Identifier
Values
[1] => [1,0] => [.,.] => 0
[1,1] => [1,0,1,0] => [.,[.,.]] => 0
[2] => [1,1,0,0] => [[.,.],.] => 0
[1,1,1] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => 0
[1,2] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 0
[2,1] => [1,1,0,0,1,0] => [[.,[.,.]],.] => 0
[3] => [1,1,1,0,0,0] => [[.,.],[.,.]] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [.,[[.,[.,.]],.]] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [[.,[.,[.,.]]],.] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [[.,[[.,.],.]],.] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [[.,.],[.,[.,.]]] => 0
[4] => [1,1,1,1,0,0,0,0] => [[[.,.],.],[.,.]] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [.,[.,[[.,[.,.]],.]]] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[.,.],[.,.]]]] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [.,[[.,[.,[.,.]]],.]] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [.,[[.,[[.,.],.]],.]] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [.,[[.,.],[.,[.,.]]]] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[.,.],.],[.,.]]] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[.,[.,[.,[.,.]]]],.] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[.,[.,[[.,.],.]]],.] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[.,[[.,.],[.,.]]],.] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[.,.],[.,[.,[.,.]]]] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[.,.],[.,[[.,.],.]]] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[[.,.],.],[.,[.,.]]] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,.]] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[[.,[.,.]],.]]]] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[.,.],[.,.]]]]] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[[.,[.,[.,.]]],.]]] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[[.,[[.,.],.]],.]]] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[[.,.],[.,[.,.]]]]] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[.,.],.],[.,.]]]] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [.,[[.,[.,[.,[.,.]]]],.]] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [.,[[.,[.,[[.,.],.]]],.]] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [.,[[.,[[.,[.,.]],.]],.]] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [.,[[.,[[.,.],[.,.]]],.]] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,.]]]]] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [.,[[.,.],[.,[[.,.],.]]]] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [.,[[[.,.],.],[.,[.,.]]]] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[.,.],[.,.]],[.,.]]] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[.,[.,[.,[.,[.,.]]]]],.] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[.,[.,[.,[[.,.],.]]]],.] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[.,[.,[[.,[.,.]],.]]],.] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[.,[.,[[.,.],[.,.]]]],.] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[.,[[.,[.,[.,.]]],.]],.] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[.,[[.,[[.,.],.]],.]],.] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[.,[[.,.],[.,[.,.]]]],.] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[.,[[[.,.],.],[.,.]]],.] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,.]]]]] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[.,.],[.,[.,[[.,.],.]]]] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[.,.],[.,[[.,[.,.]],.]]] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[.,.],[.,[[.,.],[.,.]]]] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,.]]]] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[.,.],.],[.,[[.,.],.]]] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[[.,.],[.,[.,.]]],[.,.]] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[.,.],[.,.]],[[.,.],.]] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[.,[[.,[.,.]],.]]]]] => 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[[.,.],[.,.]]]]]] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[.,[[.,[.,[.,.]]],.]]]] => 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[.,[[.,[[.,.],.]],.]]]] => 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[.,[[.,.],[.,[.,.]]]]]] => 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[[[.,.],.],[.,.]]]]] => 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [.,[.,[[.,[.,[.,[.,.]]]],.]]] => 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [.,[.,[[.,[.,[[.,.],.]]],.]]] => 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [.,[.,[[.,[[.,[.,.]],.]],.]]] => 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [.,[.,[[.,[[.,.],[.,.]]],.]]] => 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,[.,.]]]]]] => 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [.,[.,[[.,.],[.,[[.,.],.]]]]] => 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [.,[.,[[[.,.],[.,.]],[.,.]]]] => 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [.,[[.,[.,[.,[.,[.,.]]]]],.]] => 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [.,[[.,[.,[.,[[.,.],.]]]],.]] => 0
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [.,[[.,[.,[[.,[.,.]],.]]],.]] => 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [.,[[.,[.,[[.,.],[.,.]]]],.]] => 0
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [.,[[.,[[.,[.,[.,.]]],.]],.]] => 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [.,[[.,[[.,[[.,.],.]],.]],.]] => 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [.,[[.,[[.,.],[.,[.,.]]]],.]] => 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [.,[[.,[[[.,.],.],[.,.]]],.]] => 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,[.,.]]]]]] => 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [.,[[.,.],[.,[.,[[.,.],.]]]]] => 0
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [.,[[.,.],[.,[[.,[.,.]],.]]]] => 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => 0
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [.,[[[.,.],.],[.,[.,[.,.]]]]] => 0
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [.,[[[.,.],.],[.,[[.,.],.]]]] => 0
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [.,[[[.,.],[.,[.,.]]],[.,.]]] => 0
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [.,[[[.,.],[.,.]],[[.,.],.]]] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[.,[.,[.,[.,[.,[.,.]]]]]],.] => 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[.,[.,[.,[.,[[.,.],.]]]]],.] => 0
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[.,[.,[.,[[.,[.,.]],.]]]],.] => 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[.,[.,[.,[[.,.],[.,.]]]]],.] => 0
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[.,[.,[[.,[.,[.,.]]],.]]],.] => 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[.,[.,[[.,[[.,.],.]],.]]],.] => 1
>>> Load all 127 entries. <<<
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[.,[.,[[.,.],[.,[.,.]]]]],.] => 0
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[.,[.,[[[.,.],.],[.,.]]]],.] => 0
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[.,[[.,[.,[.,[.,.]]]],.]],.] => 0
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[.,[[.,[.,[[.,.],.]]],.]],.] => 0
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[.,[[.,[[.,[.,.]],.]],.]],.] => 0
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[.,[[.,[[.,.],[.,.]]],.]],.] => 0
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[.,[[.,.],[.,[.,[.,.]]]]],.] => 0
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[.,[[.,.],[.,[[.,.],.]]]],.] => 0
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[.,[[[.,.],.],[.,[.,.]]]],.] => 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[.,[[[.,.],[.,.]],[.,.]]],.] => 0
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 0
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => 0
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[.,[[.,[.,.]],.]]]] => 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => 0
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[.,.],[.,[[.,[.,[.,.]]],.]]] => 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[.,.],[.,[[.,[[.,.],.]],.]]] => 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[[[.,.],.],[.,.]]]] => 0
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,[.,.]]]]] => 0
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[.,.],.],[.,[.,[[.,.],.]]]] => 0
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[.,.],.],[.,[[.,[.,.]],.]]] => 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[.,.],.],[.,[[.,.],[.,.]]]] => 0
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[.,.],[.,[.,[.,.]]]],[.,.]] => 0
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[.,.],[.,[[.,.],.]]],[.,.]] => 0
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[.,.],[.,[.,.]]],[[.,.],.]] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[.,.],[.,.]],[[.,.],[.,.]]] => 0
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Description
The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree.
oeis:A159769 counts binary trees avoiding this pattern.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
logarithmic height to pruning number
Description
Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].