Riffs & Rotes & A061396
Jon Awbrey
jawbrey at oakland.edu
Fri Jun 22 20:36:26 CEST 2001
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F. Göbel,
On a 11Correspondence between Rooted Trees and Natural Numbers,
Journal of Combinatorial Theory, B29 (1980), pages 141143.
This correspondence is associated in one way or another
with the following sequences in the EIS online database:
A005517, A005518, A007097, A057452.
In 1980, we find the above paper published,
giving a correspondence between rooted trees
and natural numbers that is dual, in a sense,
to the one that I gave between natural numbers
and planted plane trees. In summary, we have:
1. Recur on Index : N <> Rooted Trees
2. Recur on Power : N < Planted Plane Trees
3. Recur on Both : N > Riffs <> Rotes
As given, the 2nd mapping is manytoone and onto
from planted plane trees to natural numbers, but
there is a way to make the 2nd mapping injective,
at least, I have an old note that claims this,
but I will need to read it over again first,
as I'm not as sure as I used to be about it.
I also notice that Göbel's paper is
given as "Received June 26, 1978",
so I begin to suspect that there
must have been something in the
air (or the water) that year.
Here is a cartoon synopsis of how Göbel's numbering goes:
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 1 blank blank
 2 p_1 ()
 3 p_(p_1) (())
 4 p_1^2 ()()
 5 p_(p_(p_1)) ((()))
 6 p_1 p_(p_1) ()(())
 7 p_(p_1^2) (()())
 8 p_1^3 ()()()
 9 p_(p_1)^2 (())(())
 10 p_1 p_(p_(p_1)) ()((()))

 ...

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 o
 
 o o o o o
    \ /
 o o o o o o o o o o o
   \ /  \ /  \/
 @ @ @ @ @ @ @ @

 1 2 3 4 5 6 7 8 ...

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