.,.]]],.]
[1,3,1,5] => [5,1,3,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,[.,[.,[.,[.,.]]]]],.],[.,[.,.]]],.]
[1,3,2,2,2] => [2,2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[[[[.,[.,.]],[.,.]],[.,.]],[.,[.,.]]],.]
[1,3,4,2] => [2,4,3,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [[[[.,[.,.]],[.,[.,[.,.]]]],[.,[.,.]]],.]
[1,3,6] => [6,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0] => [[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]],.]
[1,4,1,1,1,1,1] => [1,1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]],.]
[1,4,1,4] => [4,1,4,1] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[[.,[.,[.,[.,.]]]],.],[.,[.,[.,.]]]],.]
[1,4,5] => [5,4,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0] => [[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]],.]
[1,5,1,1,1,1] => [1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]],.]
[1,5,1,3] => [3,1,5,1] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[[.,[.,[.,.]]],.],[.,[.,[.,[.,.]]]]],.]
[1,5,4] => [4,5,1] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]],.]
[1,6,1,1,1] => [1,1,1,6,1] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]],.]
[1,6,1,2] => [2,1,6,1] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[.,[.,.]],.],[.,[.,[.,[.,[.,.]]]]]],.]
[1,6,3] => [3,6,1] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]],.]
[1,7,1,1] => [1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
[1,7,2] => [2,7,1] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
[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] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
[2,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[[[.,.],[.,.]],.],.],.],.],.],[.,.]]
[2,1,1,1,1,1,3] => [3,1,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[[.,[.,[.,.]]],.],.],.],.],.],[.,.]]
[2,1,1,1,1,4] => [4,1,1,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[.,[.,[.,[.,.]]]],.],.],.],.],[.,.]]
[2,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[[[.,.],.],.],[.,.]],.],.],.],[.,.]]
[2,1,1,1,5] => [5,1,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],[.,.]]
[2,1,1,6] => [6,1,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0] => [[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],[.,.]]
[2,1,3,1,2,1] => [1,2,1,3,1,2] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[[[[[.,.],[.,.]],.],[.,[.,.]]],.],[.,.]]
[2,1,4,1,2] => [2,1,4,1,2] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[[[.,[.,.]],.],[.,[.,[.,.]]]],.],[.,.]]
[2,1,7] => [7,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],[.,.]]
[2,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[[[[[[[.,.],.],.],.],.],.],[.,.]],[.,.]]
[2,2,1,1,2,2] => [2,2,1,1,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[[[[[.,[.,.]],[.,.]],.],.],[.,.]],[.,.]]
[2,2,2,1,1,1,1] => [1,1,1,1,2,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[[[.,.],.],.],.],[.,.]],[.,.]],[.,.]]
[2,2,2,2,1,1] => [1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[[.,.],.],[.,.]],[.,.]],[.,.]],[.,.]]
[2,2,2,2,2] => [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
[2,2,2,3,1] => [1,3,2,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[.,.],[.,[.,.]]],[.,.]],[.,.]],[.,.]]
[2,2,2,4] => [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]],[.,.]]
[2,2,3,2,1] => [1,2,3,2,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[[[.,.],[.,.]],[.,[.,.]]],[.,.]],[.,.]]
[2,2,3,3] => [3,3,2,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]],[.,.]]
[2,3,3,2] => [2,3,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]],[.,.]]
[2,4,4] => [4,4,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]],[.,.]]
[2,6,1,1] => [1,1,6,2] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],[.,.]]
[2,6,2] => [2,6,2] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],[.,.]]
[2,7,1] => [1,7,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
[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] => [[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]
[3,1,1,1,1,1,2] => [2,1,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[[[[[[.,[.,.]],.],.],.],.],.],[.,[.,.]]]
[3,1,6] => [6,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,[.,.]]]
[3,2,2,2,1] => [1,2,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[[[[.,.],[.,.]],[.,.]],[.,.]],[.,[.,.]]]
[3,2,2,3] => [3,2,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[[[.,[.,[.,.]]],[.,.]],[.,.]],[.,[.,.]]]
[3,3,2,2] => [2,2,3,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],[.,[.,.]]]
[3,3,3,1] => [1,3,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],[.,[.,.]]]
[3,3,4] => [4,3,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],[.,[.,.]]]
[3,4,3] => [3,4,3] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],[.,[.,.]]]
[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] => [[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]
[4,1,1,1,1,2] => [2,1,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[[[[[.,[.,.]],.],.],.],.],[.,[.,[.,.]]]]
[4,1,1,4] => [4,1,1,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[[[.,[.,[.,[.,.]]]],.],.],[.,[.,[.,.]]]]
[4,1,5] => [5,1,4] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,1,0,0,0,0] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,[.,[.,.]]]]
[4,2,2,2] => [2,2,2,4] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[[[.,[.,.]],[.,.]],[.,.]],[.,[.,[.,.]]]]
[4,2,4] => [4,2,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,[.,[.,.]]]]
[4,3,2,1] => [1,2,3,4] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[4,3,3] => [3,3,4] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,[.,[.,.]]]]
[4,4,1,1] => [1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [[[[.,.],.],[.,[.,[.,.]]]],[.,[.,[.,.]]]]
[4,4,2] => [2,4,4] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,[.,[.,.]]]]
[4,5,1] => [1,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
[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] => [[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]
[5,1,1,1,2] => [2,1,1,1,5] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[[[[.,[.,.]],.],.],.],[.,[.,[.,[.,.]]]]]
[5,4,1] => [1,4,5] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]
[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] => [[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
[6,3,1] => [1,3,6] => [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[[.,.],[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
[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] => [[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
[7,1,2] => [2,1,7] => [1,1,0,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[.,[.,.]],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
[7,2,1] => [1,2,7] => [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
[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] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
[1,10] => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
[10,1] => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
[2,2,2,2,2,2] => [2,2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
[1,2,2,2,2,2,1] => [1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],.]
[2,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
[2,8,1] => [1,8,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
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.