Developments in the Khintchine-Meinardus probabilistic method for asymptotic enumeration

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of generating functions. This allows us to prove rigorously the asymptotics of Gentile statistics and to study the asymptotics of combinatorial objects with distinct components.


Introduction
Meinardus [10] proved a theorem about the asymptotics of weighted partitions with weights satisfying certain conditions. His result was extended to the combinatorial objects called assemblies and selections in [4] and to Dirichlet generating functions for weights, with multiple singularities in [6]. In this paper, we extend Meinardus' theorem further to a general framework, which encompasses a variety of models in physics and combinatorics, including previous results.
Let f be a generating function of a nonnegative sequence {c n , n ≥ 0, c 0 = 1}: with radius of convergence 1. As an example, consider the number of weighted partitions c n of size n, determined by the generating function identity for some sequence of real numbers b k ≥ 0, k ≥ 1. When b k = 1 for all k ≥ 1, then c n is the number of integer partitions. Meinardus [10] proved a theorem giving the asymptotics of c n under certain assumptions on the sequence {b k }. The generating function in (2) may be expressed as ∞ k=1 S(z k ) b k , where S(z) = (1 − z) −1 . This observation allows the following generalization. Let f in (1) be of the form: with given sequences 0 < a k ≤ 1, b k ≥ 0, k ≥ 1, and a given function S(z). This is a particular case of the class of general multiplicative models, introduced and studied by Vershik ([15]). In the setting (3), in the case of weighted partitions, a combinatorial meaning can be attributed to the parameters a k , b k . Namely, if b k = 1, then a k can be viewed as a properly scaled number of colours for each component of size k, such that given l components of size k, the total number of colourings is a l k . On the other hand, if a k = 1, then given l components of size k, the total number of colourings equals the number of distributions of b k indistinguishable balls among l cells, so that in this model b k has a meaning of a scaled number of types prescribed to a component of size k.
Yakubovich ([17]) derived the limit shapes for models (3) in the case a k = 1, k ≥ 1, under some analytical conditions on S and b k . Note that past versions [4]- [6] of Meinardus' theorem deal with the asymptotics of c n , n → ∞, when a k = 1, k ≥ 1, for three cases of the function S, corresponding to the three classic models of statistical mechanics, which are equivalent to the three aforementioned models in combinatorics. Our objective in this paper is to derive the asymptotics c n , n → ∞, in the general framework (3).
The assumptions above (1) and (3) on the sequence c n imply that S(0) = 1, that S(z) can be expanded in a power series with radius of convergence ≥ 1 and non-negative coefficients d j : with d 0 = 1, and that log S(z) can be expanded as with radius of convergence 1. From (3) and (5) one can express the coefficients Λ k of the power series for the function log f (z), with radius of convergence 1: We define the Dirichlet generating function for the sequence Λ k : which by virtue of (6) admits the following presentation as long as ℜs is large enough so that the double Dirichlet series in (8) converges absolutely. If a k = a, 0 < a ≤ 1 for all k ≥ 1, then D(s) can be factored as where The greater generality of (3) than in previous versions of Meinardus' theorem will allow novel applications. The proof of Theorem 1, stated below, is a substantial modification of the method used in [4,5,6].
We suppose that Λ k and D(s) satisfy conditions (I) − (III), which are modifications of the three original Meinardus' conditions in [10].
Condition (I). The Dirichlet generating function D(s), s = σ + it is analytic in the half-plane σ > ρ r > 0 and it has r ≥ 1 simple poles at positions 0 < ρ 1 < ρ 2 < . . . < ρ r , with positive residues A 1 , A 2 , . . . , A r respectively. It may also happen that D(s) has a simple pole at 0 with residue A 0 . (If D(s) is analytic at 0, we take A 0 = 0). Moreover, there is a constant 0 < C 0 ≤ 1, such that the function D(s), s = σ + it, has a meromorphic continuation to the half-plane H = {s : σ ≥ −C 0 } on which it is analytic except for the above r simple poles. Condition (II). There is a constant C 1 > 0 such that The following property of the parameters a k , b k holds: where Moreover, if l 0 > 1 then for δ n as defined below in (28), for some fixed ǫ > 0 and for large enough n, where Λ k is as defined in (6). In order to state our main result, we need some more notations, which were also used in [6]. Define the finite set where we have set ρ 0 = 0 and let Z + denote the set of nonnegative integers. Let 0 < α 1 < α 2 < . . . < α |Υr| ≤ ρ r + 1 be all ordered numbers forming the setΥ r . Clearly, α 1 = 2(ρ r − ρ r−1 ), if the setΥ r is not empty. We also define the finite set observing that some of the differences ρ r − ρ k , k = 0, . . . , r − 1 may fall into the setΥ r . We let 0 < λ 1 < λ 2 < . . . < λ |Υr| be all ordered numbers forming the set Υ r .
Theorem 1 Suppose conditions (I) − (III) are satisfied. Suppose that c n has ordinary generating function of the form (3), where 0 < a k ≤ 1 and b k ≥ 0, k ≥ 1, that (11) is satisfied for a constant C 2 > 0, and that We then have, as n → ∞, where H, P l ,ĥ l and K s,l are constants. In particular, if r = 1, then K s,l = 0 for all s and l, and γ is Euler's constant.
Example This example shows that (11) is not implied by the other hypotheses of Theorem 1. Let a k = 1 for all k, let b k = k ρ 1 −1 where ρ 1 > 0, and where ζ is the Riemann zeta funtion, and D(s) = D b (s)D ξ,1 (s) has poles at ρ 1 and ρ 2 . Moreover, S(z) = exp ∞ k=1 k ρ 1 −1 z k has radius of convergence 1 and it is easy to check that (15) is satisfied.
Theorem 1 is proven in Section 2. In the remaining two sections, we focus on two novel applications implied by Theorem 1. In Sections 3 and 4 we apply our results to the asymptotic enumeration of Gentile statistics and expansive selections with a k = k −q . The latter generalizes previous results for polynomials over a finite field.

Proof of Theorem 1
As in [4]- [6], the proof of Theorem 1 is based on the Khintchine type representation( [9]) c n = e nδ f n (e −δ )P (U n = n) , n ≥ 1, where δ > 0 is a free parameter, is the truncation of (3), and the U n , n ≥ 1 are integer-valued random variables with characteristic functions defined by Khintchine established (18) for the three basic models of statistical mechanics. For general multiplicative measures (18) was stated in equation (4) of [4]. The first step in the proof is to find the asymptotics of F(δ) := log f (e −δ ), as δ → 0.
By the Laurent expansions at s = 0 of the Gamma function Γ(s) = 1 s − γ + . . . , and the function D(s) = A 0 s + Θ + · · · , the integrand δ −s D(s)Γ(s) may also have a pole at s = 0, which is a simple one with residue Θ, if A 0 = 0, Θ = 0, and is of a second order with residue Applying condition (II) shows that the integral of the integrand δ −s Γ(s)D(s), over the horizontal contour −C 0 ≤ ℜ(s) ≤ ǫ + ρ r , ℑ(s) = t, tends to zero, as t → ∞, for any fixed δ > 0. This gives the claimed formulae (21), where the remainder term M(δ; C 0 ) is the integral taken over the vertical contour In order to prove (ii), one differentiates the logarithm of (21) with respect to δ and estimates the remaining integral in the same way as above.
We will need the following bound on b k .
Proof The assumed absolute convergence of the double series in (8) implies the absolute convergence of the iterated series

The latter implies (24)
In the probabilistic approach initiated by Khintchine, the free parameter δ = δ n is chosen to be the solution of the equation The equation for δ n can be written as where the step before the last is because for the chosen δ we have nδ = Cn ρr ρr +1 → ∞, n → ∞, because of Lemma 1 (ii) and because of the fact that for k ≥ n + 1, where j 0 as in (24), while the last step follows from Lemma 1 (ii) and (24). The right hand side of (27) is > n, if C > (ρ r h r ) −(ρr+1) and ≤ n otherwise. This and (15) say that for a sufficiently large n, (26) has a unique solution δ n , which satisfies We proceed to find an asymptotic expansion for δ n by using a refinement of the scheme of Proposition 1 of [6]. We call anyδ n , such that an asymptotic solution of (26). We will show that it is sufficient for (29) that δ n obeys the condition By Lemma 1, we have Next we have for all n ≥ 1 Aplying the same argument as in (27), we derive the bound Now, (32) and (33) show that (30) implies (29). We will now demonstrate that the error of approximating the exact solution δ n by the asymptotic solutionδ n is of order o(n −1 ). By the definitions of δ n ,δ n we have Next, applying the Mean Value Theorem, we obtain By (34), the left hand side of (35) tends to 0, as n → ∞, while, by virtue of (28),(31), Combining (35) with (36), gives the desired estimate An obvious modification of the argument in (27) allows also to conclude that As a result, The latter relation will be used for derivation of the asymptotics of the second factor in (18). Define the notationsĥ l = ρ l h l , l = 1, . . . r, This is exactly the starting point of the analysis ofδ n in Proposition 1 of [6]. We may therefore apply Proposition 1 of [6] and (37) and conclude that whereK s do not depend on n, and the powers λ s are as defined in (14). We now analyze the three factors in the representation (18) when δ = δ n . (i) It follows from (39) that the first factor of (18) equals e nδn = exp ĥ r where λ s ∈ Υ r and ǫ n → 0.
(ii) By an argument similar to the one for the proof of (33) we conclude that log f n (e −δ ) (iii) The following estimate is central to our arguments.
The asymptotics of the third factor of (18) are given by a local limit theorem, using condition (III).
Proof We take δ = δ n in (20) and define We write where and The proof has two parts corresponding to evaluation of the integrals I 1 and I 2 , as n → ∞. for n fixed we have the expansion where the second equation is due to (25). By virtue of (21) and (41) we derive from (51) that the main terms in the asymptotics for B 2 n and T n depend on the rightmost pole ρ r only: where K 2 = h r ρ r (ρ r + 1) and K 3 = h r ρ r (ρ r + 1)(ρ r + 2) are obtained from (51) and Lemma 1. Therefore, Consequently, in the same way as in the proof of local theorem in [4], and it is left to show that Part 2: Integral I 2 . We rewrite the upper bound in (44) in Proposition 2 as where C > 0 is a constant and We split the interval of integration [α 0 , 1/2] into subintervals: Our goal is to bound, as n → ∞, the function V n (α) from below in each of the subintervals. Firstly, we show that on the first two subintervals for l 0 ≥ 1, the desired bound is implied by the assumption (11) in condition (III).
In the first subinterval [α 0 , (2l 0 ) −1 δ n ], l 0 ≥ 1 we will use the inequality where x denotes the distance from x to the nearest integer, i.e.

Gentile statistics
Gentile statistics are a model arising in physics [3,12,14], which counts partitions of an integer n with no part occuring more than η−1 times, where η ≥ 2 is a parameter . When η = 2, Fermi-Dirac statistics are obtained and when η = ∞, Bose-Einstein statistics, with uniform weights b k = 1, k ≥ 1 result. As far as we know, no rigorous derivation of the asymptotics of Gentile statistics has previously been given, although Theorem 3 below was anticipated in approximation (23) of [12]. In this work we derive the aforementioned theorem as a special case of our Theorem 1.
The Gentile statistics are the Taylor coefficients of the generating function We remark that there is another natural interpretation of the Gentile statistics, which is the number of integer partitions with no part size divisible by η, but where part sizes can now appear an unlimited number of times. Gentile statistics fit into the framework (3) of Theorem 1 with

Theorem 3 Gentile statistics have asymptotics
Proof We will show that all the conditions of Theorem 1 are satisfied for Gentile statistics. In order to show that (15) holds for η > 1, we calculate We have d dη where g(x) = e x (2 − x) − (2 + x). Taking the derivative of g produces g ′ (x) = e x (1 − x) − 1 < 0 for x > 0, which, together with g(0) = 0, implies that g(x) < 0 for x > 0. Combining this with the fact that d 2 dδ 2 log S(e −δ ) = 0, if η = 1, we conclude that (15) holds, for all η > 1.
It remains to be shown that conditions (I) − (III) are satisfied for the model considered. We have and so, by (6), (7) and (8), Conditions (I) and (II) are satisfied because of the analytic continuation of the Riemann zeta function and the well known bound for a constant C > 0, uniformly in x. It is easy to check that l 0 = 1 and b k a k = 1 = k ρ 1 −1 , where ρ 1 = 1, and so (11) is satisfied. Hence condition (III) is satisfied. Moreover, By the argument preceding Proposition 1 this says that the integrand δ −s n Γ(s)D(s) has a simple pole at s = 0 with residue Θ = ζ(0) log η and a simple pole at s = 1 with residue ζ(2)(1 − η −1 )δ −1 n . As a result, in the case considered δ n =ĥ 1/2 and we arrive at the claimed asymptotic formula for c n .

Asymptotic enumeration for distinct part sizes
Weighted partitions fit our framework (3) with S(z) = (1 −z) −1 , a k = 1, k ≥ 1 and weights b k . When b k = 1, k ≥ 1, Theorem 1 gives the asymptotics of the number of partitions of n obtained by Hardy and Ramanujan. If S(z) = 1 + z, a k = 1, b k = k r−1 , r > 0, k ≥ 1, then c n enumerates weighted partitions having no repeated parts, called expansive selections. The asymptotics of expansive selections were also studied in [5].
In this section, we find the asymptotics of c n induced by the generating function The model fits the setting (3) with S(z) = 1+z, b k = 1, a k = k −q , k ≥ 1 and it can be considered as a colored selection with parameter k −q proportional to the number m k of colors of a component of size k, e.g. m k = y k k −q , for some y > 1. A particular case of the model, when q = 1 was studied in Section 4.1.6 of [7] where it was proven, with the help of a Tauberian theorem, that in this case lim and it was established the rate of convergence of c n , n → ∞. Also, in [7] it was shown that c n is equal to the probability that a random polynomial of order n is a product of irreducible factors of different degrees. In [11], Section 11, it was demonstrated that c n can be treated as the probability that a random permutation on n has distinct cycle lengths, and another proof of (67) was suggested. Finally, note that in [11], (11.35), it is was shown that for q = 2, the generating function f (z) can not be analytically continued beyond the unit circle. (1 + k −q z k ), |z| < 1.
If q > 1, then, for a constant W (q) > 0 depending only on q, c n ∼ W (q)n −q , n → ∞.

Proof
The case 0 < q < 1. We will apply Theorem 1. Assumption (15) is easy to verify. We have (−1) j−1 z kj jk qj , |z| < 1 and so, by (6) and (8), We claim that the function D(s; q) allows analytic continuation to the set C excepting for poles in H q := {s = 1 − qj, j = 1, 2, . . . , q < 1}. Changing the order of summation, we write Note that where the function Φ(s; q) is analytic for s ∈ C \ H q , and moreover Φ(s; q) = O(2 −qj ) j → ∞, q > 0, uniformly in s from any compact subset of C\H q . This implies that the series converges absolutely and uniformly on any compact subset of C \ H q . By the Weirstrass convergence theorem, this implies that the series above is analytic in the above indicated domain. Since the function is analytic in C, our claim is proven. This allows to conclude that condition I of Theorem 1 holds with r = max{j ≥ 1 : 1 − qj > 0} simple poles ρ l = 1 − ql, l = 1, . . . r and with 0 < C 0 < 1 defined by C 0 = (r + 1)q − 1 − ǫ, 0 < ǫ < (r + 1)q − 1, if (r + 1)q ≤ 2 any number in (0, 1), if (r + 1)q > 2.
The case q > 1. Theorem 1 is not applicable in this case, because all poles 1−qj, j ≥ 1, q > 1 of the function D(s; q) in (68), are negative. From since the convergence of the infinite product in (69) is equivalent to the convergence of the series ∞ k=1 k −q < ∞, q > 1.
Remark: Comparing the asymptotics of c n in the cases 0 < q < 1, q = 1 and q > 1 it is clearly seen that q = 1 is a point of phase transition.
In the remainder of this section we derive representations of the function W (q) in the case of rational q > 1. The infinite product is a Weierstrass representation of an entire function F with zeroes at {−k q , k = 1, 2, . . .}. This follows from Theorem 5.12 in [1], since ∞ k=1 k −q < ∞, q > 1. Note that W (q) = f (1) = F (1), q > 1. We now show that in the case when q > 1 is a rational number, a modification of the argument in [16], p.238 allows us to decompose the value F (1) in (70) for rational q > 1. For m 2 > 1, we will consider now the functioñ Γ(z) := e Q(z) 1 zf (z) , where Q(z) is a polynomial in z that will be defined below. The preceding discussion yields thatΓ is a meromorhic function in C with simple poles at (−k 1 m 2 ), k = 0, 1, . . . ,. Now our purpose will be to obtain for the functionΓ an analog of Gauss formula for gamma function. We recall the definition of generalized Euler constants: follows: |Γ(1 − α l (q))| 2 −1 , q > 1.