# Compositions into Powers of *b*: Asymptotic Enumeration and Parameters

## Abstract

For a fixed integer base \(b\ge 2\), we consider the number of compositions of 1 into a given number of powers of *b* and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of *b*. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated.

## Keywords

Compositions Powers of 2 Infinite transfer matrices Asymptotic enumeration## 1 Introduction

*Partitions*of integers into powers of 2, i.e., representations of the form

*Mahler partitions*(see [2, 12, 16, 19]).

The number of such partitions exhibits interesting periodic fluctuations. The situation changes, however, when *compositions* into powers of 2 are considered, i.e., when the summands are arranged in an order. In other words, we consider representations of the form (1.1) without further restrictions on the exponents \(a_1,\,a_2\), ..., \(a_n\) other than being nonnegative.

*q*th root of unity and \(\tau \) is the order of 2 modulo

*q*(see [17]), Molteni [18] recently studied the maximum number of representations a positive integer can have as an ordered sum of

*n*powers of 2. More generally, fix an integer \(b\ge 2\), let

*n*powers of

*b*, and let \({{\mathcal {W}_b}({s,n})}\) be the maximum of \({{{\mathcal {U}}_b}({\ell ,n})}\) over all positive integers \(\ell \) with

*b*-ary sum of digits equal to

*s*. It was shown in [17] that

*b*. So knowledge of \({{\mathcal {W}_b}({1,n})}\) is the key to understanding \({{\mathcal {W}_b}({s,n})}\) for arbitrary

*s*.

*n*powers of 2 (

*n*fixed) is the same as the number of representations of \(\ell \) as a sum of

*n*powers of 2 for all integers

*h*if negative exponents are allowed as well (simply multiply/divide everything by \(2^h\)). Therefore, \({{\mathcal {W}_2}({1,n})}\) is also the number of solutions to the Diophantine equation

*compositions*of 1 into

*n*powers of 2. This sequence starts with

*K*. It was mainly based on an asymptotic formula for the number of

*partitions*of 1 into powers of 2, which was derived independently in different contexts, cf. [1, 7, 13] (or see the recent paper of Elsholtz et al. [5] for a detailed survey). This bound was further improved by Molteni, who gave the inequalities

### **Theorem 1**

Molteni’s argument is quite sophisticated and involves the study of the spectral radii of certain matrices. The aim of this paper will be to present a different approach to the asymptotics of \(\mathcal {W}_2(1,n)\) (and more generally, \({{\mathcal {W}_2}({s,n})}\)) by means of generating functions that allows us to obtain more precise information. Our main theorem reads as follows.

### **Theorem 2**

*s*, there exists a polynomial \({{P_s}({n})}\) with leading term

*m*. We write \(q_b(m)\) for the number of solutions [

*n*-tuples of nonnegative integers satisfying (1.5)] in this case. Note that \(q_b(m)\) is also the maximum number of representations of an arbitrary power of

*b*as an ordered sum of \(n = (b-1)m+1\) powers of

*b*. We have the following general asymptotic formula. Note that we will usually suppress the dependence on

*b*for ease of notation.

### **Theorem 3**

*b*, which is also the maximum number \({{\mathcal {W}_b}({1,n})}\) of representations of a power of

*b*as an ordered sum of

*n*powers of

*b*, satisfies

*b*-ary sum of digits

*s*as an ordered sum of \(n = (b-1)m+s\) powers of

*b*is asymptotically given by

*partition*of 1 into powers of 2 (or generally

*b*) with a weight that essentially gives the number of ways it can be permuted to a composition, and to apply the recursive approach that was used to count partitions of 1: if \(p_2(n)\) denotes the number of such partitions into

*n*summands, then the remarkable generating function identity

*b*, see the recent paper of Elsholtz et al. [5]. In our case, we do not succeed to obtain a similarly explicit formula for the generating function, but we can write it as the quotient of two determinants of infinite matrices and infer analytic information from it. The paper is organised as follows: we first describe the combinatorial argument that yields the generating function, a priori only within the ring of formal power series. We then study the expression obtained for the generating function in more detail to show that it can actually be written as the quotient of two entire functions. The rest of the proof is a straightforward application of residue calculus (using the classical Flajolet–Odlyzko singularity analysis [6]).

*n*powers of

*b*. Equivalently, it is the largest coefficient in the power series expansion of

*n*for a suitably chosen constant

*C*. The gap between the two estimates is already very small; we improve this a little further by providing the constant of exponential growth as well as a precise asymptotic formula.

### **Theorem 4**

Again, Theorem 4 holds for arbitrary integer bases \(b\ge 2\) for some constants \(\nu =\nu _b\) and \(\lambda =\lambda _b\) (it will be explained precisely how they are obtained). This is formulated as Theorem 5 in Sect. 7.

The final section contains the analysis of some parameters. We study the exponent of the largest denominator and the number of distinct parts in a composition of 1. In both cases a central limit theorem is shown; mean and variance are linear in the number of summands, cf. Theorems 6 and 7.

## 2 The Recursive Approach

For our purposes, it will be most convenient to work in the setting of compositions of 1, i.e., we are interested in the number \(q_b(m)\) of (ordered) solutions to the Diophantine equation (1.5), where \(n = (b-1)m+1\), as explained in the introduction. Our first goal is to derive a recursion for \(q_b(m)\) and some related quantities, which leads to a system of functional equations for the associated generating functions.

*n*-tuple as a “partition” (although technically the \(k_i\) are only the exponents in a partition). We denote by \(\mathsf {c}(\mathbf {k})\) the number of ways to turn it into a composition. If \(w_0\) is the number of zeros, \(w_1\) the number of ones, etc. in \(\mathbf {k}\), then we clearly have

*weight*of a partition \(\mathbf {k}\), denoted by \(\mathsf {w}(\mathbf {k})\), is now simply defined as

Our next step involves an important observation that is also used to obtain the generating function (1.6). Consider an element \(\mathbf {k}\) of \(\mathcal {P}_m\), and let *r* be the number of times the greatest element \(k_1\) occurs (i.e., \(k_1 = k_2 = \cdots = k_r > k_{r+1}\)). This number must be divisible by *b* (as can be seen by multiplying (1.5) by \(b^{k_1}\)) unless \(\mathbf {k}\) is the trivial partition, so we can replace the *r* fractions with denominator \(b^{k_1}\) by *r* / *b* fractions with denominator \(b^{k_1-1}\).

*r*times, we can replace

*s*of these fractions (\(1 \le s \le r\)) by

*bs*fractions with denominator \(b^{k_1+1}\). This recursive construction can be illustrated nicely by a tree structure as in Fig. 1 for the case \(b=2\). Each partition corresponds to a so-called canonical tree (see [5]), and vice versa. Note that if \(\mathbf {k} \in \mathcal {P}_m\), then the resulting partition \(\mathbf {k'}\) lies in \(\mathcal {P}_{m+s}\), and we clearly have

*r*times), and let \(\mathcal {C}_{m,r}\) be the set of compositions obtained by permuting the terms of an element of \(\mathcal {P}_{m,r}\). We define a generating function by

*r*not divisible by

*b*. Moreover, for all \(s \ge 1\) the recursive relation described above and in particular (2.1) yield

*b*) is given by

*T*given by (2.5) defines an entire function. This is proven in Sect. 3. The same is true (by the same argument) for

## 3 Bounds and Entireness

Note that \({{S}({x})}\) is the determinant of a matrix, which is obtained by replacing the first row of \(I-\mathbf {M}(x)\) by \(\mathbf {1}\).

We find bounds for the coefficients \(t_n\) and \(s_n\), which will be needed for numerical calculations with guaranteed error estimates as well. Further, those bounds will tell us that the two functions \({{T}({x})}\) and \({{S}({x})}\) are entire.

### **Lemma 3.1**

*T*defines an entire function. The same is true for the formal power series

*S*. More precisely, we have

### *Proof*

*x*, and note that

*a*and \(f(x) = x(\log x - 1)\) is a convex function, we have

*n*.

*q*(

*n*) of

*n*into distinct parts is asymptotically equal to \(\exp \big ( \pi \sqrt{n/3} + {{O}({\log n})} \big )\). In Robbins’s paper [20] we can find the explicit upper bound

^{1}

*h*! permutations \(\sigma \) that contribute, which can be bounded by means of Stirling’s formula (using also \(h \le \sqrt{2n}\) again). This gives

*T*is bounded (in absolute values) by

*g*(

*n*). A possible explicit bound (relevant for our numerical calculations, see Sect. 6) is

*S*. There, we split up into the summands where we have \(i_1=1\) and all other summands. For the second part (the summands with \(i_1>1\)), the terms are the same as in the determinant that defines

*T*, so it is bounded by the same expression. Each of the summands with \(i_1=1\) equals a summand of \(\det (I-\mathbf {M}(x))\) multiplied by the factor

Lemma 3.1 immediately yields a simple estimate for the tails of the power series *S* and *T*.

### **Lemma 3.2**

*c*and \({{g}({n})}\) be as in Lemma 3.1. Set

*T*. For the tails of the determinant

*S*, we have the analogous inequality

### *Proof*

## 4 Analyzing the Generating Function

Infinite systems of functional equations appear quite frequently in the analysis of combinatorial problems, see for example the recent work of Drmota, Gittenberger and Morgenbesser [3]. Alas, their very general theorems are not applicable to our situation as the infinite matrix \(\mathbf {M}\) does not represent an \(\ell _p\)-operator (one of their main requirements), due to the fact that its entries increase (and tend to \(\infty \)) along rows. However, we can adapt some of their ideas to our setting.

The main result of this section is the following lemma.

### **Lemma 4.1**

For every \(b \ge 2\), the generating function \({{Q}({x})}\) has a simple pole at a positive real point \(\rho _b\) and no other poles with modulus \(< \rho _b + \epsilon _b\) for some \(\epsilon _b > 0\).

### *Proof of Lemma 4.1*

*b*copies of \(b^{-m}\). Since there are \(\frac{((b-1)m+1)!}{((b-1)!)^{m-1}b!}\) possible ways to arrange them in an order, we know that

*Q*(

*x*) as

*Q*(

*x*) and \(Q_b(x)\) are), and its coefficients in the power series expansion are all positive (since those of \(\mathbf {1},\,\mathbf {c}\) and \(\overline{\mathbf {M}}\) are and \((\mathbf {I} - \overline{\mathbf {M}})^{-1}\) can be expanded in a geometric series). In view of the inequality (4.1), it remains bounded as \(x \rightarrow \rho _b^{-}\), so its radius of convergence must be greater than \(\rho _b\) (meaning that it is analytic in a disk of larger radius).

## 5 Getting the Asymptotics

*b*, except for the fact that different constants occur.

Truncated decimal values for the constants of Theorem 3

| \(\alpha \) | \(\gamma \) |
---|---|---|

2 | 0.296372 | 1.19268 |

3 | 0.279852 | 0.534502 |

4 | 0.236824 | 0.170268 |

5 | 0.196844 | 0.0419317 |

6 | 0.165917 | 0.00834837 |

7 | 0.142679 | 0.00138959 |

8 | 0.1249575 | 0.000198440 |

### *Proof of Theorem 3*

*Q*(

*x*) has exactly one pole \(\rho \) (which is a simple pole) inside some disk with radius \(\rho +\epsilon ,\,\epsilon >0\), around 0. Thus we can directly apply singularity analysis [6] in the meromorphic setting (cf. Theorem IV.10 of [8]) to obtain

*s*), we use the relation

## 6 Reliable Numerical Calculations

We want to calculate the constants obtained in the previous sections in a reliable way. The current section is devoted to this task. Our main tool will be interval arithmetic, which is performed by the computer algebra system Sage [21].

For the calculations, we need bounds for the tails of our infinite sums. We start with the following two remarks, which improve the bounds found in Sect. 3.

### *Remark 6.1*

The bounds of Lemma 3.1 for the determinant (2.5) can be tightened: for an explicit *n*, we can calculate \({{g}({n})}\) more precisely by using the number of partitions of *n* into distinct parts (and not a bound for that number) and similarly by using the factorial directly instead of Stirling’s formula.

*n*th coefficient of \(\det (I-\mathbf {M}(x))\) is given by

*n*or not. However, for a specific

*n*, one can calculate this bound, and it is much smaller than the general bounds obtained earlier. For example, for \(b=2\), we have \( \left| {t_{60}}\right| \le 5.96\cdot 10^{-14}\) with this method, whereas Lemma 3.1 would give the bound 0.00014.

### *Remark 6.2*

### **Lemma 6.3**

For \(b=2\), the function \({{T}({x})}\) has exactly one zero with \( \left| {x}\right| < \frac{3}{2}\). This simple zero lies at \(x_0 = 0.83845184342\dots \).

### *Remark 6.4*

Note that \(1/x_0 = \gamma = 1.192674341213\dots \), which is indeed the constant found by Molteni in [18].

### *Proof of Lemma 6.3*

Denote the polynomials consisting of the first *N* terms of \({{T}({x})}\) by \({{T_N}({x})}\). We have \( \left| {{{T}({x})} - {{T_{60}}({x})}}\right| \le B_{T_{60}}\) with \(B_{T_{60}} = 1.17\cdot 10^{-13}\), see Lemma 3.2 and Remark 6.2. On the other hand, we have \( \left| {{{T_{60}}({x})}}\right| > 0.062\) for \( \left| {x}\right| = \frac{3}{2}\) (the minimum is attained on the positive real axis) by using a bisection method together with interval arithmetic (in Sage [21]). Therefore, the functions \({{T}({x})}\) and \({{T_{60}}({x})}\) have the same number of zeros inside a disk \( \left| {x}\right|<\frac{3}{2}\) by Rouché’s theorem (\(0.062>B_{T_{60}}\)). This number equals one, since there is only one zero, a simple zero, of \({{T_{60}}({x})}\) with absolute value smaller than \(\frac{3}{2}\).

To find the exact position of that zero consider \({{T_{60}}({x})} + B_{T_{60}}I\) with the interval \(I = [-1,1]\). Again, using a bisection method (starting with \(\frac{3}{2} I\)) plus interval arithmetic, we find an interval that contains \(x_0\). From this, we can extract correct digits of \(x_0\). \(\square \)

From this result, which gives the numerical value of the dominant singularity, we can compute all the constants in Theorem 2. Numerical values of the constants in the general case of Theorem 3 are obtained analogously. The values of those constants for the first few *b* can be found in Table 1. The following remark gives some further details of the computation.

### *Remark 6.5*

As mentioned, to obtain reliable numerical values of all the constants involved in the statement of our theorems, we use the bounds obtained in Sect. 3 together with interval arithmetic.

Let \(b=2\) and denote, as above, the polynomials consisting of the first *N* terms of \({{S}({x})}\) and \({{T}({x})}\), by \({{S_N}({x})}\) and \({{T_N}({x})}\) respectively. By the methods of Lemmas 3.1 and 3.2 and Remarks 6.1 and 6.2 we get, for instance, that \( \left| {{{T'}({x})} - {{T'_{60}}({x})}}\right| \le B_{T'_{60}}\) with \(B_{T'_{60}} = 8.397\cdot 10^{-12}\). We also have \( \left| {{{S}({x})} - {{S_{60}}({x})}}\right| \le B_{S_{60}}\) with \(B_{S_{60}} = 1.848\cdot 10^{-13}\) for the function in the numerator of \({{Q}({x})}\). We plug \(x_0\) into the approximations \(S_{60}\) and \(T'_{60}\) and use these bounds to obtain precise values (with guaranteed error estimates) for all the constants that occur in our formula.

### *Remark 6.6*

## 7 Maximum Number of Representations

*n*powers of

*b*.

*W*has real, nonnegative coefficients) of the equation \({{W}({\theta })}=1\), and set \(\nu = \nu _b =1/\theta _b\) (as usual, constants depend on

*b*, but we will leave out the subscript

*b*). We prove the following theorem, which is a generalized version of Theorem 4.

### **Theorem 5**

*N*.

Values (numerical approximations) for the constants of Theorem 5

| \(\lambda \) | \(\theta \) | \(\nu =1/\theta \) | \(\mu \) | \(\sigma ^2\) |
---|---|---|---|---|---|

2 | 0.27693430 | 0.57071698 | 1.75218196 | 0.44867215 | 0.41775807 |

3 | 0.70656285 | 0.84340237 | 1.18567368 | 0.66924459 | 0.57114748 |

4 | 1.70314663 | 0.95872521 | 1.04305174 | 0.87318716 | 0.37650717 |

5 | 4.20099030 | 0.99167231 | 1.00839763 | 0.96645454 | 0.13477198 |

6 | 10.61691472 | 0.99861115 | 1.00139078 | 0.99304650 | 0.03480989 |

7 | 28.28286119 | 0.99980159 | 1.00019845 | 0.99880929 | 0.00714564 |

8 | 80.09108610 | 0.99997520 | 1.00002480 | 0.99982638 | 0.00121534 |

We start with the upper bound (7.1) of Theorem 5, which is done in the following lemma.

### **Lemma 7.1**

### *Proof*

*s*and

*n*, and taking the maximum over all \(s \ge 1\) yields

It remains to prove the asymptotic formula for \({{M}({n})}\). We first gather some properties of the solution \(x={{\theta }({u})}\) of the functional equation \({{W}({x})}=1/u\).

### **Lemma 7.2**

For \(u\in \mathbbm {C}\) with \( \left| {u}\right|\le 1\) and \( \left| {{\text {Arg}}\, u}\right| \le \frac{\pi }{b-1}\), each root *x* of \({{W}({x})}= 1/u\) satisfies the inequality \( \left| {x}\right| \ge \theta \), where equality holds only if \(x = \theta \) and \(u =1\).

### *Proof*

*u*be as stated in the lemma. By the nonnegativity of the coefficients of

*W*and the triangle inequality, we have

*W*is increasing on the positive real line. It remains to determine when equality holds, so we assume in the following that \( \left| {x}\right| = \theta \).

*V*are indeed positive, the power series

*V*is aperiodic.

^{2}Therefore, the inequality \( \left| { {{V}\big ({x^{b-1}}\big )} }\right| \le {{V}\big ({ \Big | {x^{b-1}}\Big |}\big )}\) is strict, i.e., we have \( \left| { {{V}\big ({x^{b-1}}\big )} }\right| < {{V}\big ({ \Big | {x^{b-1}}\Big |}\big )}\) (which would yield a contradiction to the assumption that \( \left| {x}\right|=\theta \)) unless \(x^{b-1}\) is real and positive, which means that \(x^{b-1} = \theta ^{b-1}\). When this is the case, we have

*x*is itself real and positive, which implies that \(x = \theta \) and \(u = 1\). \(\square \)

The following lemma tells us that the single dominant root of \({{W}({x})}=1\) is the simple zero \(\theta \).

### **Lemma 7.3**

There exists exactly one root of \({{W}({x})}=1\) with \( \left| {x}\right|\le \theta \), namely \(\theta \). Further, \(\theta \) is a simple root, and there exists an \(\epsilon >0\) such that \(\theta \) is the only root of \({{W}({x})}=1\) with absolute value less than \(\theta +\epsilon \).

### *Proof*

By Lemma 7.2 with \(u=1\), the positive real \(\theta \) is the unique root of \({{W}({x})}=1\) with minimal absolute value. This proves the first part of the lemma.

*r*of

*W*is at least \(1/\gamma ^{1/(b-1)}>\theta \), and so

*W*is holomorphic inside a disk that contains \(\theta \). Since zeros of holomorphic functions do not accumulate, the existence of a suitable \(\epsilon >0\) as desired follows.

The root \(\theta \) is simple, since *W*(*x*) is strictly increasing on (0, *r*). \(\square \)

*s*fulfil a local limit law (as

*n*tends to \(\infty \)). The maximum is then attained close to the mean.

### *Proof of Theorem 5*

Now we check that all requirements for applying the quasi-power theorem are fulfilled. By Lemma 7.3, the function \({{G}({x,1})}\) has a dominant simple pole at \(x=\theta \) and no other singularities with absolute values smaller than \(\theta +\epsilon \). The denominator \(1-u {{W}({x})}\) is analytic and not degenerated at \((x,u)=(\theta ,1)\); the latter since its derivative with respect to *x* is \({{W'}({\theta })}\ne 0\) (\(\theta \) is a simple root of *F*) and its derivative with respect to *u* is \(-{{W}({\theta })}=-1\ne 0\).

Thus the function \({{\theta }({u})}\) which gives the solution to the equation \(W(\theta (u)) = 1/u\) with smallest modulus has the following properties: it is analytic at \(u=1\), it fulfils \(\theta (1)=\theta \), and for some \(\epsilon > 0\) and *u* in a suitable neighbourhood of 1, there is no \(x \ne \theta (u)\) with \(W(x) = 1/u\) and \(|x| \le \theta + \epsilon \).

*u*in a suitable neighbourhood of 1.

*v*, which satisfies \( \left| {{\text {Arg}}\,v^{1/(b-1)}}\right| \le \frac{\pi }{b-1}\).

Since \({{\theta }({u})} \ne 0\) for *u* in a suitable neighbourhood of 0, the function *B* is analytic at zero, and so is the function *A* (since *W* is analytic in a neighbourhood of \(\theta (1) = \theta \) as well and has a nonzero derivative there). Moreover, we can use the fact that \( \left| {{{\theta }({e^{i\varphi }})} }\right|\) has a unique minimum at \(\varphi =0\) if we assume that \( \left| {\varphi }\right| \le \frac{\pi }{b-1}\) (which follows from Lemma 7.2).

*t*outside of the central region can be treated by standard tail estimates. Mean and variance can be calculated as follows. We have

The value \({{\mathcal {W}_b}({s,n})}/n!\) is maximal with respect to *s* when \(s = \mu n + {{O}({1})}\). Its asymptotic value can then be calculated by (7.3). \(\square \)

## 8 The Largest Denominator and the Number of Distinct Parts

In this last section we analyze some parameters of our compositions of 1. In particular, we will see that the exponent of the largest denominator occurring in a random composition into a given number of powers of *b* and the number of distinct summands are both asymptotically normally distributed and that their means and variances are of linear order.

Let us start with the largest denominator, for which we obtain the following theorem. Note again that we suppress the dependence on *b* in all constants.

### **Theorem 6**

The exponent of the largest denominator in a random composition of 1 into \(m = (b-1)n+1\) powers of *b* is asymptotically normally distributed with mean \(\mu _\ell n + {{O}({1})}\) and variance \(\sigma _\ell ^2 n + {{O}({1})}\).

| \(\mu _\ell \) | \(\sigma _\ell ^2\) | \(\mu _d\) | \(\sigma _d^2\) |
---|---|---|---|---|

2 | 0.81885148 | 2.38703164 | 0.71440975 | 2.13397882 |

3 | 0.93352696 | 0.53468588 | 0.93318787 | 0.53600822 |

4 | 0.97869416 | 0.15390515 | 0.97869416 | 0.15390519 |

5 | 0.99366804 | 0.04335760 | 0.99366804 | 0.04335760 |

6 | 0.99819803 | 0.01180985 | 0.99819803 | 0.01180985 |

7 | 0.99950066 | 0.00315597 | 0.99950066 | 0.00315597 |

8 | 0.99986404 | 0.00083471 | 0.99986404 | 0.00083471 |

### *Sketch of proof of Theorem 6*

*x*for

*y*in a suitable neighbourhood of 1. Thus our bivariate generating function belongs to the meromorphic scheme as described in Section IX.6 of [8], and the asymptotics of mean and variance are obtained by standard tools of singularity analysis. Asymptotic normality follows by Hwang’s quasi-power theorem [11]. \(\square \)

For the number of distinct parts we prove the following result.

### **Theorem 7**

The number of distinct parts in a random composition of 1 into \(m = (b-1)n+1\) parts is asymptotically normally distributed with mean \(\mu _d n + {{O}({1})}\) and variance \(\sigma _d^2 n + {{O}({1})}\).

Approximations of the constants can be found in Table 3. Again we only sketch the proof, since it uses the same ideas.

### *Sketch of proof of Theorem 7*

*y*now marks the number of distinct parts, i.e., we use

## Footnotes

## References

- 1.Boyd, D.W.: The asymptotic number of solutions of a diophantine equation from coding theory. J. Combin. Theory Ser. A
**18**, 210–215 (1975)MathSciNetCrossRefzbMATHGoogle Scholar - 2.de Bruijn, N.G.: On Mahler’s partition problem. Nederl. Akad. Wetensch. Proc.
**51**, 659–669 (1948). (Indagationes Math. 10, 210–220 (1948))MathSciNetzbMATHGoogle Scholar - 3.Drmota, M., Gittenberger, B., Morgenbesser, J.: Infinite systems of functional equations and Gaussian limiting distributions. In: 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA’12), volume AQ of DMTCS Proceedings, pp. 453–478 (2012)Google Scholar
- 4.Eaves, R.E.: A sufficient condition for the convergence of an infinite determinant. SIAM J. Appl. Math.
**18**, 652–657 (1970)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Elsholtz, C., Heuberger, C., Prodinger, H.: The number of Huffman codes, compact trees, and sums of unit fractions. IEEE Trans. Inf. Theory
**59**, 1065–1075 (2013)MathSciNetCrossRefGoogle Scholar - 6.Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math.
**3**, 216–240 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Flajolet, P., Prodinger, H.: Level number sequences for trees. Discrete Math.
**65**(2), 149–156 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
- 9.Giorgilli, A., Molteni, G.: Representation of a 2-power as sum of \(k\) 2-powers: a recursive formula. J. Number Theory
**133**(4), 1251–1261 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Hwang, H.-K.: Large deviations of combinatorial distributions. II. Local limit theorems. Ann. Appl. Probab.
**8**(1), 163–181 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Hwang, H.-K.: On convergence rates in the central limit theorems for combinatorial structures. Eur. J. Combin.
**19**, 329–343 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Knuth, D.E.: An almost linear recurrence. Fibonacci Q.
**4**, 117–128 (1966)MathSciNetzbMATHGoogle Scholar - 13.Komlós, J., Moser, W., Nemetz, T.: On the asymptotic number of prefix codes. Mitt. Math. Sem. Giessen
**165**, 35–48 (1984)MathSciNetzbMATHGoogle Scholar - 14.Krenn, D., Wagner, S.: The number of compositions into powers of \(b\). In: 25th International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA’14), volume BA of DMTCS Proceedings, pp. 241–252 (2014)Google Scholar
- 15.Lehr, S., Shallit, J., Tromp, J.: On the vector space of the automatic reals. Theor. Comput. Sci.
**163**(1–2), 193–210 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Mahler, K.: On a special functional equation. J. Lond. Math. Soc.
**15**, 115–123 (1940)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Molteni, G.: Cancellation in a short exponential sum. J. Number Theory
**130**(9), 2011–2027 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Molteni, G.: Representation of a 2-power as sum of \(k\) 2-powers: the asymptotic behavior. Int. J. Number Theory
**8**(8), 1923–1963 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Pennington, W.B.: On Mahler’s partition problem. Ann. Math.
**2**(57), 531–546 (1953)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Robbins, N.: A simply-obtained upper bound for \(q(n)\). Ann. Univ. Sci. Bp. Sect. Comp.
**27**, 39–43 (2007)MathSciNetzbMATHGoogle Scholar - 21.Stein, W.A., et al.: Sage Mathematics Software (Version 6.3). The Sage Development Team. http://www.sagemath.org (2015)
- 22.The On-Line Encyclopedia of Integer Sequences. http://oeis.org (2015)

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