The asymptotic number of weighted partitions with a given number of parts

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Introduction and background
This study is devoted to the asymptotic formula for the quantity c n,m which denotes the number of weighted integer partitions of n, having exactly 1 ≤ m ≤ n parts. The weights are a sequence of real numbers b k , k ≥ 1 and the ordinary bivariate generating function f (y, z) for the sequence c n,m is (i) Let s = σ + it. For constants r > 0 and 1 < C 0 < 2 the Dirichlet series D(s) converges in the half-plane σ > r > 0 and the function D(s) has an analytic continuation to the half-plane on which it is analytic except for a simple pole at s = r with residue A > 0. (ii) There is a constant C 1 > 0 such that uniformly in s ∈ H. (iii) There is a constant C > 0 such that The first two conditions are similar to assumptions of Meinardus [1], although we have assumed 1 < C 0 < 2 in the second condition rather than the slightly weaker assumption of Meinardus [1] that 0 < C 0 < 1. Meinardus' third condition did not make any direct assumptions on the b k . He assumed (iii)' There are constants C 2 > 0 and ν > 0, such that the function g(x) = ∞ k=1 b k e −kx , x = δ + 2πiα, α real and δ > 0 satisfies (g(x)) − g(δ) ≤ −C 2 δ −ν , | arg(x)| > π/4, 0 = |α| ≤ 1.2, for small enough values of δ.
Meinardus [1] introduced his conditions in an analysis of c n = n m=1 c n,m with generating function f (1, z). Granovsky et al. [7] weakened condition (iii) and obtained the asymptotics of c n under (iii)" For small enough δ > 0 and any μ > 0, Let ξ be a random variable having distribution Haselgrove and Tempereley [2] obtained an expression for c n,m under several conditions, one of which implies r < 2 and conjectured that ξ n should have a limiting Gaussian distribution for r > 2. Of particular interest is the case b k = k for which c n is the number of plane partitions of n and ξ n is the number of the sum of the diagonal parts; see [3]. Under conditions (i) (with 0 < C 0 < 1), (ii), and (iii) , Mutafchiev [4] found the limiting distribution of ξ n for all r > 0. The non-Gaussian distributions for r < 2 had been discovered previously, as is explained in [4]. The Gaussian distributions for r ≥ 2 confirmed the conjecture of [2]. Hwang [5] studied the number of components in a randomly chosen selection, partitions having no repeated parts, assuming Meinardus-type conditions and an analysis of a bivariate generating function analogous to (1).
In this paper we will find asymptotics of c n,m through an analysis of the bivariate function (1) which adapts the methods used in Granovsky et. al. [6][7][8][9] for finding the asymptotics of the coefficients of univariate functions including f (1, z). The initiator of the method was Meinardus [1]. Our main result is stated in terms of functions of n and m defined in (8) and (9). Let be the polylogarithmic function of order s and define The asymptotic which holds uniformly for all s ∈ H results in Λ(μ) ∼ e −μ . The identity which holds for all s implies that where the implicit constant in the O(·) term depends on r . Therefore, for some μ 0 > 0, Λ(μ) decreases monotonically to 0 for when restricted to μ > μ 0 . Taking now Λ restricted to (μ 0 , ∞), it follows that Λ has an inverse Λ −1 . Assuming that m = and letting h r = AΓ (r ), for n large enough define μ n,m = Λ −1 r r m r +1 n −r h r (8) and Note that μ n,m → ∞ as n → ∞ and and δ n,m ∼ rm/n (11) as n → ∞.
The assumption (4) can probably be weakened to, say, b k k r −1 and an approximation to c n,m still obtained, but doing so with the methods of this paper would require at least the derivation or imposition of a lower bound on the left-hand side of (45).

A fundamental identity
We will establish an expression for c n,m which is fundamental for our analysis of c n,m . Define a truncation of f (y, z) by Let X k have p.d.f.
a negative binomial distribution with parameters b k and e −μ−δk , where the parameters μ > 0, δ > 0 are arbitrary, and let Lemma 1 For any μ > 0 and δ > 0 we have Proof Observe that It follows that c n,m = e mμ+nδ For |α| ≤ 1/2 and |β| ≤ 1/2 we have Therefore, the joint characteristic function of Z n and Y n is We now combine (20) and (21) to obtain c n,m = e mμ+nδ f n (e −μ , e −δ ) In proving Theorem 1 we take μ = μ n,m and δ = δ n,m given by (8) and (9), giving In Sect. 3 we estimate f n (e −μ n,m , e −δ n,m ) and in Sect. 4 we estimate P(Y n = m, Z n = n).

Asymptotics for the truncated generating function
We first find the asymptotics of f (e −μ , e −δ ).

Lemma 2
We have where h r is given by (15), Proof Substituting the expression of e −δ as the inverse Mellin transform of the Gamma function: The function Li s+1 (e −μ ) defined in (5) is analytic for all complex s for each μ > 0 while by condition (i) the function D(s) is assumed to be holomorphic in H, with a unique simple positive pole r with a positive residue A. The gamma function has simple poles at s = 0 and s = −1 with residues 1 and −1, respectively. We will shift the contour of integration in (24) from {s : In performing this shift we use (3), the fact that and the bound for a constant C 2 > 0; see [3]. The Cauchy residue theorem produces (23).
We now are able to find the asymptotics of the second factor of (22). and so, by (1) and (19), We estimate where we have used nδ n,m → ∞ which follows from (11) and (13).

The local limit theorem
We have found asymptotics for the first two factors of (22) and we now will find them for the third factor. The proof of the following Local Limit Lemma is similar in places to one in [6].

It follows from (21) that
Therefore, (1), (19), (23) and an estimate similar to one in the proof of Lemma 3 imply and similarly  (8) and (9) that and We also have to estimate the |ρ s |. We have Let us define the matrix Σ n by so that where T denotes transpose. Since Σ n is positive definite and symmetric it has a square root √ Σ n . Define the variables u and v by Change of variables gives where S n is the image of R n under the map √ Σ n . Under the assumption (iii) that b k ∼ Ck r −1 , and using nδ n,m → ∞, we have We therefore have and If we show that lim inf n→∞ S n = R 2 , then will be an immediate consequence and will have been shown. Let ∂ R n and ∂ S n denote the boundaries of R n and S n . In view of the identity where | · | 2 represents L 2 distance, and the fact that (0, 0) ∈ S n , if we show that the right-hand side of (44) converges to ∞ as n → ∞, then lim inf n→∞ S n = R 2 will follow. Observe that The last infimum occurs when We check using (27) where {x} is defined to be the fractional part of x and b k e −μ n,m −δ n,m k α + βk 2 .
We will find lower bounds for V n (α, β) on four regions which partition R n .