Identifier
Values
=>
Cc0010;cc-rep
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[[[.,.],.],[[[.,.],[.,.]],.]]=>0 [[[.,.],.],[[[.,[.,.]],.],.]]=>0 [[[.,.],.],[[[[.,.],.],.],.]]=>0 [[.,[.,[.,.]]],[.,[.,[.,.]]]]=>3 [[.,[.,[.,.]]],[.,[[.,.],.]]]=>2 [[.,[.,[.,.]]],[[.,.],[.,.]]]=>2 [[.,[.,[.,.]]],[[.,[.,.]],.]]=>1 [[.,[.,[.,.]]],[[[.,.],.],.]]=>1 [[.,[[.,.],.]],[.,[.,[.,.]]]]=>2 [[.,[[.,.],.]],[.,[[.,.],.]]]=>1 [[.,[[.,.],.]],[[.,.],[.,.]]]=>1 [[.,[[.,.],.]],[[.,[.,.]],.]]=>0 [[.,[[.,.],.]],[[[.,.],.],.]]=>0 [[[.,.],[.,.]],[.,[.,[.,.]]]]=>2 [[[.,.],[.,.]],[.,[[.,.],.]]]=>1 [[[.,.],[.,.]],[[.,.],[.,.]]]=>1 [[[.,.],[.,.]],[[.,[.,.]],.]]=>0 [[[.,.],[.,.]],[[[.,.],.],.]]=>0 [[[.,[.,.]],.],[.,[.,[.,.]]]]=>2 [[[.,[.,.]],.],[.,[[.,.],.]]]=>1 [[[.,[.,.]],.],[[.,.],[.,.]]]=>1 [[[.,[.,.]],.],[[.,[.,.]],.]]=>0 [[[.,[.,.]],.],[[[.,.],.],.]]=>0 [[[[.,.],.],.],[.,[.,[.,.]]]]=>2 [[[[.,.],.],.],[.,[[.,.],.]]]=>1 [[[[.,.],.],.],[[.,.],[.,.]]]=>1 [[[[.,.],.],.],[[.,[.,.]],.]]=>0 [[[[.,.],.],.],[[[.,.],.],.]]=>0 [[.,[.,[.,[.,.]]]],[.,[.,.]]]=>3 [[.,[.,[.,[.,.]]]],[[.,.],.]]=>2 [[.,[.,[[.,.],.]]],[.,[.,.]]]=>2 [[.,[.,[[.,.],.]]],[[.,.],.]]=>1 [[.,[[.,.],[.,.]]],[.,[.,.]]]=>2 [[.,[[.,.],[.,.]]],[[.,.],.]]=>1 [[.,[[.,[.,.]],.]],[.,[.,.]]]=>1 [[.,[[.,[.,.]],.]],[[.,.],.]]=>0 [[.,[[[.,.],.],.]],[.,[.,.]]]=>1 [[.,[[[.,.],.],.]],[[.,.],.]]=>0 [[[.,.],[.,[.,.]]],[.,[.,.]]]=>2 [[[.,.],[.,[.,.]]],[[.,.],.]]=>1 [[[.,.],[[.,.],.]],[.,[.,.]]]=>1 [[[.,.],[[.,.],.]],[[.,.],.]]=>0 [[[.,[.,.]],[.,.]],[.,[.,.]]]=>1 [[[.,[.,.]],[.,.]],[[.,.],.]]=>0 [[[[.,.],.],[.,.]],[.,[.,.]]]=>1 [[[[.,.],.],[.,.]],[[.,.],.]]=>0 [[[.,[.,[.,.]]],.],[.,[.,.]]]=>2 [[[.,[.,[.,.]]],.],[[.,.],.]]=>1 [[[.,[[.,.],.]],.],[.,[.,.]]]=>1 [[[.,[[.,.],.]],.],[[.,.],.]]=>0 [[[[.,.],[.,.]],.],[.,[.,.]]]=>1 [[[[.,.],[.,.]],.],[[.,.],.]]=>0 [[[[.,[.,.]],.],.],[.,[.,.]]]=>1 [[[[.,[.,.]],.],.],[[.,.],.]]=>0 [[[[[.,.],.],.],.],[.,[.,.]]]=>1 [[[[[.,.],.],.],.],[[.,.],.]]=>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 [[.,[[[.,[.,[.,.]]],.],.]],.]=>1 [[.,[[[.,[[.,.],.]],.],.]],.]=>0 [[.,[[[[.,.],[.,.]],.],.]],.]=>0 [[.,[[[[.,[.,.]],.],.],.]],.]=>0 [[.,[[[[[.,.],.],.],.],.]],.]=>0 [[[.,.],[.,[.,[.,[.,.]]]]],.]=>3 [[[.,.],[.,[.,[[.,.],.]]]],.]=>2 [[[.,.],[.,[[.,.],[.,.]]]],.]=>2 [[[.,.],[.,[[.,[.,.]],.]]],.]=>1 [[[.,.],[.,[[[.,.],.],.]]],.]=>1 [[[.,.],[[.,.],[.,[.,.]]]],.]=>2 [[[.,.],[[.,.],[[.,.],.]]],.]=>1 [[[.,.],[[.,[.,.]],[.,.]]],.]=>1 [[[.,.],[[[.,.],.],[.,.]]],.]=>1 [[[.,.],[[.,[.,[.,.]]],.]],.]=>1 [[[.,.],[[.,[[.,.],.]],.]],.]=>0 [[[.,.],[[[.,.],[.,.]],.]],.]=>0 [[[.,.],[[[.,[.,.]],.],.]],.]=>0 [[[.,.],[[[[.,.],.],.],.]],.]=>0 [[[.,[.,.]],[.,[.,[.,.]]]],.]=>2 [[[.,[.,.]],[.,[[.,.],.]]],.]=>1 [[[.,[.,.]],[[.,.],[.,.]]],.]=>1 [[[.,[.,.]],[[.,[.,.]],.]],.]=>0 [[[.,[.,.]],[[[.,.],.],.]],.]=>0 [[[[.,.],.],[.,[.,[.,.]]]],.]=>2 [[[[.,.],.],[.,[[.,.],.]]],.]=>1 [[[[.,.],.],[[.,.],[.,.]]],.]=>1 [[[[.,.],.],[[.,[.,.]],.]],.]=>0 [[[[.,.],.],[[[.,.],.],.]],.]=>0 [[[.,[.,[.,.]]],[.,[.,.]]],.]=>2 [[[.,[.,[.,.]]],[[.,.],.]],.]=>1 [[[.,[[.,.],.]],[.,[.,.]]],.]=>1 [[[.,[[.,.],.]],[[.,.],.]],.]=>0 [[[[.,.],[.,.]],[.,[.,.]]],.]=>1 [[[[.,.],[.,.]],[[.,.],.]],.]=>0 [[[[.,[.,.]],.],[.,[.,.]]],.]=>1 [[[[.,[.,.]],.],[[.,.],.]],.]=>0 [[[[[.,.],.],.],[.,[.,.]]],.]=>1 [[[[[.,.],.],.],[[.,.],.]],.]=>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
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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