Jon Awbrey • Forest Primeval
Author: Jon Awbrey
Contents
Forest Primeval • Correspondence
Jon Awbrey to Martin Gardner • 76.08.16
page 1, typewritten
760816 Mr. Martin Gardner Scientific American Dear Sir: On the following sheet find the beginnings of a correspondence between integers and petrified trees, those so rooted and embedded in their strata as to have a fixed order of limbs at each node. Read blank areas as nodes, the outside being counted first, and enclosure relations as branches. The 0 is merely an orthographic variant for a final parenthesis, reading inward. If this is not old news to you, maybe you would enjoy taking it as a problem to uncover the motivation of this arrangement. If not the time, there is a brief explanation under the stapled section below. Jon Awbrey 404 Michigan Ave. East Lansing, Michigan 48823 This arrangement of trees I have dubbed "the forest primeval". It is developed by expressing each integer in its prime factorization, and then carrying this process to its extremes, factoring each exponent into powers of primes, and each of these powers in turn, quitting only when zero (or empty sets of) exponents are reached, and given the convention that unit exponents be expressed as 2⁰. This suggests a system of numerals with the primes in order as place values, the placeholders being exponents instead of multipliers, the held values being multiplied instead of added together, and with the same multitude of placevalues implied in each place.
o / o o o \ / / o o \ / @ 
I have therefore sometimes thought of this as a notation in which numerals are wholly expressed by means of placevalue, there being but one symbol, whose only office is to reserve a place; but I'm not sure how much this means, since as much might be said of simple tally marks, with alot less complexity taken for granted.
page 2, hand written

Jon Awbrey to Martin Gardner • 78.08.04
page 1, typewritten
August 4, 1978 Martin Gardner Scientific American 415 Madison Avenue New York, New York Dear Sir: The attached short essay will update my letter of August 16, 1976 on the "forest primeval" or "petrified trees", which were just planted plane trees with the planter omitted. The natural correspondence of integers to trees came about as an accidental hybrid of Gödel numbers with Zermelo's integers. The problem of getting a onetoone correspondence was suggested by the various binary encodings of trees in your column of June 1976, and was worked out later that year while reading Frazer's 'Golden Bough'. However, the time since then has brought forth no obvious use for this correspondence, i.e. no theorems of graph theory seem transferable to number theory (or vice versa) by virtue of it. As a note of possible interest connecting two of your recent columns, Charles Peirce used Bell numbers to count the different forms of nplace relations, giving a complicated expression and a triangular array for calculating them; but I don't know how original these may have been with him. See his 'Collected Papers': 3.229. As a last incidental, Peirce's logical graphs have received a modern echo in George SpencerBrown's 'Laws of Form', which accents what Peirce called the entitative interpretation of the graphs. SpencerBrown derives from the algebra of graphs its arithmetic of truth values; and just as Mandelbrot is led to fractional dimensions by a consideration of selfsimilar curves, SpencerBrown finds imaginary truthvalues useful in evaluating infinitely selfsimilar graphs. Yours Truly, Jon Awbrey 404 Michigan Ave. East Lansing, Michigan 48823
page 2, typewritten
The forest primeval: a correspondence between natural numbers and planted plane trees. Consider the primes factorization of a number and represent it as the sequence of its exponents separated by parentheses, thus, N = (n_1) (n_2) (n_3) ... (n_m), ignoring for now the allzero exponents after some point m. e.g.: 1 = 2^0 = (), 2 = 2^2^0 = (()), 3 = 2^0 3^2^0 = ()(()) Repeat this process with each exponent in the factorization and with each further exponent in 'its' factorization and so on, until zero exponents are reached along every path of development. A zero exponent, having no representation in primes, may simply be omitted on the next pass. For example: 360 = (3)(2)(1) = ((0)(1)) ((1)) ((0)) = (()((0))) (((0))) (()) = (()(())) ((())) (()) = 2^(2^0 3^2^0) 3^2^2^0 5^2^0 Thus, parentheses suffice to represent this "involved factorization" of a number. But they also represent planted plane trees, if we interpret the exterior space as the planter, parenthetical containments as further edges, and contained spaces as nodes, e.g.: o o   o o o o \   o o o \  / \/ o  360 = @ In this way, every number may be assigned a unique minimal tree, but many trees will share the same number when the "insignificant digits" of a factorization are taken into account. E.g., trees of the following form, where an ellipsis represents an arbitrary number of terminal edges, all have the number 360: o … o … / / o o … o … o … \/ / / o o o \  / \  / \/ o…  @ = 2^(2^0 3^(2^0 3^0 …) 5^0 …) 3^(2^(2^0 3^0 …) 3^0 …) 5^(2^0 …) 7^0 … The correspondence becomes onetoone by the following transformation. To the rightmost "offspring", when not the only offspring, of any node of a given planted plane tree, assign an extra unit of significance: such that, if terminal, that node represents a unit exponent; if it is not terminal, the unit may be "inherited", but only by a singleton offspring of that node, with a like condition entailing any further inheritance, until a terminal node reached; otherwise, the significance is lost. E.g., 360 is now uniquely represented by the following tree, where primed nodes count for unit exponents under the above rule: o  o o' o \  o o o' \  / \/ o  360 = @ e.g., we now have: o o o  \ / o o o    @ = () = 1, @ = (()) = 2, @ = ()() = 3
page 2, reformatted
The Forest Primeval • A Correspondence Between Natural Numbers And Planted Plane Trees
Consider the primes factorization of a positive integer and represent it as the sequence of its exponents separated by parentheses, thus, ignoring for now the allzero exponents after some point
For example:
Repeat this process with each exponent in the factorization and with each further exponent in its factorization and so on, until zero exponents are reached along every path of development. A zero exponent, having no representation in primes, may simply be omitted on the next pass.
For example:

Thus, parentheses suffice to represent this “involved factorization” of a number. But they also represent planted plane trees, if we interpret the exterior space as the planter, parenthetical containments as further edges, and contained spaces as nodes.
For example,
o o   o o o o \   o o o \  / \/ o  360 = @ 
In this way, every number may be assigned a unique minimal tree, but many trees will share the same number when the “insignificant digits” of a factorization are taken into account.
For example, trees of the following form, where an ellipsis represents an arbitrary number of terminal edges, all have the number 360:
o … o … / / o o … o … o … \/ / / o o o \  / \  / \/ o…  @  
The correspondence becomes onetoone by the following transformation. To the rightmost “offspring”, when not the only offspring, of any node of a given planted plane tree, assign an extra unit of significance: such that, if terminal, that node represents a unit exponent; if it is not terminal, the unit may be “inherited”, but only by a singleton offspring of that node, with a like condition entailing any further inheritance, until a terminal node is reached; otherwise, the significance is lost.
For example, 360 is now uniquely represented by the following tree, where primed nodes count for unit exponents under the above rule:
o  o o' o \  o o o' \  / \/ o  360 = @ 
In this way, we now have:
o o o  \ / o o o    @ = () = 1, @ = (()) = 2, @ = ()() = 3 
Jon Awbrey to N.J.A. Sloane • 80.08.04 {?}
hand written, 2 pages, reformatted here
N.J.A. Sloane Dear Mr. Sloane, I don't know of any published references to the sequence 1, 2, 6, 20, 73, 281, 1124, 4488*, … or to anything resembling riffs, but there are a few items of context I can supply. Unfortunately, I have just gone through a change of address and can't find notes or exact references on anything. I haven't worked out the sequence any further, but am returning to the problem with some fancier methods and will let you know if I have any luck. The rest of this is less immediate. Riffs came out of an investigation with several heads: either ‘numbertheoretic codings of graphs’, or ‘graphtheoretic models of the integers’, or ‘measures of complexity on the integers’. From a musing together of Gödel numberings and the Zermelo–Von Neumann construction of integers, I was led to the following map of integers into planted plane trees, based on the multiplicative sequence: 
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 a set of ordered pairs, where then 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 point immediately above the root there is a unique line leading to an odd tree, and this is the exponent. Yours truly, Jon Awbrey 521 Sunrise Court 
Jon Awbrey to H.W. Gould • 81.12.03
page 1, hand written
81/12/03 H.W. Gould Dept. of Mathematics West Virginia University Morgantown, W. Virginia 26506 Dear Sir: I thought you might be interested in this oneone correspondence between natural numbers and planted plane trees, which I worked out after reading Martin Gardner's June 1976 column on Catalan numbers. I have left its description in the somewhat whimsical form which his article originally induced. It seems that A.A. Mullin had suggested the idea of such continued factorizations of exponents as far back as 1954. He called the resulting expressions 'mosaics'. His several articles can be traced through Math Reviews. Recently, F. Göbel has given a 11 correspondence between natural numbers and rooted (free) trees, based on continued factorization of the ordinal indices of the primes in a factorization; see Discrete Math, Aug 1980. I have also enumerated (generating function and first few terms only) the expressions resulting from continued factorization of both exponents and indices. These 'riffs' have 11 correspondences with special subsets of forests of oriented rooted trees and rooted odd trees.
# of Riffs (by # of p's) = # of Rotes (by # of Γ's) = 1, 1, 2, 6, 20, 73, 281, 1124, 4488[*] Yours truly, Jon Awbrey 521 Sunrise, E. Lansing, Mich. 48823
[*] Correct value is 4618. See A061396.
Forest Primeval • Old Notes
Note 1
Subj: Riffs & Rotes & A061396 Date: Fri, 22 Jun 2001 08:42:01 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> I would like to step back a bit from my retrospective and try to make some connections with my present work, specifically for the reason that I think that this is where some new enumerative problems might be found to arise, all of which I no longer have the proper tools to tackle, but I think that some of this community is almost bound to. During the 80's my work shifted its focus from the enumerative to the computational. Yes, there was a distinction then, strange to say, and yet what I meant was more toward theoremproving programs. All along I had been keeping an eye out for relations among arithmetic (number theory) and algebra (groups), geometry (graphs), and logic  the "lambda point" as one of my professors (Davis) called it then, counting the first two as one. So the little bit of graph and group theory that I picked up was already ready to be converted into datastructures and algorithms able to take advantage of their symmetries, all in support of the new logical mission. But I had to let go of what I earnestly would have kept, all the joys of counting. From my present perspective, what all of these graphs and other sorts of structures have in common is their socalled "reflective" or "selfdocumenting" property. The archetype of these "reflective or autodocumental" data structures is, of course, the usual construction of natural numbers as sets  {}, {{}}, {{}{{}}}, ...  which builds its own distinctive character by keeping a record of its history as it goes. In a similar way, I recall that I first began to think of riffs and all their kin by asking myself a question about molecular tagging: What would be the limit of a tagging tactic? For now, I do not think that I can explain myself any better, but I hope the sense of it will clear in time. Enough flashforward  back to the story ... The earliest indication of this line of thinking, that I can find at present among my personal records, is contained in the following letter to Martin Gardner:  19760816   Mr. Martin Gardner  Scientific American   Dear Sir:   On the following sheet find the beginnings of a correspondence  between integers and petrified trees, those so rooted and embedded  in their strata as to have a fixed order of limbs at each node.  Read blank areas as nodes, the outside being counted first, and  enclosure relations as branches. The 0 is merely an orthographic  variant for a final parenthesis, reading inward.   ...   This arrangement of trees I have dubbed "the forest primeval".  It is developed by expressing each integer in its prime factorization,  and then carrying this process to its extremes, factoring each exponent  into powers of primes, and each of these powers in turn, quitting only  when zero (or empty sets of) exponents are reached, and given the convention  that unit exponents be expressed as 2^0. This suggests a system of numerals  with primes in order as place values, the placeholders being exponents instead  of multipliers, the held values being multiplied instead of added together,  and with the same multitude of placevalues implied in each place.   o  /  o o o  \ / /  o o  \ /  Example: 24 = 2^3 3^1 = 2^(2^0 3^(2^0)) 3^(2^0) = (()(()))(()) = @   I have therefore sometimes thought of this as a notation in which numerals are  wholly expressed by means of placevalue, there being but one symbol, whose only  office is to reserve a place; but I'm not sure how much this means, since as much  might be said of simple tally marks, with alot less complexity taken for granted.   [Enclosure:]   1 0  2 (0)  3 0(0)  4 ((0))  5 00(0)  6 (0)(0)  7 000(0)  8 (0(0))  9 0((0))  10 (0)0(0)  11 0000(0)  12 ((0))(0)  13 00000(0)  14 (0)00(0)  15 0(0)(0)  16 (((0)))   ... ... Continuing apologies for dragging you through my memoirs this way, but there are just too many important details and incidental sidelights that I will otherwise forget.
Note 2
Subj: Topic Maps As A Fool For Inquiry Date: Mon, 02 Jul 2001 14:34:18 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Arisbe <arisbe@stderr.org>, SemioCom <semiocom@listbot.com>, Standardize Unto Others <standardupperontology@ieee.org> Paul Prueitt wrote: > > Please look at : > > http://www.ontologystream.com/IRRTest/Evaluation/ARLReport.htm > > and make a comment to me, if you wish. Paul, Initial Impression: Very Interesting! Here are some links to my own musements: The Forest Primeval: http://www.research.att.com/~njas/sequences/a061396.txt http://www.research.att.com/~njas/sequences/a061396a.txt http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A061396 http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A062504 http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A062537 http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A062860 Inquiry Driven Systems: http://www.chss.montclair.edu/inquiry/fall95/awbrey.html http://www.door.net/arisbe/menu/library/aboutcsp/awbrey/integrat.htm http://www.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ > > Original Message > > From: Jon Awbrey [mailto:jawbrey@oakland.edu] > > Sent: Sunday, July 01, 2001 12:30 AM > > To: topicmapmail@infoloom.com > > Cc: W.M. Jaworski; Paul Prueitt; Steve Pepper > > Subj: [topicmapmail] Topic Maps As A KM Fool > > > > > > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ > > > > W.M. Jaworski wrote: > > > > > > [Paul Prueitt] > > > Steve, > > > Might someone take a part of > > > http://www.antarcti.ca > > > say: > > > http://maps.map.net/cat/Games/Roleplaying?ms=10;ap=0 > > > and write this out as a Topic Map, i.e. as a special XML document? > > > > > > [WMJ] > > > I did try to generate a context map (aka 'topic map') for the > > > WebPage but our harvester is not processing special scripts. > > > Of course the conversion can be done manually, but why move > > > to a foreign context of > > > > > > http://maps.map.net/cat/Games/Roleplaying?ms=10;ap=0 > > ? > > > My desire to enrich receiver context (aka deliver knowledge) > > > and my "human action perception cycle" is responsible for > > > suggesting a similar (better?) task namely 'rewriting' > > > > > > http://www.ontologystream.com (or http://www.ontopia.net ) > > > > > > website into a kind of 'topic map'. Do we know what valueadded should > > > the 'topic map' deliver? What new views provided by 'topic map' of the > > > website content are expected or required by 'owners' or 'navigators'? > > > Are the websites good content sources for the 'topic maps'? > > > > Hi Folks, > > > > Something in this note led me to wonder what > > the 'Machines And People' (MAP's) under the > > influence of a Topical Aesthetic would do > > with a large database site like NJA Sloane's > > "Encyclopedia of Integer Sequences" (EIS): > > > > http://www.research.att.com/~njas/sequences/Seis.html > > > > I ask this merely as a way of introducing myself to your aims and means > > in the medium of a concrete example. > > > > Thanks In Prospect, > > > > Jon Awbrey
Note 3
Subj: This Is The Forest Primeval ... Date: Wed, 04 Jul 2001 07:40:06 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Topic Map Mail <topicmapmail@infoloom.com> CC: Arisbe <arisbe@stderr.org> Here is another Epistle to the SeqFanFolders that may serve as a bit of background to my revival of my work on 'graph theoretical data structures' (GTDS)  I am fond of acronyms, but not those that lack the vowels to be vocalizable, so I will just refer to all these structures as "gits"  with especial reference to ilks of 'reflective or autodocumental' (ROAD) gits. ¤~~~~~~~~~¤~~~~~~~~~¤~MEMOIR~¤~~~~~~~~~¤~~~~~~~~~¤ SeqFanTasians, I would like to step back a bit from my retrospective and try to make some connections with my present work, specifically for the reason that I think that this is where some new enumerative problems might be found to arise, all of which I no longer have the proper tools to tackle, but I think that some of this community is almost bound to. During the 80's my work shifted its focus from the enumerative to the computational. Yes, there was a distinction then, strange to say, and yet what I meant was more toward theoremproving programs. All along I had been keeping an eye out for relations among arithmetic (number theory) and algebra (groups), geometry (graphs), and logic  the "lambda point" as one of my professors (Davis) called it then, counting the first two as one. So the little bit of graph and group theory that I picked up was already ready to be converted into datastructures and algorithms able to take advantage of their symmetries, all in support of the new logical mission. But I had to let go of what I earnestly would have kept, all the joys of counting. From my present perspective, what all of these graphs and other sorts of structures have in common is their socalled "reflective" or "selfdocumenting" property. The archetype of these "reflective or autodocumental" data structures is, of course, the usual construction of natural numbers as sets  {}, {{}}, {{}{{}}}, ...  which builds its own distinctive character by keeping a record of its history as it goes. In a similar way, I recall that I first began to think of riffs and all their kin by asking myself a question about molecular tagging: What would be the limit of a tagging tactic? For now, I do not think that I can explain myself any better, but I hope the sense of it will clear in time. Enough flashforward  back to the story ... The earliest indication of this line of thinking, that I can find at present among my personal records, is contained in the following letter to Martin Gardner:  19760816   Mr. Martin Gardner  Scientific American   Dear Sir:   On the following sheet find the beginnings of a correspondence  between integers and petrified trees, those so rooted and embedded  in their strata as to have a fixed order of limbs at each node.  Read blank areas as nodes, the outside being counted first, and  enclosure relations as branches. The 0 is merely an orthographic  variant for a final parenthesis, reading inward.   ...   This arrangement of trees I have dubbed "the forest primeval".  It is developed by expressing each integer in its prime factorization,  and then carrying this process to its extremes, factoring each exponent  into powers of primes, and each of these powers in turn, quitting only  when zero (or empty sets of) exponents are reached, and given the convention  that unit exponents be expressed as 2^0. This suggests a system of numerals  with primes in order as place values, the placeholders being exponents instead  of multipliers, the held values being multiplied instead of added together,  and with the same multitude of placevalues implied in each place.   o  /  o o o  \ / /  o o  \ /  Example: 24 = 2^3 3^1 = 2^(2^0 3^(2^0)) 3^(2^0) = (()(()))(()) = @   I have therefore sometimes thought of this as a notation in which numerals are  wholly expressed by means of placevalue, there being but one symbol, whose only  office is to reserve a place; but I'm not sure how much this means, since as much  might be said of simple tally marks, with alot less complexity taken for granted.   [Enclosure:]   1 0  2 (0)  3 0(0)  4 ((0))  5 00(0)  6 (0)(0)  7 000(0)  8 (0(0))  9 0((0))  10 (0)0(0)  11 0000(0)  12 ((0))(0)  13 00000(0)  14 (0)00(0)  15 0(0)(0)  16 (((0)))   ... ... Continuing apologies for dragging you through my memoirs this way, but there are just too many important details and incidental sidelights that I will otherwise forget.
Note 4
Subj: This Is The Forest Primeval ... Date: Thu, 05 Jul 2001 10:00:01 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SemioCom <semiocom@listbot.com>, Standardize Unto Others <standardupperontology@ieee.org> Here is another Epistle to the SeqFanFolders that may serve as a bit of background to my revival of my work on 'graph theoretical data structures' (GTDS)  I am fond of acronyms, but not those that lack the vowels to be vocalizable, so I will just refer to all these structures as "gits"  with especial reference to ilks of 'reflective or autodocumental' (ROAD) gits. ¤~~~~~~~~~¤~~~~~~~~~¤~MEMOIR~¤~~~~~~~~~¤~~~~~~~~~¤ SeqFanTasians, I would like to step back a bit from my retrospective and try to make some connections with my present work, specifically for the reason that I think that this is where some new enumerative problems might be found to arise, all of which I no longer have the proper tools to tackle, but I think that some of this community is almost bound to. During the 80's my work shifted its focus from the enumerative to the computational. Yes, there was a distinction then, strange to say, and yet what I meant was more toward theoremproving programs. All along I had been keeping an eye out for relations among arithmetic (number theory) and algebra (groups), geometry (graphs), and logic  the "lambda point" as one of my professors (Davis) called it then, counting the first two as one. So the little bit of graph and group theory that I picked up was already ready to be converted into datastructures and algorithms able to take advantage of their symmetries, all in support of the new logical mission. But I had to let go of what I earnestly would have kept, all the joys of counting. >From my present perspective, what all of these graphs and other sorts of structures have in common is their socalled "reflective" or "selfdocumenting" property. The archetype of these "reflective or autodocumental" data structures is, of course, the usual construction of natural numbers as sets  {}, {{}}, {{}{{}}}, ...  which builds its own distinctive character by keeping a record of its history as it goes. In a similar way, I recall that I first began to think of riffs and all their kin by asking myself a question about molecular tagging: What would be the limit of a tagging tactic? For now, I do not think that I can explain myself any better, but I hope the sense of it will clear in time. Enough flashforward  back to the story ... The earliest indication of this line of thinking, that I can find at present among my personal records, is contained in the following letter to Martin Gardner:  19760816   Mr. Martin Gardner  Scientific American   Dear Sir:   On the following sheet find the beginnings of a correspondence  between integers and petrified trees, those so rooted and embedded  in their strata as to have a fixed order of limbs at each node.  Read blank areas as nodes, the outside being counted first, and  enclosure relations as branches. The 0 is merely an orthographic  variant for a final parenthesis, reading inward.   ...   This arrangement of trees I have dubbed "the forest primeval".  It is developed by expressing each integer in its prime factorization,  and then carrying this process to its extremes, factoring each exponent  into powers of primes, and each of these powers in turn, quitting only  when zero (or empty sets of) exponents are reached, and given the convention  that unit exponents be expressed as 2^0. This suggests a system of numerals  with primes in order as place values, the placeholders being exponents instead  of multipliers, the held values being multiplied instead of added together,  and with the same multitude of placevalues implied in each place.   o  /  o o o  \ / /  o o  \ /  Example: 24 = 2^3 3^1 = 2^(2^0 3^(2^0)) 3^(2^0) = (()(()))(()) = @   I have therefore sometimes thought of this as a notation in which numerals are  wholly expressed by means of placevalue, there being but one symbol, whose only  office is to reserve a place; but I'm not sure how much this means, since as much  might be said of simple tally marks, with alot less complexity taken for granted.   [Enclosure:]   1 0  2 (0)  3 0(0)  4 ((0))  5 00(0)  6 (0)(0)  7 000(0)  8 (0(0))  9 0((0))  10 (0)0(0)  11 0000(0)  12 ((0))(0)  13 00000(0)  14 (0)00(0)  15 0(0)(0)  16 (((0)))   ... ... Continuing apologies for dragging you through my memoirs this way, but there are just too many important details and incidental sidelights that I will otherwise forget.
Note 5
Subj: Gits & Gambits Date: Thu, 05 Jul 2001 12:34:01 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Arisbe <arisbe@stderr.org>, SemioCom <semiocom@listbot.com>, Standardize Unto Others <standardupperontology@ieee.org>, Topic Map Mail <topicmapmail@infoloom.com> I would that I could not be held responsible for the hasty way that I will have to be pudding this today, but, of course, nobody ever escapes responsibility for anything, and especially the halfbaked loaves. I have been trying to rely as much as I can on my 'archives', as I was so much sharper then than I'll ever be again, but I do not see how I can avoid treading on the haziness of trying to force a fresh thought on this point or edge. Aside to Graham: I do not see much chance that I can do anything about most of the annoyances and irritations that you mention, as they are parts of what I need to keep even this barest semblance of organization, myself, but I do see that a continuing fraction of the discontinuity you experienced is due to the circumstance that I have always been disposed to think and to work in what one might call "overlapping neighborhoods of thought" (ONOT's), as a consequence of which I'll often be trying to make some contributary to one "watershed of thinking" (WOT) when out of the blue it will appear to me that some of it might be pertinet to some other group that is weighing on my mind  I am often disieved by my illusions of pertinets, sure, but the raveling of that is always a post hoc matter  anyway, with that prerumble, here is the message that preceded the one you seized up on  oh, wait, it will probably be less distracting to emit it separately:  Cf: "This Is The Forest Primeval ..."  Now Serving At Two Locations, No Waiting:   http://stderr.org/pipermail/arisbe/2001July/000690.html  http://www.infoloom.com/pipermail/topicmapmail/2001q3/001161.html To continue, I want now to introduce the family of 'graphtheoretical data structures' ("gits") that I know as "gambits", for reasons that will soon be manifest. Structures like these always come in two forms, the text and the graph forms, like the phenotypic 'alternation of generations' that I remember from a biology course lang syne. I will present the two variations on this theme in the parallel paradigmatics that follows here. Nota Bene. The atsign "@" denotes a root node. ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤  Definition  ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤     Blank " " is a gambit.  @ is a gambit.     ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤     If i is a gambit and  If i is a gambit and   if j is a gambit,  if j is a gambit,       i j   then ij is one too.  then @ is one too.     ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤     If i is a gambit and  If i is a gambit and   if j is a gambit,  if j is a gambit,       i j    oo       then (i(j)) is one too.  then @ is one too.     ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤ In Step 1 of this recursive scheme, the notation "ij" denotes the concatenation of i and j, id est, just the writing of string i and string j one after another. The associated graphical notation means that one joins tree i and tree j at the root in order to construct the new tree. In Step 2 of this scheme, i is called the "index" and j is called the "exponent" of the gambit that is constructed there. The notation "(i(j))" says to place the strings for i and j in the indicated places of the frame "(_(_))". The parallel graph notation enjoins one to graft the tree i and then to graft the tree j at the pair of sites that are severally indicated, merging their roots with the nodes that are currently sited in the underlying gambit, in passing erasing their status as roots. The glyphing of that basic constructor, the rooted path of length 2, in the form of a 'gamma' is the source of the family name, but that is little more than a conventional way of scribing it that is often convenient. In actuality, except for their roots, gambits are free trees, not embedded in the plane or unnaturally crooked or anything. That is the basic construction of gambits, contingent on my remembering it all right.
Note 6
Subj: Gits & Gambits Date: Thu, 05 Jul 2001 12:48:16 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> I would that I could not be held responsible for the hasty way that I will have to be pudding this today, but, of course, nobody ever escapes responsibility for anything, and especially the halfbaked loaves. I have been trying to rely as much as I can on my 'archives', as I was so much sharper then than I'll ever be again, but I do not see how I can avoid treading on the haziness of trying to force a fresh thought on this point or edge. Aside to Graham: I do not see much chance that I can do anything about most of the annoyances and irritations that you mention, as they are parts of what I need to keep even this barest semblance of organization, myself, but I do see that a continuing fraction of the discontinuity you experienced is due to the circumstance that I have always been disposed to think and to work in what one might call "overlapping neighborhoods of thought" (ONOT's), as a consequence of which I'll often be trying to make some contributary to one "watershed of thinking" (WOT) when out of the blue it will appear to me that some of it might be pertinet to some other group that is weighing on my mind  I am often disieved by my illusions of pertinets, sure, but the raveling of that is always a post hoc matter  anyway, with that prerumble, here is the message that preceded the one you seized up on  oh, wait, it will probably be less distracting to emit it separately:  Cf: "This Is The Forest Primeval ..."  Now Serving At Two Locations, No Waiting:   http://stderr.org/pipermail/arisbe/2001July/000690.html  http://www.infoloom.com/pipermail/topicmapmail/2001q3/001161.html To continue, I want now to introduce the family of 'graphtheoretical data structures' ("gits") that I know as "gambits", for reasons that will soon be manifest. Structures like these always come in two forms, the text and the graph forms, like the phenotypic 'alternation of generations' that I remember from a biology course lang syne. I will present the two variations on this theme in the parallel paradigmatics that follows here. Nota Bene. The atsign "@" denotes a root node. ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤  Definition  ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤     Blank " " is a gambit.  @ is a gambit.     ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤     If i is a gambit and  If i is a gambit and   if j is a gambit,  if j is a gambit,       i j   then ij is one too.  then @ is one too.     ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤     If i is a gambit and  If i is a gambit and   if j is a gambit,  if j is a gambit,       i j    oo       then (i(j)) is one too.  then @ is one too.     ¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~¤ In Step 1 of this recursive scheme, the notation "ij" denotes the concatenation of i and j, id est, just the writing of string i and string j one after another. The associated graphical notation means that one joins tree i and tree j at the root in order to construct the new tree. In Step 2 of this scheme, i is called the "index" and j is called the "exponent" of the gambit that is constructed there. The notation "(i(j))" says to place the strings for i and j in the indicated places of the frame "(_(_))". The parallel graph notation enjoins one to graft the tree i and then to graft the tree j at the pair of sites that are severally indicated, merging their roots with the nodes that are currently sited in the underlying gambit, in passing erasing their status as roots. The glyphing of that basic constructor, the rooted path of length 2, in the form of a 'gamma' is the source of the family name, but that is little more than a conventional way of scribing it that is often convenient. In actuality, except for their roots, gambits are free trees, not embedded in the plane or unnaturally crooked or anything. That is the basic construction of gambits, contingent on my remembering it all right.
Note 7
Subj: Göbel = Matula Numbering. (Neé: Riffs & Rotes & A061396) Date: Fri, 06 Sep 2002 00:30:42 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Antti Karttunen <karttu@megabaud.fi> SeqFan <seqfan@ext.jussieu.fr> i will see if i can find the references later, but i remember neil or maybe nordhaus telling me about some similar ideas that a.a. mullin (?) called 'mosaics', some of which got published in the nat.acad.sci., and i think that there were related constructions going back to a goodstein or goldstein in the 20's, also some connections to mirimanoff. all very fuzzy now. my initial interest in riffs and rotes came out of a search for "autodocumenting data structures" and "selftagging molecules", plus links between additive and multiplicative number theory that i associated with the question: "how much of the ordering of the integers is purely combinatorial", whatever that means. so, yes, some flavor of generalized primes comes into play  if you ask: is there a "purely combinatorical" reason that (p_2)^1 < (p_1)^2? viewed as rotes: oo oo   oo oo   @ <?< @ (p_2)^1 <?< (p_1)^2 fun & games ... links here to dana scott's d_infinity and similar constructions. Antti Karttunen wrote: > > Jon Awbrey wrote Fri, 22 Jun 2001 14:36:26: > > > > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ > > > > 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: > > > > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ > >  > >  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)) ()((())) > >  > >  ... > >  > > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ > >  > >  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 ... > >  > > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ > > Something in the air ten years earlier ? > > D. W. Matula, Washington University, St. Louis, Missouri. > A Natural Rooted Tree Enumeration by Prime Factorization. > SIAM Rev. 10 (1968), p. 273. > > A onetoone correspondence between the space of rooted > trees and the positive integers is given by the following > recursive procedure. Let the positive integer n correspond > to the root node of a tree, the prime factors of n > (each occurring as often as its multiplicity) correspond > to the edges incident to the root node, and for each such > edge incident to the root which corresponds to the prime p > let the branch of the tree incident to that edge be the > rooted tree corresponding to the number pi(p) (the prime > number function pi). The terminal condition of this > recursion arises quite naturally as unity (which has no > prime factors) is thus associated with each pendent vertex. > On the other hand, the number associated with a given > rooted tree may be recursively determined as follows: > If the numbers associated with the branches of a rooted > tree T are n1, n2, ..., nm, then the unique number > associated with T is the product p(n1) x p(n2) x ... x p(nm), > where p(ni) is the ni:th prime. Certain interesting > relationships between the theories of primes and trees > will be developed utilizing this "natural" correspondence. > > Compare with the beginning of Göbel's paper: > > In the sequel, we suppose that p(1) = 2, p(2), p(3), ... > is the sequence of primes in ascending order. > Let T be a rooted tree, r its root. The connected components > of T  r are denoted by T1, ..., Tv, where v is the degree > of r. The graphs Tj (j = 1,...,v) obviously are trees, which > we transform into rooted trees by defining as the the root > of Tj the vertex of Tj which is adjacent to r in T. We now > assign a natural number phi(T) to the rooted tree T. > > Definition. If T has only one vertex, then phi(T) = 1. > If the number of vertices of T is greater than 1, then > > phi(T) = product_{i=1..v} p(phi(Ti)), > > where T1, ..., Tv are the rooted trees defined above. > > ......... > > There's at least one online document about Matulanumbers: > > Ivan Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, > Publications de L'institut Mathématique (Beograd), Vol. 53(67), pp. 1722 (1993). > URL: http://www.emis.de/journals/PIMB/067/n067p017.pdf > > It would be nice to compute the functions f1  f10 (corresponding > to the properties P1  P10) defined in that paper. > I don't know how much you can find connections from there > to the classical elementary number theory, but as a simple > example, the degree of the rooted tree with Matula/Göbel number n > is bigomega(n) (A001222), and "the primeth recurrence" A007097 > gives the trees with just one leaf. > > > Now, I started thinking why we must use specifically > the ordinary primes. Why not use instead the binary > encoding of irreducible Z2[x] polynomials, i.e. > A014580 (2,3,7,11,13,19,25,31,...) instead of > A000040 (2,3,5,7,11,13,17,19,23,...) as the function 'p' > in the above definition given by Göbel? > Of course then we should use the carryless Z2[x] multiplication > instead of the ordinary one, to acquire a onetoone map. > So, 5 = 101 in binary (corresponding to x^2 + 1 = x+1 * x+1 > in Z2[X]) corresponds now to the tree: > > \ / > \/ > > which in original Matula/Göbel scheme is associated with the number 9. > (Deducing the properties of the trees using this scheme > might be slightly easier than with the original one?) > > Combining the mapping N <> rooted trees, and > rooted trees <> Z2[x] polynomials we obtain a map > between the multiplicative semigroup (or specifically: > commutative monoid) of natural numbers > and the multiplicative semigroup found in > the ring of Z2[x] polynomials. > Let's call it khi. Unless I'm completely zombie, > it obeys > khi(1) = 1 > and > khi(ab) = khi(a) X khi(b) > where X means multiplication of Z2[x] polynomials, > so it is a homomorphism between these semigroups. > > Using the standard binary encoding of Z2[x] > polynomials we can represent it in OEIS as > yet another permutation of natural numbers. > Among other things, it maps A000040 and A014580 > termbyterm to each other. > (How does it map the corresponding multiplication > tables A004247 and A048720 to each other?) > > Then we can get even wilder, because any domain > with unique factorization can be used, so why not to > use, say Gaussian integers, or some other "double ring"? > I remember that Marc spoke some months ago about storing > these ones to OEIS with the "bitsinterleaved" method. > > Terveisin, > > Antti Karttunen
Note 8
Subj: Göbel = Matula Numbering. (Neé: Riffs & Rotes & A061396) Date: Tue, 10 Sep 2002 09:16:23 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Antti Karttunen <karttu@megabaud.fi>, SeqFan <seqfan@ext.jussieu.fr> AK = Antti Karttunen JA = Jon Awbrey some incidental remarks on riffs & rotes, from the standpoint of ordered domains. suppose that we don't know anything at all about the number theory connection, but are simply given a species of trees defined by the following recursion:  @ is a rote.   if f is a finite function from rotes to rotes,  say, f : index_j ~> exponent_j for j = 1 to k,  where the "indices" and "exponents" are rotes,  then the following tree is a rote:   i_1 e_1 i_k e_k  oo oo  \ ... /  \  /  \  /  \/  @   where the rotes for the "indices" (source elements) i_j  and the "exponents" (target elemnts) e_j are attached  at the places indicated. so the question is, what is a natural way of partial ordering this species? then, can we specify enough information to make it a total linear ordering? JA:  Göbel, F., "On a 11Correspondence between Rooted Trees and Natural Numbers", 'Journal of Combinatorial Theory', B29 (1980), pages 141143. JA: This correspondence is associated in one way or another with the following sequences in the EIS online database: A005517, A005518, A007097, A057452. JA: 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 JA: 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. JA: 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. JA: Here is a cartoon synopsis of how Göbel's numbering goes: ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤   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)) ()((()))   ...  ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤   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 ...  ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ AK: Something in the air ten years earlier ?  D.W. Matula, Washington University, St. Louis, Missouri.  A Natural Rooted Tree Enumeration by Prime Factorization.  SIAM Rev. 10 (1968), p. 273.   A onetoone correspondence between the space of rooted  trees and the positive integers is given by the following  recursive procedure. Let the positive integer n correspond  to the root node of a tree, the prime factors of n  (each occurring as often as its multiplicity) correspond  to the edges incident to the root node, and for each such  edge incident to the root which corresponds to the prime p  let the branch of the tree incident to that edge be the  rooted tree corresponding to the number pi(p) (the prime  number function pi). The terminal condition of this  recursion arises quite naturally as unity (which has no  prime factors) is thus associated with each pendent vertex.  On the other hand, the number associated with a given  rooted tree may be recursively determined as follows:  If the numbers associated with the branches of a rooted  tree T are n1, n2, ..., nm, then the unique number  associated with T is the product p(n1) x p(n2) x ... x p(nm),  where p(ni) is the ni:th prime. Certain interesting  relationships between the theories of primes and trees  will be developed utilizing this "natural" correspondence. AK: Compare with the beginning of Göbel's paper:  In the sequel, we suppose that p(1) = 2, p(2), p(3), ...  is the sequence of primes in ascending order.  Let T be a rooted tree, r its root. The connected components  of T  r are denoted by T1, ..., Tv, where v is the degree  of r. The graphs Tj (j = 1,...,v) obviously are trees, which  we transform into rooted trees by defining as the the root  of Tj the vertex of Tj which is adjacent to r in T. We now  assign a natural number phi(T) to the rooted tree T.   Definition. If T has only one vertex, then phi(T) = 1.  If the number of vertices of T is greater than 1, then   phi(T) = product_{i=1..v} p(phi(Ti)),   where T1, ..., Tv are the rooted trees defined above.   ... AK: There's at least one online document about Matulanumbers:  Ivan Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers,  Publications de L'institut Mathématique (Beograd), Vol. 53(67), pp. 1722 (1993).  URL: http://www.emis.de/journals/PIMB/067/n067p017.pdf AK: It would be nice to compute the functions f1  f10 (corresponding to the properties P1  P10) defined in that paper. I don't know how much you can find connections from there to the classical elementary number theory, but as a simple example, the degree of the rooted tree with Matula/Göbel number n is bigomega(n), (A001222), and "the primeth recurrence" A007097 gives the trees with just one leaf. AK: Now, I started thinking why we must use specifically the ordinary primes. Why not use instead the binary encoding of irreducible Z2[x] polynomials, i.e. A014580 (2,3,7,11,13,19,25,31,...) instead of A000040 (2,3,5,7,11,13,17,19,23,...) as the function 'p' in the above definition given by Göbel? Of course then we should use the carryless Z2[x] multiplication instead of the ordinary one, to acquire a onetoone map. AK: So, 5 = 101 in binary (corresponding to x^2 + 1 = x+1 * x+1 in Z2[X]) corresponds now to the tree: \ / \/ which in original Matula/Göbel scheme is associated with the number 9. (Deducing the properties of the trees using this scheme might be slightly easier than with the original one?) AK: Combining the mapping N <> rooted trees, and rooted trees <> Z2[x] polynomials we obtain a map between the multiplicative semigroup (or specifically: commutative monoid) of natural numbers and the multiplicative semigroup found in the ring of Z2[x] polynomials. Let's call it khi. Unless I'm completely zombie, it obeys khi(1) = 1 and khi(ab) = khi(a) X khi(b) AK: where X means multiplication of Z2[x] polynomials, so it is a homomorphism between these semigroups. AK: Using the standard binary encoding of Z2[x] polynomials we can represent it in OEIS as yet another permutation of natural numbers. Among other things, it maps A000040 and A014580 termbyterm to each other. AK: (How does it map the corresponding multiplication tables A004247 and A048720 to each other?) AK: Then we can get even wilder, because any domain with unique factorization can be used, so why not to use, say Gaussian integers, or some other "double ring"? I remember that Marc spoke some months ago about storing these ones to OEIS with the "bitsinterleaved" method. JA: i will see if i can find the references later, but i remember neil or maybe nordhaus telling me about some similar ideas that a.a. mullin (?) called 'mosaics', some of which got published in the nat.acad.sci., and i think that there were related constructions going back to a goodstein or goldstein in the 20's, also some connections to mirimanoff. all very fuzzy now. JA: my initial interest in riffs and rotes came out of a search for "autodocumenting data structures" and "selftagging molecules", plus links between additive and multiplicative number theory that i associated with the question: "how much of the ordering of the integers is purely combinatorial", whatever that means. so, yes, some flavor of generalized primes comes into play  if you ask: JA: is there a "purely combinatorical" reason that (p_2)^1 < (p_1)^2? JA: viewed as rotes:  oo oo     oo oo     @ <?< @   (p_2)^1 <?< (p_1)^2 JA: fun & games ... JA: links here to dana scott's d_infinity and similar constructions.
Note 9
Subj: Riffs & Rotes & A061396 Date: Wed, 18 Sep 2002 08:32:31 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> memories like strawberries ... grow fuzzy with time ... i remember now that it was bonnie stewart who first told me about mullin's work, only he said "mulligan", so that threw me off the trail for a while, and it was only later when r.w. robinson gave me the correct name that i was able to find his work on "mosaics". here are the references that i can find at present:  mullin, a.a., 'zeitschrift f. math. logik und grundlagen d. mathk.',  bd. 10, p.159 & p.199, 1964.  mullin, a.a., 'notre dame j. of formal logic',  vol. 8, no. 4, p. 353, oct. 1967. a basic reference on dana scott's "d_infinity" (model of lambda calculus) is this:  stoy, j.e., 'denotational semantics: the scottstrachey approach to programming language theory',  mit press, cambridge, ma, 1977.
Note 10
Subj: Gödel, Göbel, Matula, Mullin, ... Date: Wed, 18 Sep 2002 10:14:25 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> AK = Antti Karttunen JA = Jon Awbrey catching up on old correspondence, so this may be a bit out of order. will try to pare the redundancies. JA: 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 JA: 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. AK: Look at A075166, giving a bijection between rooted plane trees and natural numbers > 0. http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=075166  The rooted plane trees encoded here are:  .....................o...............o.........o...o..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......9..... AK: was it that what you had in mind then? oh, strictly speaking, we should be calling these "planted plane trees", as otherwise you can rotate the branches around the root and transform, for example, 6 into 9. i found the two letters that i wrote to martin gardner in 76 & 78, and will try to extract the gist of them later, but here is a bit:  The forest primeval: a correspondence between natural numbers and planted plane trees.   Consider the primes factorization of a number and represent it as the sequence of its  exponents separated by parentheses, thus, N = (n_1) (n_2) (n_3) ... (n_m), ignoring  for now the allzero exponents after some point m.   e.g.: 1 = 2^0 = (), 2 = 2^2^0 = (()), 3 = 2^0 3^2^0 = ()(())   Repeat this process with each exponent in the factorization and with each further  exponent in 'its' factorization and so on, until zero exponents are reached along  every path of development. A zero exponent, having no representation in primes,  may simply be omitted on the next pass. For example:   360 = (3)(2)(1) = ((0)(1)) ((1)) ((0)) = (()((0))) (((0))) (())   = (()(())) ((())) (())   = 2^(2^0 3^2^0) 3^2^2^0 5^2^0   Thus, parentheses suffice to represent this "involved factorization" of a number.  But they also represent planted plane trees, if we interpret the exterior space  as the planter, parenthetical containments as further edges, and contained  spaces as nodes, e.g.:   o o     o o o o  \    o o o  \  /  \/  o    360 = @   In this way, every number may be assigned a unique minimal tree, but  many trees will share the same number when the "insignificant digits" of  a factorization are taken into account. E.g., trees of the following form,  where an ellipsis represents an arbitrary number of terminal edges, all have  the number 360:   o … o …  / /  o o … o … o …  \/ / /  o o o  \  /  \  /  \/  o…    @   = 2^(2^0 3^(2^0 3^0 …) 5^0 …) 3^(2^(2^0 3^0 …) 3^0 …) 5^(2^0 …) 7^0 …
Note 11
Subj: Tree Terminology, Terminable & Interminable Date: Thu, 19 Sep 2002 13:48:55 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> oh, i understand that dialects in graph theory differ, especially when you have to interface with computer sci, i am merely going by the particular usage that i learned long ago, and there's always a way of finding the proper translations between different terminologies. "planted plane tree" was a classical usage for the trees that are counted by catalans. different folks would use different strokes like these to indicate a "planter": o OR o OR o ^    @ and sometimes they would count the root and sometimes not. the point here is to prevent the rotations around the root within the plane, not to block the reflections, which are not allowed anyway. e.g. "the five planted plane trees of order 5" o  o o o o o  \ /   o o o o o o o o o   \ / \ / \/ o o o o o      @ @ @ @ @ p.67 in Harary & Palmer, 'Graphical Enumeration', 1973. i was in the middle of writing another note slightly related to this, but i think that all of this fuss about the planters may have been one of the reasons why i eventually shifted to cacti, which are much happier without the planters, the desert sand being far too forbidding of these underground rotations in any case. AK:  The rooted plane trees encoded here are:  .....................o...............o.........o...o..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......9..... JA: oh, strictly speaking, we should be calling these "planted plane trees" as otherwise you can rotate the branches around the root and transform, for example, 6 into 9. what i meant is that we have to add a planter to the above trees in order to get a correspondence with parenthetical expressions. AK: I'm following Stanley's usage in his "Exercises on Catalan and Related Numbers" http://wwwmath.mit.edu/~rstan/ec/catalan.pdf AK: that is, the above trees belong to his manifestation 19 (e) "plane trees with n + 1 vertices" while the next manifestation (f) is given as "planted (i.e., root has degree one) trivalent plane trees with 2n + 2 vertices" AK: so I have thought that the term "planted" means exactly that, that the tree (whether binary (= trivalent) or general), is "planted" on a top of the single "stem"  _not_ that it cannot be reflected as a whole, as you indicate. (Because being "planar", it is already forbidden). AK: And also, I assume the terms "plane", "planar", and "oriented" (cf. A000108) are synonyms, as well as their antonyms "nonplanar" and "nonoriented" (cf. A000081). AK: Am I correct? AK: I guess we could create a "tree matrix", where we have all the combinations of labeled/unlabeled, rooted/nonrooted, planar/nonplanar, binary/general trees, with each entry pointing to the corresponding enumerating EISsequence. (will do that, when I have spare time ...)
Note 12
Subj: The Forest Primeval Date: Thu, 19 Sep 2002 14:16:32 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> Here is what the initial mapping of numbers into parenthetical expressions looked like: 1 2^0 o () 2 2^2^0 (o) (()) 3 2^0 3^2^0 o(o) ()(()) 4 2^2^2^0 ((o)) ((())) 5 2^0 3^0 5^2^0 oo(o) ()()(()) 6 2^2^0 3^2^0 (o)(o) (())(()) 7 2^0 3^0 5^0 7^2^0 ooo(o) ()()()(()) 8 2^[2^0 3^2^0] (o(o)) (()(())) 9 2^0 3^2^2^0 o((o)) ()((())) 10 2^2^0 3^0 5^2^0 (o)o(o) (())()(()) 11 2^0 3^0 5^0 7^0 11^2^0 oooo(o) ()()()()(()) 12 2^2^2^0 3^2^0 ((o))(o) ((()))(()) 13 2^0 3^0 5^0 7^0 11^0 13^0 ooooo(o) ()()()()()(()) 14 2^2^0 3^0 5^0 7^2^0 (o)oo(o) (())()()(()) 15 2^0 3^2^0 5^2^0 o(o)(o) ()(())(()) 16 2^2^2^2^0 (((o))) (((()))) The "o" is just an easier to read variant for an empty pair of terminal parentheses. Here is the second half of that note I wrote, where I "supposedly" got a 11 correspondence:  The correspondence becomes onetoone by the following transformation.  To the rightmost "offspring", when not the only offspring, of any node  of a given planted plane tree, assign an extra unit of significance:  such that, if terminal, that node represents a unit exponent; if it  is not terminal, the unit may be "inherited", but only by a singleton  offspring of that node, with a like condition entailing any further  inheritance, until a terminal node reached; otherwise, the significance  is lost. E.g., 360 is now uniquely represented by the following tree,  where primed nodes count for unit exponents under the above rule:   o    o o' o  \   o o o'  \  /  \/  o    360 = @ Subsequently, I wrote this marginal note:  e.g., now:   o o o   \ /  o o o      @ = () = 1, @ = (()) = 2, @ = ()() = 3 Well, that is what I wrote, but to be honest it looks wrong somehow. Maybe I muffed up the initial conditions in my later note. Or maybe it can be fixed by deleting the "when not the only offspring" clause in the tokenpassing transformation. Let me give it another try ... What we are really interested in are sequences of exponents, that we turn into sequences of trees, and we use the gimmick of a graphtheoretical "planter" to convert the sequences of trees into planted plane trees. In the general setting, the planter look like this: ^ ^ ^ \/ o  @ But it's a quirky property of this setup that the empty sequence and the singleton sequences require planters that appear to be excessively large: Empty sequence <>: o  @ Singleton sequence <k>: k  o  @ So maybe I should have written:  o' o o'   \ /  o o o      @ = _ = 1, @ = () = 2, @ = ()() = 3 So far this seems to suggest that the buckpassing rule should apply to every node but the root, but now I will have to rethink the whole thing. Maybe I'll just leave this one to the current inhabitants of Fair Catalania, and return to the Realm Of Rote Arithmetic.
Note 13
Subj: The Lambda Point Date: Thu, 19 Sep 2002 15:14:15 0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SeqFan <seqfan@ext.jussieu.fr> AK = Antti Karttunen AK: So far I haven't found/constructed any ranking scheme which would be practical to implement as a computer algorithm, in a sense that the sequence {max integer mapped to the size n tree} wouldn't grow too steeply. I reckon this is also the problem with Matula/Göbel encoding of rooted unoriented trees. Ok, there are lots and lots of mappings between any two countable sets, so I guess that the "interesting" ones are relative to one's interest, one's endinview, and most likely one's personal sense of aesthetics. I was initially interested in the correspondences that might serve as mathematical "laborsaving devices", allowing us to transfer theorems, what we know and can prove, among different domains, especially among algebra, geometry, and logic  what one of my teachers used to call the "lambda point", after the critical triple point of transitions among gas, liquid, and solid phases. From this perspective, it's not a problem, may even be a feature, that some relatively big numbers have some interesting aspects of their structure captured by some relatively small graphs. I haven't said anything about the logical side of this yet, but remember that it was folks like Gödel, not to mention Peirce, who lobbed us into this particular briar patch.
Note 14
Residual Questions ... Here is how I last defined rotes: JA:  @ is a rote.   If f is a finite function from rotes to rotes,  say, f : index_j ~> exponent_j for j = 1 to k,  where the "indices" and "exponents" are rotes,  then the following tree is a rote:   i_1 e_1 i_k e_k  oo oo  \ ... /  \  /  \  /  \/  @   where the rotes for the "indices" (source elements) i_j  and the "exponents" (target elemnts) e_j are attached  at the places indicated. Strictly speaking, I probably should have stipulated that each f is a "finite partial function" from rotes to rotes, but since the set of rotes is infinite the partiality of f might be taken as implicit in the fact of its being finite. Let $N$ ("Script N") be the set of rotes. Let (X > Y) be the set of functions from X to Y. Let [X > Y] be the set of finite partial functions from X to Y. Then we have the isomorphism $N$ ~=~ [$N$ > $N$], that is, every rote k : $N$ is uniquely analyzed as a finite partial function k : $N$ > $N$, and every such function gets a unique gödel number in $N$. Notice the polymorphism of types. 1 = {} = the empty function 2 = (p_1)^1 = {<1, 1>} 3 = (p_2)^1 = {<2, 1>} 4 = (p_1)^2 = {<1, 2>} 5 = (p_3)^1 = {<3, 1>} 6 = (p_1)^1 (p_2)^1 = {<1, 1>, <2, 1>} As rooted trees, these look like this: oo  oo oo oo oo    / oo oo oo oo oo oo     \ / @ @ @ @ @ @ 1 2 3 4 5 6 Anyway, I think this is right, but I've learned not to trust stuff like this until several other people have a chance to go over it.
Note 15
More Residual Questions ... JA: so the question is, what is a natural way of partial ordering this species? then, can we specify enough information to make it a total linear ordering? AK: I delve into this later in autumn, after I have read more about posets and lattices, and also when I have grokked your rotes & riffs (has to find the folder where I put all your mails I printed out then ...) AK: Anyways, I doubt in the following: JA: my initial interest in riffs and rotes came out of a search for "autodocumenting data structures" and "selftagging molecules", plus links between additive and multiplicative number theory that I associated with the question: "how much of the ordering of the integers is purely combinatorial", whatever that means. so, yes, some flavor of generalized primes comes into play  if you ask: is there a "purely combinatorical" reason that (p_2)^1 < (p_1)^2? AK: that you could find with such methods a total order that would correspond with our usual linear order 1 < 2 < 3 < 4 < 5 < 6 < ... because it's based on the additive property of integers, and these N <> various trees schemes are based on the multiplicative properties (more or even less). (And I'm afraid that the connection with primes and such is just a red herring ...)