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Identifier

St000118: Binary trees ⟶ ℤ

Values

[.,.] => 0

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

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

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

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

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

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

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

[.,[.,[.,[.,.]]]] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,.]]]]] => 3

[.,[.,[.,[[.,.],.]]]] => 2

[.,[.,[[.,.],[.,.]]]] => 2

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

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

[.,[[.,.],[.,[.,.]]]] => 2

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

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

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

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

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

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

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

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

[[.,.],[.,[.,[.,.]]]] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,[.,[.,[.,.]]]],.] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => 4

[.,[.,[.,[.,[[.,.],.]]]]] => 3

[.,[.,[.,[[.,.],[.,.]]]]] => 3

[.,[.,[.,[[.,[.,.]],.]]]] => 2

[.,[.,[.,[[[.,.],.],.]]]] => 2

[.,[.,[[.,.],[.,[.,.]]]]] => 3

[.,[.,[[.,.],[[.,.],.]]]] => 2

[.,[.,[[.,[.,.]],[.,.]]]] => 2

[.,[.,[[[.,.],.],[.,.]]]] => 2

[.,[.,[[.,[.,[.,.]]],.]]] => 2

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

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

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

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

[.,[[.,.],[.,[.,[.,.]]]]] => 3

[.,[[.,.],[.,[[.,.],.]]]] => 2

[.,[[.,.],[[.,.],[.,.]]]] => 2

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

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

[.,[[.,[.,.]],[.,[.,.]]]] => 2

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

[.,[[[.,.],.],[.,[.,.]]]] => 2

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

[.,[[.,[.,[.,.]]],[.,.]]] => 2

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

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

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

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

[.,[[.,[.,[.,[.,.]]]],.]] => 2

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

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

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

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

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

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

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

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

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Description

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree.
oeis:A001006 counts binary trees avoiding this pattern.

References

[1] Donaghey, R., Shapiro, L. W. Motzkin numbers MathSciNet:0505544
[2] Rowland, E. S. Pattern avoidance in binary trees MathSciNet:2645188 arXiv:0809.0488

Code

(* in Mathematica *)
tree = {{{{}, {}}, {{}, {}}}, {{{}, {}}, {{}, {}}}};
Count[tree, {{___}, {{___}, {{___}, {___}}}}, {0, Infinity}]

Created

Jun 18, 2013 at 16:59 by Eric Rowland

Updated

Oct 17, 2015 at 11:05 by Christian Stump