- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees

Images

[1] => [1,0] => [[]] => [.,.]

[1,1] => [1,0,1,0] => [[],[]] => [.,[.,.]]

[2] => [1,1,0,0] => [[[]]] => [[.,.],.]

[1,1,1] => [1,0,1,0,1,0] => [[],[],[]] => [.,[.,[.,.]]]

[1,2] => [1,0,1,1,0,0] => [[],[[]]] => [.,[[.,.],.]]

[2,1] => [1,1,0,0,1,0] => [[[]],[]] => [[.,[.,.]],.]

[3] => [1,1,1,0,0,0] => [[[[]]]] => [[[.,.],.],.]

[1,1,1,1] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => [.,[.,[.,[.,.]]]]

[1,1,2] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => [.,[.,[[.,.],.]]]

[1,2,1] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => [.,[[.,[.,.]],.]]

[1,3] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => [.,[[[.,.],.],.]]

[2,1,1] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => [[.,[.,[.,.]]],.]

[2,2] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => [[.,[[.,.],.]],.]

[3,1] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => [[[.,[.,.]],.],.]

[4] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => [[[[.,.],.],.],.]

[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]]

[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]]

[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => [.,[.,[[.,[.,.]],.]]]

[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]]

[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => [.,[[.,[.,[.,.]]],.]]

[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => [.,[[.,[[.,.],.]],.]]

[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => [.,[[[.,[.,.]],.],.]]

[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => [.,[[[[.,.],.],.],.]]

[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => [[.,[.,[.,[.,.]]]],.]

[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => [[.,[.,[[.,.],.]]],.]

[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => [[.,[[.,[.,.]],.]],.]

[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => [[.,[[[.,.],.],.]],.]

[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.]

[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => [[[.,[[.,.],.]],.],.]

[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => [[[[.,[.,.]],.],.],.]

[5] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => [[[[[.,.],.],.],.],.]

[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]]

[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]]

[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => [.,[.,[.,[[.,[.,.]],.]]]]

[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]]

[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => [.,[.,[[.,[.,[.,.]]],.]]]

[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => [.,[.,[[.,[[.,.],.]],.]]]

[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => [.,[.,[[[.,[.,.]],.],.]]]

[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]]

[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => [.,[[.,[.,[.,[.,.]]]],.]]

[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => [.,[[.,[.,[[.,.],.]]],.]]

[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[]] => [.,[[.,[[.,[.,.]],.]],.]]

[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[[]]]] => [.,[[.,[[[.,.],.],.]],.]]

[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => [.,[[[.,[.,[.,.]]],.],.]]

[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[],[[[]]],[[]]] => [.,[[[.,[[.,.],.]],.],.]]

[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[],[[[[]]]],[]] => [.,[[[[.,[.,.]],.],.],.]]

[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[],[[[[[]]]]]] => [.,[[[[[.,.],.],.],.],.]]

[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[]] => [[.,[.,[.,[.,[.,.]]]]],.]

[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[[]]] => [[.,[.,[.,[[.,.],.]]]],.]

[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[[]],[],[[]],[]] => [[.,[.,[[.,[.,.]],.]]],.]

[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[[]],[],[[[]]]] => [[.,[.,[[[.,.],.],.]]],.]

[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[[]],[[]],[],[]] => [[.,[[.,[.,[.,.]]],.]],.]

[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[[]],[[]],[[]]] => [[.,[[.,[[.,.],.]],.]],.]

[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[[]],[[[]]],[]] => [[.,[[[.,[.,.]],.],.]],.]

[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[[]],[[[[]]]]] => [[.,[[[[.,.],.],.],.]],.]

[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[[]]],[],[],[]] => [[[.,[.,[.,[.,.]]]],.],.]

[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[[[]]],[],[[]]] => [[[.,[.,[[.,.],.]]],.],.]

[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[[[]]],[[]],[]] => [[[.,[[.,[.,.]],.]],.],.]

[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[[[]]],[[[]]]] => [[[.,[[[.,.],.],.]],.],.]

[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[[]]]],[],[]] => [[[[.,[.,[.,.]]],.],.],.]

[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[[[]]]],[[]]] => [[[[.,[[.,.],.]],.],.],.]

[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[[]]]]],[]] => [[[[[.,[.,.]],.],.],.],.]

[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[[]]]]]]] => [[[[[[.,.],.],.],.],.],.]

[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]

[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[],[[]]] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]

[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[],[[]],[]] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]

[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[],[[[]]]] => [.,[.,[.,[.,[[[.,.],.],.]]]]]

[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[],[[]],[],[]] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]

[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[],[[]],[[]]] => [.,[.,[.,[[.,[[.,.],.]],.]]]]

[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[],[[[]]],[]] => [.,[.,[.,[[[.,[.,.]],.],.]]]]

[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[],[[[[]]]]] => [.,[.,[.,[[[[.,.],.],.],.]]]]

[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[],[],[[]],[],[],[]] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]

[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[],[],[[]],[],[[]]] => [.,[.,[[.,[.,[[.,.],.]]],.]]]

[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[],[],[[]],[[]],[]] => [.,[.,[[.,[[.,[.,.]],.]],.]]]

[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[],[],[[]],[[[]]]] => [.,[.,[[.,[[[.,.],.],.]],.]]]

[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[],[],[[[]]],[],[]] => [.,[.,[[[.,[.,[.,.]]],.],.]]]

[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[],[],[[[]]],[[]]] => [.,[.,[[[.,[[.,.],.]],.],.]]]

[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[],[],[[[[]]]],[]] => [.,[.,[[[[.,[.,.]],.],.],.]]]

[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[],[],[[[[[]]]]]] => [.,[.,[[[[[.,.],.],.],.],.]]]

[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[],[[]],[],[],[],[]] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]

[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[],[[]],[],[],[[]]] => [.,[[.,[.,[.,[[.,.],.]]]],.]]

[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[],[[]],[],[[]],[]] => [.,[[.,[.,[[.,[.,.]],.]]],.]]

[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[],[[]],[],[[[]]]] => [.,[[.,[.,[[[.,.],.],.]]],.]]

[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[],[[]],[[]],[],[]] => [.,[[.,[[.,[.,[.,.]]],.]],.]]

[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]],[[]]] => [.,[[.,[[.,[[.,.],.]],.]],.]]

[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[],[[]],[[[]]],[]] => [.,[[.,[[[.,[.,.]],.],.]],.]]

[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[],[[]],[[[[]]]]] => [.,[[.,[[[[.,.],.],.],.]],.]]

[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[],[[[]]],[],[],[]] => [.,[[[.,[.,[.,[.,.]]]],.],.]]

[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[],[[[]]],[],[[]]] => [.,[[[.,[.,[[.,.],.]]],.],.]]

[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[],[[[]]],[[]],[]] => [.,[[[.,[[.,[.,.]],.]],.],.]]

[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[],[[[]]],[[[]]]] => [.,[[[.,[[[.,.],.],.]],.],.]]

[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[],[[[[]]]],[],[]] => [.,[[[[.,[.,[.,.]]],.],.],.]]

[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[],[[[[]]]],[[]]] => [.,[[[[.,[[.,.],.]],.],.],.]]

[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[],[[[[[]]]]],[]] => [.,[[[[[.,[.,.]],.],.],.],.]]

[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[],[[[[[[]]]]]]] => [.,[[[[[[.,.],.],.],.],.],.]]

[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[],[]] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]

[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[],[[]]] => [[.,[.,[.,[.,[[.,.],.]]]]],.]

[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[[]],[],[],[[]],[]] => [[.,[.,[.,[[.,[.,.]],.]]]],.]

[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[[]],[],[],[[[]]]] => [[.,[.,[.,[[[.,.],.],.]]]],.]

[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[[]],[],[[]],[],[]] => [[.,[.,[[.,[.,[.,.]]],.]]],.]

[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[[]],[],[[]],[[]]] => [[.,[.,[[.,[[.,.],.]],.]]],.]

>>> Load all 310 entries. <<<

[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[[]],[],[[[]]],[]] => [[.,[.,[[[.,[.,.]],.],.]]],.]

[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[[]],[],[[[[]]]]] => [[.,[.,[[[[.,.],.],.],.]]],.]

[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[[]],[[]],[],[],[]] => [[.,[[.,[.,[.,[.,.]]]],.]],.]

[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[[]],[[]],[],[[]]] => [[.,[[.,[.,[[.,.],.]]],.]],.]

[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[[]],[]] => [[.,[[.,[[.,[.,.]],.]],.]],.]

[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[[]],[[]],[[[]]]] => [[.,[[.,[[[.,.],.],.]],.]],.]

[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[[]],[[[]]],[],[]] => [[.,[[[.,[.,[.,.]]],.],.]],.]

[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[[]],[[[]]],[[]]] => [[.,[[[.,[[.,.],.]],.],.]],.]

[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[[]],[[[[]]]],[]] => [[.,[[[[.,[.,.]],.],.],.]],.]

[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[[]],[[[[[]]]]]] => [[.,[[[[[.,.],.],.],.],.]],.]

[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[[]]],[],[],[],[]] => [[[.,[.,[.,[.,[.,.]]]]],.],.]

[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[[[]]],[],[],[[]]] => [[[.,[.,[.,[[.,.],.]]]],.],.]

[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[[[]]],[],[[]],[]] => [[[.,[.,[[.,[.,.]],.]]],.],.]

[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[[[]]],[],[[[]]]] => [[[.,[.,[[[.,.],.],.]]],.],.]

[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[[[]]],[[]],[],[]] => [[[.,[[.,[.,[.,.]]],.]],.],.]

[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[[]]],[[]],[[]]] => [[[.,[[.,[[.,.],.]],.]],.],.]

[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[[[]]],[[[]]],[]] => [[[.,[[[.,[.,.]],.],.]],.],.]

[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[[]]],[[[[]]]]] => [[[.,[[[[.,.],.],.],.]],.],.]

[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[[]]]],[],[],[]] => [[[[.,[.,[.,[.,.]]]],.],.],.]

[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[[[]]]],[],[[]]] => [[[[.,[.,[[.,.],.]]],.],.],.]

[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[[[]]]],[[]],[]] => [[[[.,[[.,[.,.]],.]],.],.],.]

[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[[[]]]],[[[]]]] => [[[[.,[[[.,.],.],.]],.],.],.]

[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[[[[]]]]],[],[]] => [[[[[.,[.,[.,.]]],.],.],.],.]

[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[[[[]]]]],[[]]] => [[[[[.,[[.,.],.]],.],.],.],.]

[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[[[[]]]]]],[]] => [[[[[[.,[.,.]],.],.],.],.],.]

[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[[[[[[]]]]]]]] => [[[[[[[.,.],.],.],.],.],.],.]

[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]

[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[],[],[[]]] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]

[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[],[],[[]],[]] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]

[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[],[],[[[]]]] => [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]

[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[],[],[[]],[],[]] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]

[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[],[],[[]],[[]]] => [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]

[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[],[],[[[]]],[]] => [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]

[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[],[],[[[[]]]]] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]

[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[],[],[],[[]],[],[],[]] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]

[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[],[],[],[[]],[],[[]]] => [.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]

[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[],[],[],[[]],[[]],[]] => [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]

[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[],[],[],[[]],[[[]]]] => [.,[.,[.,[[.,[[[.,.],.],.]],.]]]]

[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[],[],[],[[[]]],[],[]] => [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]

[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[],[],[],[[[]]],[[]]] => [.,[.,[.,[[[.,[[.,.],.]],.],.]]]]

[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[],[],[],[[[[]]]],[]] => [.,[.,[.,[[[[.,[.,.]],.],.],.]]]]

[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[],[],[],[[[[[]]]]]] => [.,[.,[.,[[[[[.,.],.],.],.],.]]]]

[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[],[],[[]],[],[],[],[]] => [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]

[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[],[],[[]],[],[],[[]]] => [.,[.,[[.,[.,[.,[[.,.],.]]]],.]]]

[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[],[[]],[]] => [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]

[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[],[],[[]],[],[[[]]]] => [.,[.,[[.,[.,[[[.,.],.],.]]],.]]]

[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[[]],[],[]] => [.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]

[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]],[[]]] => [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]

[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[],[],[[]],[[[]]],[]] => [.,[.,[[.,[[[.,[.,.]],.],.]],.]]]

[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[],[],[[]],[[[[]]]]] => [.,[.,[[.,[[[[.,.],.],.],.]],.]]]

[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[],[],[[[]]],[],[],[]] => [.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]

[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[],[],[[[]]],[],[[]]] => [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]

[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[],[],[[[]]],[[]],[]] => [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]

[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[],[],[[[]]],[[[]]]] => [.,[.,[[[.,[[[.,.],.],.]],.],.]]]

[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[],[],[[[[]]]],[],[]] => [.,[.,[[[[.,[.,[.,.]]],.],.],.]]]

[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[],[],[[[[]]]],[[]]] => [.,[.,[[[[.,[[.,.],.]],.],.],.]]]

[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[],[],[[[[[]]]]],[]] => [.,[.,[[[[[.,[.,.]],.],.],.],.]]]

[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[],[],[[[[[[]]]]]]] => [.,[.,[[[[[[.,.],.],.],.],.],.]]]

[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[],[[]],[],[],[],[],[]] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]

[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[],[[]],[],[],[],[[]]] => [.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]

[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[],[[]],[],[],[[]],[]] => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]

[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[],[[]],[],[],[[[]]]] => [.,[[.,[.,[.,[[[.,.],.],.]]]],.]]

[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[[]],[],[]] => [.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]

[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[],[[]],[],[[]],[[]]] => [.,[[.,[.,[[.,[[.,.],.]],.]]],.]]

[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[],[[]],[],[[[]]],[]] => [.,[[.,[.,[[[.,[.,.]],.],.]]],.]]

[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[],[[]],[],[[[[]]]]] => [.,[[.,[.,[[[[.,.],.],.],.]]],.]]

[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[[]],[],[],[]] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]

[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[[]],[],[[]]] => [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]

[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[[]],[]] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]

[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[]],[[[]]]] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]

[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[],[[]],[[[]]],[],[]] => [.,[[.,[[[.,[.,[.,.]]],.],.]],.]]

[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[],[[]],[[[]]],[[]]] => [.,[[.,[[[.,[[.,.],.]],.],.]],.]]

[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[],[[]],[[[[]]]],[]] => [.,[[.,[[[[.,[.,.]],.],.],.]],.]]

[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[],[[]],[[[[[]]]]]] => [.,[[.,[[[[[.,.],.],.],.],.]],.]]

[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[],[[[]]],[],[],[],[]] => [.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]

[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[],[[[]]],[],[],[[]]] => [.,[[[.,[.,[.,[[.,.],.]]]],.],.]]

[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[],[[[]]],[],[[]],[]] => [.,[[[.,[.,[[.,[.,.]],.]]],.],.]]

[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[],[[[]]],[],[[[]]]] => [.,[[[.,[.,[[[.,.],.],.]]],.],.]]

[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[],[[[]]],[[]],[],[]] => [.,[[[.,[[.,[.,[.,.]]],.]],.],.]]

[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[],[[[]]],[[]],[[]]] => [.,[[[.,[[.,[[.,.],.]],.]],.],.]]

[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[[[]]],[]] => [.,[[[.,[[[.,[.,.]],.],.]],.],.]]

[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[],[[[]]],[[[[]]]]] => [.,[[[.,[[[[.,.],.],.],.]],.],.]]

[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[],[[[[]]]],[],[],[]] => [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]

[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[],[[[[]]]],[],[[]]] => [.,[[[[.,[.,[[.,.],.]]],.],.],.]]

[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[],[[[[]]]],[[]],[]] => [.,[[[[.,[[.,[.,.]],.]],.],.],.]]

[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[],[[[[]]]],[[[]]]] => [.,[[[[.,[[[.,.],.],.]],.],.],.]]

[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[],[[[[[]]]]],[],[]] => [.,[[[[[.,[.,[.,.]]],.],.],.],.]]

[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[],[[[[[]]]]],[[]]] => [.,[[[[[.,[[.,.],.]],.],.],.],.]]

[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[],[[[[[[]]]]]],[]] => [.,[[[[[[.,[.,.]],.],.],.],.],.]]

[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[],[[[[[[[]]]]]]]] => [.,[[[[[[[.,.],.],.],.],.],.],.]]

[2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[],[],[]] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]

[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[],[],[[]]] => [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]

[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[]],[],[],[],[[]],[]] => [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]

[2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[[]],[],[],[],[[[]]]] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]

[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[[]],[],[],[[]],[],[]] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]

[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[[]],[],[],[[]],[[]]] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]

[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[[]],[],[],[[[]]],[]] => [[.,[.,[.,[[[.,[.,.]],.],.]]]],.]

[2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[[]],[],[],[[[[]]]]] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]

[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[[]],[],[[]],[],[],[]] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]

[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]],[],[[]]] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]

[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[[]],[],[[]],[[]],[]] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]

[2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[[]],[],[[]],[[[]]]] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]

[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[[]],[],[[[]]],[],[]] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]

[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[[]],[],[[[]]],[[]]] => [[.,[.,[[[.,[[.,.],.]],.],.]]],.]

[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[]],[],[[[[]]]],[]] => [[.,[.,[[[[.,[.,.]],.],.],.]]],.]

[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[[]],[],[[[[[]]]]]] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]

[2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[[]],[[]],[],[],[],[]] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]

[2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[[]],[[]],[],[],[[]]] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]

[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[[]],[[]],[],[[]],[]] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]

[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[[]],[[]],[],[[[]]]] => [[.,[[.,[.,[[[.,.],.],.]]],.]],.]

[2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[[]],[[]],[[]],[],[]] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]

[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[]],[[]],[[]],[[]]] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]

[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[[]],[[]],[[[]]],[]] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]

[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[[]],[[]],[[[[]]]]] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]

[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[]],[[[]]],[],[],[]] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]

[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[[]],[[[]]],[],[[]]] => [[.,[[[.,[.,[[.,.],.]]],.],.]],.]

[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[[]],[[[]]],[[]],[]] => [[.,[[[.,[[.,[.,.]],.]],.],.]],.]

[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[]],[[[]]],[[[]]]] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]

[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[[]],[[[[]]]],[],[]] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]

[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[[]],[[[[]]]],[[]]] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]

[2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[]],[[[[[]]]]],[]] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]

[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[[]],[[[[[[]]]]]]] => [[.,[[[[[[.,.],.],.],.],.],.]],.]

[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[]]],[],[],[],[],[]] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]

[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[]]],[],[],[],[[]]] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]

[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0] => [[[[]]],[],[],[[]],[]] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]

[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0] => [[[[]]],[],[],[[[]]]] => [[[.,[.,[.,[[[.,.],.],.]]]],.],.]

[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0] => [[[[]]],[],[[]],[],[]] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]

[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0] => [[[[]]],[],[[]],[[]]] => [[[.,[.,[[.,[[.,.],.]],.]]],.],.]

[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0] => [[[[]]],[],[[[]]],[]] => [[[.,[.,[[[.,[.,.]],.],.]]],.],.]

[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0] => [[[[]]],[],[[[[]]]]] => [[[.,[.,[[[[.,.],.],.],.]]],.],.]

[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0] => [[[[]]],[[]],[],[],[]] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]

[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0] => [[[[]]],[[]],[],[[]]] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]

[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0] => [[[[]]],[[]],[[]],[]] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]

[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => [[[[]]],[[]],[[[]]]] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]

[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => [[[[]]],[[[]]],[],[]] => [[[.,[[[.,[.,[.,.]]],.],.]],.],.]

[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[[]]],[[]]] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]

[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [[[[]]],[[[[]]]],[]] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]

[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [[[[]]],[[[[[]]]]]] => [[[.,[[[[[.,.],.],.],.],.]],.],.]

[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[]]]],[],[],[],[]] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]

[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0] => [[[[[]]]],[],[],[[]]] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]

[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0] => [[[[[]]]],[],[[]],[]] => [[[[.,[.,[[.,[.,.]],.]]],.],.],.]

[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0] => [[[[[]]]],[],[[[]]]] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]

[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0] => [[[[[]]]],[[]],[],[]] => [[[[.,[[.,[.,[.,.]]],.]],.],.],.]

[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [[[[[]]]],[[]],[[]]] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]

[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [[[[[]]]],[[[]]],[]] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]

[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [[[[[]]]],[[[[]]]]] => [[[[.,[[[[.,.],.],.],.]],.],.],.]

[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [[[[[[]]]]],[],[],[]] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]

[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [[[[[[]]]]],[],[[]]] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]

[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [[[[[[]]]]],[[]],[]] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]

[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [[[[[[]]]]],[[[]]]] => [[[[[.,[[[.,.],.],.]],.],.],.],.]

[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [[[[[[[]]]]]],[],[]] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]

[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [[[[[[[]]]]]],[[]]] => [[[[[[.,[[.,.],.]],.],.],.],.],.]

[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[[[[[[[]]]]]]],[]] => [[[[[[[.,[.,.]],.],.],.],.],.],.]

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[[[[[[[[]]]]]]]]] => [[[[[[[[.,.],.],.],.],.],.],.],.]

[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]

[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[],[],[],[[]]] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]

[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[],[],[],[[[]]]] => [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]

[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[],[],[],[[]],[[]]] => [.,[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]]

[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] => [[],[],[],[[]],[],[],[],[]] => [.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]

[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[[]],[],[],[],[],[]] => [.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]

[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[],[],[[[[[[[]]]]]]]] => [.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]

[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] => [[],[[]],[],[],[],[],[[]]] => [.,[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]]

[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] => [[],[[]],[[]],[],[],[],[]] => [.,[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]]

[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[],[[[]]],[],[],[],[],[]] => [.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]]

[1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [[],[[[[[[]]]]]],[[]]] => [.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]

[1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[],[[[[[[[]]]]]]],[]] => [.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]

[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[],[[[[[[[[]]]]]]]]] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]

[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] => [[[]],[],[],[],[],[],[],[]] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]

[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] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[]],[],[],[],[],[[]],[]] => [[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]

[2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[]],[[[[[[]]]]]],[]] => [[.,[[[[[[.,[.,.]],.],.],.],.],.]],.]

[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[]],[[[[[[[]]]]]]]] => [[.,[[[[[[[.,.],.],.],.],.],.],.]],.]

[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] => [[[[]]],[],[],[],[],[],[]] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]

[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[[[]]],[[[[[[]]]]]]] => [[[.,[[[[[[.,.],.],.],.],.],.]],.],.]

[4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[]]]],[],[],[],[],[]] => [[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]

[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [[[[[]]]],[[[[[]]]]]] => [[[[.,[[[[[.,.],.],.],.],.]],.],.],.]

[5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[]]]]],[],[],[],[]] => [[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]

[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [[[[[[]]]]],[[[[]]]]] => [[[[[.,[[[[.,.],.],.],.]],.],.],.],.]

[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => [[[[[[[]]]]]],[],[],[]] => [[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]

[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [[[[[[[]]]]]],[[[]]]] => [[[[[[.,[[[.,.],.],.]],.],.],.],.],.]

[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [[[[[[[[]]]]]]],[],[]] => [[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]

[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [[[[[[[[]]]]]]],[[]]] => [[[[[[[.,[[.,.],.]],.],.],.],.],.],.]

[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[]]]]]]]],[]] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],.]

[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] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]

[1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[],[],[],[],[[]]] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]

[1,1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[],[],[],[],[],[[]],[],[],[]] => [.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]

[1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [[],[[[[[[[[]]]]]]]],[]] => [.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]

[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [[],[[[[[[[[[]]]]]]]]]] => [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]

[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] => [[[]],[],[],[],[],[],[],[],[]] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]

[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] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[[]],[[[[[[[[]]]]]]]]] => [[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]

[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] => [[[[]]],[],[],[],[],[],[],[]] => [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]

[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[[]]],[[[[[[[]]]]]]]] => [[[.,[[[[[[[.,.],.],.],.],.],.],.]],.],.]

[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] => [[[[[]]]],[],[],[],[],[],[]] => [[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]

[4,6] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[]]]],[[[[[[]]]]]]] => [[[[.,[[[[[[.,.],.],.],.],.],.]],.],.],.]

[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] => [[[[[[]]]]],[],[],[],[],[]] => [[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]

[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] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0] => [[[[[[[]]]]]],[],[],[],[]] => [[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]

[6,4] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => [[[[[[[]]]]]],[[[[]]]]] => [[[[[[.,[[[[.,.],.],.],.]],.],.],.],.],.]

[7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0] => [[[[[[[[]]]]]]],[],[],[]] => [[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]

[7,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0] => [[[[[[[[]]]]]]],[[[]]]] => [[[[[[[.,[[[.,.],.],.]],.],.],.],.],.],.]

[8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => [[[[[[[[[]]]]]]]],[],[]] => [[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]

[8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => [[[[[[[[[]]]]]]]],[[]]] => [[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.]

[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[[]]]]]]]]],[]] => [[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]

[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [[[[[[[[[[[]]]]]]]]]]] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]

[1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [[],[[[[[[[[[[]]]]]]]]]]] => [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]

[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[[[]]]]]]]]]],[]] => [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]

[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] => [[],[],[],[],[],[],[],[],[],[[]]] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to ordered tree

Description

Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path

$D$ of semilength

$n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths

$D_1,\dots,D_k$ of respective semilengths

$n_1,\dots,n_k$ (so one has

$n = n_1 + \dots n_k$ ) each of which has no returns.
Denote by

$\tilde D_i$ the path of semilength

$n_i-1$ obtained from

$D_i$ by removing the initial up- and the final down-step.
This map then sends

$D$ to the tree

$T$ having a root note with ordered children

$T_1,\dots,T_k$ which are again ordered trees computed from

$D_1,\dots,D_k$ respectively.
The unique path of semilength

$1$ is sent to the tree consisting of a single node.

Map

Zeilberger's Strahler bijection

Description

Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.