Riffs & Rotes & A061396
Jon Awbrey
jawbrey at oakland.edu
Thu Jun 21 17:44:39 CEST 2001
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SeqFan Addicts,
I would like to give you some account of what I know
about the background and the context of this family
of structures and sequences. It appears from my
notes that I hardly ever wrote up a reasonably
concise summary of my results unless I had a
particular audience or correspondent in mind
at the moment. I find an old flyer in my
archive that informs me that NJAS visited
East Lansing in June of 1979 and gave two
talks at a conference on groups, geometry,
and combinatorics, and this seems to have
inspired me to pass on something like that
summary of work in my previous note to him.
On his follow-up inquiry for references in
the literature, I sent this smattering of
what I knew at the time, dated 1980-08-04.
| N.J.A. Sloane
| Bell Labs
| 600 Mountain Ave
| Murray Hill, New Jersey 07974
|
| Dear Mr. Sloane,
|
| i don't know of any published references to the sequence
| 1, 2, 6, 20, 73, 281, 1124, 4488 [sic], ... or to anything
| resembling riffs, but there are a few items of context i can
| supply. ...
|
| riffs came out of an investigation with several heads:
| either "number-theoretic codings of graphs",
| or "graph-theoretic models of the integers",
| or "measures of complexity on the integers".
| from a musing together of Gödel numberings &
| the Zermelo-VonNeumann construction of integers,
| i was led to the following map of integers into
| planted plane trees, based on their multiplicative
| structure:
|
| o
| |
| 1 = 2^0 = () = @
|
| o
| |
| o
| |
| 2 = 2^1 = (()) = @
|
| o
| |
| o o
| \ /
| o
| |
| 3 = 2^0 3^1 = ()(()) = @
|
| o
| |
| o
| |
| o
| |
| 4 = 2^2 = ((())) = @
|
| o
| |
| o o o
| \|/
| o
| |
| 5 = 2^0 3^0 5^1 = ()()(()) = @
|
| o o
| | |
| o o
| \ /
| o
| |
| 6 = 2^1 3^1 = (())(()) = @
|
| etc.
|
| by a proper consideration of rightmost paths out of each node,
| this map can be changed into a one-one onto correspondence;
| but of course, counting planted plane trees does not
| a new sequence make.
|
| i have since found out that one Albert A. Mullin had already analyzed
| integers this way, getting patterns of primes like 10,000 = 2^2^2 5^2^2
| that he called "mosaics"; and in one place he mentions a graphical
| correspondence but gives no example. articles of his that i can find
| are in Zeitschr. f. math. Logik und Grundlagen d. Mathk. Bd.10 p.159 and
| p.199 (1964) and Notre Dame J. of Form. Logic VIII #4 p.353 (Oct 1967).
| you may want to check with him to see if he knows this sequence or has
| got something similar to riffs. the latest address i've seen is in
| AMS Notices 26 #2 p.A195 (Feb 1979). using the number of pts in the
| graph of an integer as a measure of its complexity is a study related
| to the measures of "roundness" discussed in Hardy & Wright.
|
| by way of the Zermelo-VonNeumann construction of
| integers (as 0 = {}, 1 = {{}}, 2 = {0, 1}, etc.),
| riffs are also a multiplicative analog of Conway's
| surreal numbers & game trees. ([note in margin]
| it would be nice if we could add these representations
| as easily as Conway multiplies his.) there we have a
| class of games G recursively defined as ordered pairs
| of sets of G's, and a special subclass of numbers with
| nothing on the left >= anything on the right. here we
| have a class of forts (forests of oriented rooted trees) F
| defined as sets of ordered pairs of F's, and a special class of
| riffs (rooted index-functional forests) in which the riffs above
| lines 'into' a pt are all distinct. another class of graphs in this
| correspondence, with the same counting sequence, & somewhat easier to
| draw, are called rotes (rooted odd trees with only exponent symmetries),
| o---o
| |
| which are a subset of rooted trees formed from @ 's & called gambits.
| pictures --
|
| integer factorization riff r.i.f.f. rote --> in parentheses
| ---------------------------------------------------------------------------
|
| 1 blank blank blank @ blank
|
| o---o
| |
| 2 p_1^1 p @ @ (())
|
| o---o
| |
| o---o
| 3 p_2^1 = |
| p_(p_1)^1 p_p @ @ ((())())
| ^
| \
| o
|
| o---o
| o |
| ^ o---o
| 4 p_1^2 = / |
| p_1^(p_1) p^p @ @ (((())))
|
| o---o
| |
| o---o
| |
| 5 p_3 = o---o
| p_(p_2) = |
| p_(p_(p_1)) p_(p_p) @ @ (((())())())
| ^
| \
| o
| ^
| \
| o
|
| o-o
| /
| o-o o-o
| 6 p_1 p_2 = \ /
| p_1 p_(p_1) p p_p @ @ @ (())((())())
| ^
| \
| o
|
| 360 = p_1^3 p_2^2 p_3^1 = p^(p_p) p_p^p p_(p_p)
|
| o-o o-o
| \ /
| o-o o-o o-o o-o
| \ | | /
| o-o o---o o-o
| \ | /
| \ | /
| \ | /
| o o \ | /
| ^ ^ ^ \ | /
| / \ / \|/
| = @ o @ @ = @
| ^ ^
| \ \
| o o
| ^
| \
| o
|
| note that if looked at as set of ordered pairs, where (A, B) = (A(B)),
| then 360 = (1(3))(2(2))(3(1)), which can be filled in recursively to
| get the parenthetical form of the corresponding rote. also note that
| rotes are free of any plane embedding as long as the root stays marked;
| for each pt immediately above the root there is a unique line leading
| to an odd tree, & this is the exponent.
|
| yours truly,
|
| Jon Awbrey
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