TL;DR
This paper explores multiple methods to enumerate positive rationals, introduces new constructions, and generalizes these using circle packings, revealing fundamental limitations and intriguing sequence properties.
Contribution
It presents several novel and existing enumeration techniques for rationals, extends them via circle packings, and analyzes their mathematical properties and limitations.
Findings
Three enumeration methods are proven to be the only possibilities using circle packings.
The paper introduces new constructions involving negative continued fractions and Chebyshev polynomials.
Sequences derived from these enumerations exhibit remarkable similarities and differences.
Abstract
In 1999, Neil Calkin and Herbert Wilf wrote "Recounting the rationals" which gave an explicit bijection between the positive integers and the positive rationals. We find several different (some new) ways to construct this enumeration and thus create pointers for generalizing. Next, we use circle packings to generalize and find two other enumerations. Surprisingly, the three enumerations are all that are possible by using this technique. The proofs involve, among other things, "negative" continued fractions, Chebyshev polynomials, Euler's totient function, and generalizations of Stern's diatomic sequence. Finally we look at some of the remarkable similarities -- and differences -- of these sequences.
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Re3counting the Rationals
Sam Northshield
Department of Mathematics
SUNY-Plattsburgh
Plattsburgh, NY 12901
Abstract.
In 1999, Neil Calkin and Herbert Wilf wrote “Recounting the rationals” which gave an explicit bijection between the positive integers and the positive rationals. We find several different (some new) ways to construct this enumeration and thus create pointers for generalizing. Next, we use circle packings to generalize and find two other enumerations. Surprisingly, the three enumerations are all that are possible by using this technique. The proofs involve, among other things, “negative” continued fractions, Chebyshev polynomials, Euler’s totient function, and generalizations of Stern’s diatomic sequence. Finally we look at some of the remarkable similarities – and differences – of these sequences.
1. Introduction
In 1999, Neil Calkin and Herbert Wilf wrote their charming “Recounting the rationals” [1] which gave an explicit listing of the positive rationals beginning thus:
[TABLE]
Throughout this paper, we label this sequence .
There are many ways to arrive at this sequence. In Section 2, we present several. In particular, we recall the Calkin-Wilf tree which arranges the positive rationals on the vertices of a rooted binary tree. To show that all positive rationals appear on the tree, we introduce the “Tree Theorem”, a useful tool for enumeration that we also use later in Section 3. Another natural setting for is in terms of Stern’s diatomic array and in terms of Stern’s diatomic sequence. We are quickly led to a “semi-recursive” formula
[TABLE]
where is the usual valuation function that counts the number of times 2 divides .
From this, we find a new source for in terms of a greedy algorithm: for a given list of rationals , append defined as the least not already on the list. Starting with , iterates of this algorithm gives .
This leads to “minus continued fractions”: for
[TABLE]
[TABLE]
and also
[TABLE]
Finally, since Stern’s sequence has closed formulas – both a Binet formula and a formula as a sum across Pascal’s triangle mod 2 [12] – we can create closed formulas for .
Through all these ways of constructing we are led to many possible directions to generalize. Our initial choice actually goes further afield to varieties of circle packings. In Section 3, we look at the family of Ford circles and how they are parameterized by the rationals. It turns out that Stern’s sequence appears here. More general circle packings are introduced and, from them, we construct analogues of Stern’s sequence. From these, we construct two other sequences of the positive rationals that begin:
[TABLE]
and
[TABLE]
Surprisingly, the three enumerations are all that are possible by using the “circle” technique – all others necessarily contain irrational numbers as well. A fourth is introduced as well; here we get rationals in (we conjecture that we get all of them).
In Section 4, we look at some of the remarkable similarities – and differences – of these sequences. In particular, semi-recursive formulas, tree inducing functions, greedy algorithms, and generating functions are all studied. Further, we look at the “degree” of each sequence (the critical exponent for which is finite).
Ponton [16] has recently found these three sequences as well as some corresponding trees that generate them. His paper and this one are still quite complementary.
2. The basic enumeration, and several ways to get it
In this section, we consider the enumeration of the positive rationals
[TABLE]
The reader is invited to find a pattern for generating this sequence before reading further.
Our first way of constructing this sequence is by arranging the positive rationals on a tree and then reading off the entries.
2.1. Tree Theorem
A function from a countable set to itself can be thought of as inducing a digraph, with vertex set , where one assigns to each vertex a directed edge from that vertex to its image under . It is generally possible to get loops (corresponding to fixed points of ) and cycles. The following theorem gives a condition that guarantees that the digraph is actually a union of disjoint rooted trees.
Theorem 2.1** (Tree Theorem).**
If is a countable set, with set of fixed points , and such that, for all ,
[TABLE]
then canonically arranges the elements of on the vertices of a collection of rooted directed trees (with set of respective roots ).
Proof.
On the vertex set , let be the set of all directed edges where ranges over all elements of and define to be the directed graph .
Let denote the th iterate of . For any , the sequence is a decreasing sequence of positive integers but, by the well-ordering of , cannot be strictly decreasing. Hence, there exists (a unique) and so that . Let denote the minimum such . It follows that there is a path of length from to . Since is a function (so the out-degree of every vertex is at most 1), that path is unique. Hence, the connected component of containing is a directed tree with root . The result follows. ∎
This theorem is a discrete version of the “contraction mapping theorem” which states that, no matter the starting point, the iterates of any strict contraction on a set must converge to a fixed point.
2.2. Calkin-Wilf Tree
We introduce a tree that first appeared in “Recounting the rationals” [1] by Calkin and Wilf. Our approach to showing it contains all the positive rationals will be by the Tree Theorem.
Consider and . It is easy to see that for , and so, by the Tree Theorem, every positive rational appears exactly once on the tree.
The map may be inverted in order to get a procedure for actually constructing the tree:
[TABLE]
Reading this tree like a book, but right to left, gives the enumeration of the positive rationals:
[TABLE]
2.3. Stern’s diatomic array and sequence
Probably the clearest setting for understanding the sequence is Stern’s diatomic array, sometimes thought of as “Pascal’s triangle with memory”, which begins thus:
[TABLE]
It is defined recursively as follows: Start with row 1 1. Then, given the th row, define the next one by copying the numbers on the th row but inserting, in each gap, the sum of the two numbers above.
The numbers in the diatomic array, read like a book (but deleting the right-most column of 1s), form what is known as Stern’s diatomic sequence which begins:
[TABLE]
It is easy to see that it is defined recursively by
[TABLE]
Every pair of consecutive numbers in the diatomic array corresponds, via , to the rationals appearing on the Calkin-Wilf tree and so
[TABLE]
See [12] and its references, and [17], sequence A002478, for information about this exceptional, and exceptionally well studied, sequence.
2.4. A semi-recursive formula
A “semi-recursive” formula for is a formula of the form where is some fixed sequence and is some function.
Let
[TABLE]
be the number of times divides .
Since
[TABLE]
and
[TABLE]
it follows by induction on that
[TABLE]
From this and equation (2), it follows that
[TABLE]
2.5. A recursive formula
Note that and so . Further, if , then . By induction,
[TABLE]
Substituting this into equation (3), we find the recursive formula
[TABLE]
which can be rewritten as
[TABLE]
where denotes the fractional part of . This remarkable result is equivalent to one first noticed by Moshe Newman (see Ponton [16]).
Finally, in terms of Stern’s sequence, we may rewrite this as
[TABLE]
2.6. A greedy algorithm
Define a list of numbers inductively as follows: given a list , append , defined as the smallest number of the form () not already on the list. Starting with the list , we get
[TABLE]
To see that this sequence is the same as , note that by equations (1) and (2), for all ,
[TABLE]
Hence, for every odd , and .
Let be the smallest number for which . By equation (4), for some integer and, for some ,
[TABLE]
Hence
[TABLE]
is already on the list and thus .
By induction, the greedy algorithm gives the sequence .
2.7. Minus continued fractions
Keeping with the nomenclature of Katok [8], we call expressions of the following form “minus continued fractions”:
[TABLE]
By equation (3), the positive rationals
[TABLE]
can be written in terms of minus continued fractions:
[TABLE]
with the sequence in plain sight.
Minus continued fractions are not unique and we have an alternative formulation. Given a formal list , define and . Starting with the empty list , define
[TABLE]
We get the sequence of lists
[TABLE]
When turned into minus continued fractions:
[TABLE]
we get our enumeration again (exercise for the reader).
2.8. Closed formulas
It is easy to see that, by equation (1) (see [12], section 3), the generating function of satisfies
[TABLE]
Note: This implies that counts the number of “hyperbinary representations” of (i.e., the number of ways to represent as a sum of powers of two with no power appearing more than twice).
This leads to a closed formula: letting be 0 or 1 according to whether , share non-zero digits in their respective binary expansions or not,
[TABLE]
By a well-known theorem of Kummer ( [6], [9] ) stating that is the number of carries when adding and base ,
[TABLE]
and we have the analogous formulas: the first is the representation of Fibonacci numbers as diagonal sums in Pascal’s triangle, the second, due to Carlitz [2],
[TABLE]
Here is a further similarity with Fibonacci numbers: Recalling Binet’s formula
[TABLE]
we have
[TABLE]
where is a primitive sixth root of unity and is the number of ones in the binary expansion of (see [12], Prop. 4.4).
Therefore we have two closed formulas for :
[TABLE]
and
[TABLE]
2.9. Ways to view
In conclusion, we summarize some of the places that the enumeration
[TABLE]
occurs:
- •
a tree defined by a contraction on ,
- •
where is Stern’s diatomic sequence,
- •
a semi-recursive formula ,
- •
a recursive formula
- •
a greedy algorithm,
- •
minus continued fractions (two different ways), and
- •
closed formulas based on those for .
3. Circle packings and some new enumerations
For real , let be the circle of center and radius . Every such circle is thus above and tangent to the horizontal axis. By the Pythagorean theorem, and are tangent to each other – we write – if and only if .
A Möbius transformation is a function of the form
[TABLE]
where the defining matrix is non-singular. Möbius transformations preserve circles and tangencies. Further, it is not hard to see ([13], Lemma 6) that when ,
[TABLE]
3.1. Ford circles
Consider the array of circles
[TABLE]
Given circles and we formed a new one, , by componentwise addition. For tangent circles , there is a Möbius transformation taking and to and , respectively, and thus taking to . We are thus led to the iterative procedure
[TABLE]
Starting with our initial configuration, successive iterations give us the “Ford circles” (see [5] and [14]).
A Ford circle is thus a circle of the form where are relatively prime integers. Clearly, any two Ford circles are either tangent to each other or do not touch at all. Further each is tangent to the horizontal axis at a positive rational number and, in fact, for every positive rational number in lowest terms there is a Ford circle .
The locations of tangency for these circles are
and Stern’s sequence makes a new appearance: the locations of tangency of the circles in the th iterate are
Recall, with , the function is easily seen to satisfy
[TABLE]
and so, by the Tree Theorem, the tree induced by on the positive rationals coincides with the tree with values ; i.e., enumerates the positive rationals.
3.2. Another circle packing, diatomic array, and diatomic sequence
In this subsection, and with subsection 3.1 as a guide, the array of circles
[TABLE]
indicates the recursive process
[TABLE]
which leads to a function
[TABLE]
that, by the Tree Theorem, induces an enumeration of the postive rationals.
Here are the details. Guettler and Mallows proposed a new type of Ford circle in [7]. Their form of Ford circles is different from ours since, for a given pair of tangent circles, they add two new tangent circles. To ensure uniqueness, they do that according to the rule that the tangent points of the four circles with each other all lie on some circle.
One can also construct this analogue of the set of Ford circles (with tangents between 0 and ) by, starting with and , recursively appending to every pair of tangent circles , the two other circles and .
The first and second coordinates of these circles satisfy a recursion defined by
[TABLE]
and we get another diatomic array:
[TABLE]
and another diatomic sequence (formed, again, by deleting the right-most column of 1s):
[TABLE]
Its recursive definition is
[TABLE]
As with Stern’s sequence, this sequence satisfies a 3-term recurrence (Th. 18 of [13]), namely
[TABLE]
Hence, for ,
[TABLE]
Theorem 3.1**.**
Every positive rational is of the form for some .
Proof.
Let , , and define by, for all ,
[TABLE]
With, as before, when is in lowest terms, it is easy to verify that for all . By the Tree Theorem, the positive rationals can be arranged in two rooted trees (here we identify with ):
\Tree
[.(1,2) [.(1,4) ] [.(2,3) ] [.(3,2) ] ].(1,2) \Tree[.(1,1) [.(1,3) ] [.(3,4) ] [.(2,1) ] ].(1,1)
By the definition of , we can verify
[TABLE]
and so, for all ,
[TABLE]
Hence every appears exactly once on the pair of trees and thus is an enumeration of the positive rationals. ∎
Corollary 3.2**.**
The iterates of , starting at 2, span the entire set of positive rational numbers.
Note. This result appeared [15] and [10] as a problem and solution in the American Mathematical Monthly. The sequence first appeared in [13] where most of the material in this section also appeared as well.
3.3. Another example
Here we have another array of circles:
[TABLE]
giving rise to the recursive process
[TABLE]
which leads to
[TABLE]
and thus to another enumeration of rationals.
Here are some details; our analogue of Stern’s sequence here is
[TABLE]
Let
[TABLE]
Theorem 3.3**.**
Every positive rational is of the form for some .
Proof.
As above, we can use the Tree Theorem with
[TABLE]
fixed points , and as before.∎
As above, we can find a recursive formula: and
[TABLE]
and we have the semi-recursive formula
[TABLE]
3.4. A limit to generality
The circular arrays are all examples of “necklaces”: the points of tangency are all on a circle. In particular, the iterates of the Möbius transformation
[TABLE]
are periodic for \alpha=1,\sqrt{2},\phi\text{ (the golden ratio \frac{\sqrt{5}+1}{2})}, and (we have necklaces of length 3,4,5, and 6 respectively).
For a real , we can start a chain:
[TABLE]
This is a sequence where is a sequence of polynomials that satisfy a recurrence similar to those of Chebyshev polynomials and, in fact, where are Chebyshev polynomials of the second kind. Furthermore, if is the largest root of for some , then, and only then, the chain terminates with . Such necklaces give rise to an enumeration of the positive rationals via the recursion
[TABLE]
only when is rational and thus must have largest irreducible factor of degree at most 2. It is known [11] that has largest irreducible factor of degree where is the Euler totient function. Starting with , the sequence goes ; the only quadratic or linear cases correspond to . The cases for are spoken for above, leaving the as the last quadratic case.
The three enumerations of rationals we have found are the same as those Ponton [16] found, independently, by using analogues of Calkin-Wilf trees. Ponton further concludes, as did we, that these are the only three possibilities.
3.5. The case
Based on the array
we have the necklace
[TABLE]
which indicates the recursive process
[TABLE]
and the analogue of Stern’s sequence and, for
[TABLE]
The sequence begins
and, letting we have a sequence of elements in that begins:
By the definition of and ,
[TABLE]
and so
[TABLE]
satisfies for all .
Since is a UFD, we may define, as above, where is in lowest terms over . It is then easy to check that
[TABLE]
for all . It is not clear, however, if every element of appears on this list since the range of is not well-ordered – the problem being that and all its integer powers are units in . We thus can only conjecture that the sequence is an enumeration of . Note. A proof of this conjecture has resisted several attempts by the author. Although computer evidence indicates it is true, it is of course possible for a counterexample to exist.
As above, we can find a recursive formula:
[TABLE]
and we also have a semi-recursive formula
[TABLE]
4. Similarities and Differences
In this section, we summarize some of the similarities and differences between our various sequences.
4.1. Trees
The four sequences we have studied each could be arranged on the vertices of trees defined, via the Tree Theorem, by a certain function. Here our functions are:
- •
- •
- •
- •
A method of finding them is as follows. Given , let be the Möbius transformation defined by , and let where . The corresponding is given by
[TABLE]
4.2. Analogues of Stern’s sequence
For , our sequences all have similar 3-term recurrences:
- •
- •
- •
- •
From these, our sequences of rationals are as follows (the last, of course, in ):
- •
- •
- •
- •
4.3. Semi-recursive formulas
The sequences in the text all have semi-recursive formulas involving valuation functions .
Starting with ,
- •
- •
- •
- •
In fact, they are all of the form
[TABLE]
where and 4 respectively. The corresponding sequences for surely share many of the properties of the sequences we have studied.
4.4. Recurrence formulas
Let
[TABLE]
The 3-term recurrences above give:
- •
- •
- •
- •
That is, all are of the form
[TABLE]
for respectively.
4.5. Greedy algorithms
The sequences above all can be obtained by greedy algorithms:
- •
- •
- •
- •
where, in each case, is the least positive integer making the elements in the list distinct.
4.6. Degrees
The degree of a sequence is a real number such that for all ,
[TABLE]
The term comes from the fact that, for a polynomial of degree , the sequence has degree . For a general sequence, this number need not exist. However, if it does exist, then it is unique.
- •
has degree . This result follows from a stronger result by Coons and Tyler [3]. It is also fairly easy to prove from basic properties of Stern’s sequence [12] .
- •
has degree . This result follows from a stronger result by Coons [4]. It is also fairly easy to prove from basic properties of that can be found in [13].
- •
has degree . This has been proved by the author using methods similar to those used in the previous cases (unpublished).
- •
seems to have degree . This result seems to follow by the same methods as above. The result is surprising since one might expect the value to between those for and (i.e., between 0.802 and 0.819).
4.7. Generating functions
The generating functions for these sequences each have product formulas easily deduced from the following expressions. By observation, the polynomials involved have each of their zeros equal to some root of unity (we call them “primary roots”).
- •
primary roots , .
- •
primary roots , .
- •
primary roots , .
- •
primary roots , .
We see that the sets of primary roots are, for respectively,
[TABLE]
4.8. Closed formulas
Letting be 1 or 0 according as the -ary expansions of share no non-zero digits, recall by equation (10)
[TABLE]
This is equivalent to the fact that diagonal sums across Pascal’s triangle mod 2 give (where the unmodded sums give ).
By [13], Th. 10
[TABLE]
and it is not hard to see that
[TABLE]
Recall Binet’s formula:
[TABLE]
where are zeros of and .
The sequences and have Binet type formulas: by equation (11),
[TABLE]
where are zeros of and is the number of ones in the binary expansion of .
By [13], Th. 12
[TABLE]
where are zeros of and is the number of ones in the ternary expansion of .
The proofs are from the formulas for the generating functions; it is not clear if and have similar formulas.
4.9. Singular functions
As noted in [12], is Conway’s box function with , Minkowski’s question mark function defined by
[TABLE]
As noted in [13], has a singular inverse function satisfying
[TABLE]
Based on these results, it seems likely that has a singular inverse function satisfying
[TABLE]
and that has a singular inverse function satisfying
[TABLE]
4.10. Minus continued Fractions
“Minus continued fractions” are of the form
[TABLE]
where .
Every positive rational is a minus continued fraction of the form
[TABLE]
where .
Every positive rational is a minus continued fraction of the form
[TABLE]
where and is 1 or 2, depending on the parity of .
Every positive rational is a minus continued fraction of the form
[TABLE]
where and is 1 or 3, depending on the parity of .
As with equation (8), we can also express our rational sequences in terms of minus continued fractions defined by lists. Let be the empty list and, for a given , let
[TABLE]
For our given and its corresponding lists , let and define a sequence in terms of minus continued fractions
[TABLE]
These sequences agree with when respectively.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] N. Calkin and H. Wilf, Recounting the rationals , Amer. Math. Monthly 107 (2000), no. 4, 360–363.
- 2[2] L. Carlitz, Single variable Bell polynomials , Collect. Math 14 (1960) 13–25.
- 3[3] M. Coons and J. Tyler, The maximal order of Stern’s diatomic sequence , Mosc. J. Comb. Number Theory 4 (2014), no. 3, 3–14.
- 4[4] M. Coons, Proof of Northshield’s conjecture concerning an analogue of Stern’s sequence for ℤ [ 2 ] ℤ delimited-[] 2 {\mathbb{Z}}[\sqrt{2}] , Australas. J. Combin. 71 (2018) 113–120.
- 5[5] L. R. Ford, Fractions , Amer. Math. Monthly, 45 (1938) 586–601.
- 6[6] A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. Organic mathematics (Burnaby, BC, 1995), 253–276, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997.
- 7[7] G. Guettler and C. Mallows, A generalization of Apollonian packing of circles , J. Comb. 1 (2010), no. 1, 1–27.
- 8[8] S. Katok, Continued fractions, hyperbolic geometry and quadratic forms , course notes for MATH 497A, REU Program, Summer 2001 (revised November, 2001), Pennsylvania State University (https://math.psu.edu/katok_s/pub/reu-book.pdf).