Asymptotic density and the asymptotics of partition functions
Melvyn B. Nathanson
设A是gcd（A）= 1的正整数集，p_A（n）是A的分拆函数.令c = π \sqrt（2/3）。设α> 0。证明了log p_A（n）~ c \sqrt（αn）当且仅当集合A具有渐近密度α.
Let A be a set of positive integers with gcd(A) = 1, and let p_A(n) be the partition function of A. Let c = π\sqrt(2/3). Let α> 0. It is proved that log p_A(n) ~ c\sqrt(αn) if and only if the set A has asymptotic density α.
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Asymptotics of abelian group-partitions and associated Dirichlet series
对有限阿贝尔群引入群划分的概念，它是标准划分的推广。为了得到群划分的渐近分布，我们利用平面划分的母函数研究了群划分的狄利克雷级数.
On the Asymptotic Formula for the Number of Plane Partitions of Positive Integers
本文讨论了大正整数平面分拆数的渐近公式。
We establish the asymptotic behavior of the Andrews $G_k(q)$ function as $q\to 1.$
Asymptotic formulae for partition ranks
利用Wright圆方法的推广，我们得到了类似于F.戴森和最近证明的第一作者和K。布林曼
Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions
本文通过推导ℓ-正则分拆与ℓ-互异分拆中钩长为1、2、3的钩的总数的渐近公式，建立了这两类分拆之间及其内部的钩长不等式，并证明相应钩数比值趋于依赖ℓ和t的常数。
Inequalities and asymptotics for hook numbers in restricted partitions
本文研究了受限划分中钩数的渐进行为，通过渐近公式证明了奇数部分划分在平均意义上比不同部分划分有更多钩数。
Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large $n$, and the stronger conjectures of Ulas are proved. The asymptotics of Wright's generalized Bessel functions are also treated.
Partition Polynomials: Asymptotics and Zeros
令$F_n(x)$为划分多项式$\sum_{k=1}^n p_k(n) x^k$，其中$p_k(n)$为具有$k$部分的$n$的划分的数目。我们强调的计算实验使用度高达$70,000$，发现这些多项式的渐近。令人惊讶的是，$F_n(x)$的渐近有两个规模的订单$n$和$\sqrt{n}$和在三个不同的制度内的单位磁盘。因此，零点收敛到由双对数给出的单位圆盘内的曲线网络。
Asymptotics of partition parts in arithmetic progressions
研究算术级数中分拆部分的分布，通过分析生成函数中的非模艾森斯坦级数，获得捕获所有指数增长项的新近结果。
Asymptotics for $d$-fold partition diamonds and related infinite products
本文证明了$n$的$d$重分拆钻石及其Schmidt型对应物数量的渐近公式，通过研究无穷乘积的渐近行为，并指出了与数学物理和Bloch群的潜在联系。
Asymptotics and inequalities for the distinct partition function
本文研究了偏移不同划分函数的渐近分析与不等式，推导出四次二元形式不变量及相关不等式的精确阈值。
Let $j,n$ be even positive integers, and let $\overline{p}_j(n)$ denote the number of partitions with BG-rank $j$, and $\overline{p}_j(a,b;n)$ to be the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. We give asymptotics for both statistics, and show that $\overline{p}_j(a,b;n)$ is asymptotically equidistributed over the congruence classes modulo $b$. We also show that each of $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy all higher-order Turán inequalities.
Asymptotics of parity biases for partitions into distinct parts via Nahm sums
本文推广了Kim等人的奇偶偏差研究，利用圆法给出互异分拆的模N偏差渐近结果，同时提供了分析Nahm型q-超几何级数渐近性的独立有用技术。
For primes $p$, we study the maximal possible size of a $p$-core $p'$-partition (a partition with no hook lengths or parts divisible by $p$). McDowell recently proved that the maximum is attained by a unique partition, say $Λ_p$. Using his graph theoretic description of $Λ_p$, we prove for $p > 10^6$ that \[\frac{1}{24}p^6 - p^5\sqrt{p} < |Λ_p| < \frac{1}{24}p^6 - \frac{1}{200}p^5\sqrt{p},\] which shows that $|Λ_p| \sim p^6/24$ as $p \to \infty$.
On Erdos's elementary method in the asymptotic theory of partitions
设m是正整数，A是属于r个模m不同同余类并的所有正整数的集合.我们假设A的元素是互素的，即gcd（A）= 1。设p_A（n）表示n分成属于A的部分的个数。我们得到了logp_A（n）~ π \sqrt（2 rn/3 m）的渐近公式.证明是基于Erdos的初等方法，以获得通常的分拆函数p（n）的渐近公式。
Asymptotics of various partitions
研究了各种划分的个数的渐近公式，如3色划分、凹划分、某些平面划分、无小部分划分、p-环的个数。
Asymptotics for moments of the minimal partition excludant in congruence classes
本文研究整数划分中频率小于给定数的最小正整数的矩，针对特定同余类建立渐近公式，并证明固定模数下各矩渐近相等。
On the asymptotic behavior for partitions separated by parity
本文运用 Ingham 的 Tauberian 定理，为 Andrews 在奇偶分隔分拆研究中提出的八个生成函数分别给出渐近主项。
We prove asymptotic formulas for the complex coefficients of $(ζq;q)_\infty^{-1}$, where $ζ$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions of $n$ with largest part congruent $a$ modulo $b$. Our results imply that, as $n \to \infty$, the difference $p(a_1,b,n)-p(a_2,b,n)$ for $a_1 \neq a_2$ oscillates like a cosine, when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of $\{p(a,b,n)\}_{1 \leq a \leq b}$.
Asymptotics of Reciprocal Supernorm Partition Statistics
本文研究整数划分上超范数倒数之和的统计量，在大小、周长和最大部分三个系综下，证明其累积统计量均渐近于 e^γ log n。
Asymptotics for partitions over the Fibonacci numbers and related sequences
本文结合经典分拆理论与 Fibonacci Dirichlet 级数的新结果，首次给出整数表示为非互异斐波那契数之和的方式数 $p_F(n)$ 的渐近公式，并将类似结果推广到一般线性递推序列。
We deduce from the strong form of the Hardy--Ramanujan asymptotics for the partition function $p(n)$ an asymptotics for $p_{-S}(n)$, the number of partitions of $n$ that do not use parts from a finite set $S$ of positive integers. We apply this to construct highly oscillating partition ideals.
Asymptotic Formula for $(t+1)$-Regular Partitions
本文基于 saddle point 方法，推导了 (t+1) 正则划分数 p(N,t) 的渐近公式，并讨论了其在对称群特征表零点估计中的应用。
Asymptotics of the number of partitions into p-cores and some trigonometric sums
一个渐近公式的分区成p-核心的数量。作为一个副产品，一些整数值三角和发现
On asymptotics for lacunary partition functions
本研究针对缺项分拆函数相关幂级数开展渐近分析，同时提供若干基本超几何级数的新分拆理论阐释作为实例，丰富了分拆函数的渐近研究内容。
In 1918, Hardy and Ramanujan published a seminal paper which included an asymptotic formula for the partition function. In their paper, they also claim without proof an asymptotic equivalence for $p^k(n)$, the number of partitions of a number $n$ into $k$-th powers. In this paper, we provide an asymptotic formula for $p^k(n)$, using the Hardy-Littlewood Circle Method. We also provide a formula for the difference function $p^k(n+1)-p^k(n)$. As a necessary step in the proof, we obtain a non-trivial bound on exponential sums of the form $\sum_{m=1}^q e(\frac{am^k}{q})$.
On the asymptotics of certain colored partitions
本文证明了某些双色分区对数的无限族渐近公式，其中无限子族验证了Guadalupe提出的猜想。
Plane Partition Polynomial Asymptotics
平面分区多项式$Q_n(x)$是次数为$n$的多项式，其系数计算由其迹索引的$n$的平面分区的数目。扩展E.M. Wright，我们使用圆方法在单位圆盘内发展这些多项式的渐近性。
Asymptotics for $k$-crank of $k$-colored partitions
本文利用Hardy-Ramanujan圆法研究k-着色分拆的k-crank，推广了Fu和Tang关于M_k(a,c;n)的不等式结果，并进一步建立了对数凹性与对数次可加性等新不等式关系。
A partition $α$ is said to contain another partition (or pattern) $μ$ if the Ferrers board for $μ$ is attainable from $α$ under removal of rows and columns. We say $α$ avoids $μ$ if it does not contain $μ$. In this paper we count the number of partitions of $n$ avoiding a fixed pattern $μ$, in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $μ$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic $p$-groups. We f…
Asymptotics for rank partition functions
本文得到了一类无限秩生成函数的渐近公式。作为应用，我们解决了Andrews和Lewis关于秩间不等式的一个猜想.
In this paper, we investigate $π(m,n)$, the number of partitions of the \emph{bipartite number} $(m,n)$ into \emph{steadily decreasing} parts, introduced by L.Carlitz ['A problem in partitions', Duke Math Journal 30 (1963), 203--213]. We give a relation between $π(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, some uniform asymptotic formulas for $π(m,n)$ are established.
The asymptotic behavior of the rectangle partition function $p(m,n)$
本文研究矩形分拆函数$p(m,n)$的渐近行为，通过初等方法推导其对数渐近公式，证实相关猜想并推广整数分拆的Hardy-Ramanujan公式。
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<\dots <s_k$ and $\ell_1, \ell_2,\dots , \ell_k$. Let $g(\mathbf{s}, \mathbf{l}, n)$ be the number of $\ell$-colored pa…
In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $ n $ large enough, the two quantities are different and which of the two is bigger depends on the parity of $ n. $ This solves a recent conjecture formulated by Bringmann and Mahlburg (2012).
The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function.
Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. He…
Let $κ$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be given. Let $p_{κ, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(\lfloor \ell^κ\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{κ, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution.
Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.
Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the g…
We study the Dyson rank function $N(r,t;n)$, the number of partitions with rank congruent to $r$ modulo $t$. We first show that it is monotonic in $n$, and then show that it equidistributed as $n \rightarrow \infty$. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of $N(r,t;n)$.
In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, they introduce generalized partition functions $p(n)_{\textbf{e}}$. The group theory motivates the study of the asymptotics for these functions. Moreover, Bacher and de la Harpe also conjecture over 200 congruences for these functions which are analogous to the Ramanujan congruences for the unrestricted partition function $p(n)$. We obtain asymptotic formulas for all of the $p(n)_{\textbf{e}…
Improved asymptotics for moments of reciprocal sums for partitions into distinct parts
针对互异整数分拆中的倒数和与倒数平方和，本文改进了 $s_1(n)$ 和 $s_2(n)$ 的渐近估计，并处理了非模、非乘积生成函数带来的技术困难。
Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We also study the Witten zeta function $ζ_{\mathfrak{so…
Asymptotics for the reciprocal and shifted quotient of the partition function
本文研究分拆函数平移商 $p(n+k)/p(n)$ 的高阶渐近展开与误差界，并通过分别分析 $p(n+k)$ 和 $1/p(n)$ 推广了已有结果。
We prove an asymptotic formula for the number of partitions of $n$ into distinct parts where the largest part is at most $t\sqrt{n}$ for fixed $t \in \mathbb{R}$. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres' asymptotic formula for distinct parts partitions when instead the number of parts is bounded by $t\sqrt{n}$. Although equivalent to a circle method/saddle-point method calculation, the probabilistic approach predicts the shape of the asymptotic formula, to some degree.
Asymptotics for $t$-Core Partitions and Stanton's Conjecture
本文证明了Stanton猜想，通过建立$t$-核心划分数目的鞍点渐进行为公式，揭示了$t^2$与$N$大小关系对划分行为的影响。
On asymptotic formula of the partition function $p_A(n)$
针对有限正整数集$A$对应的分拆函数$p_A(n)$的渐近公式，利用$A$有限时$p_A(n)$是拟多项式的性质，给出该公式的纯组合新证明。
Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under hypotheses on the roots of $f-f(0)$, we use the Hardy--Littlewood circle method, a polylogarithm identity, and the Matsumoto--Weng zeta function to derive asymptotic formulae for $p_{\mathcal{A}_f}(n)$ as $n$ tends to infinity. This generalises asymptotic formulae for the number of partitions into perfect $d$th powers, established by Vaughan for $d=2$, and Gafni …
We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two- and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in w…
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.
We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our formulas for the $M_2$-rank at roots of unity lead to asymptotics for certain combinations of $N2(r,m,n)$ (the number of partitions without repeated odd parts of $n$ with $M_2$-rank congruent to $r$ modulo $m$). This allows us to deduce inequalities among certain combinations of $N2(r,m,n)$. In particular, we resolve a few conjectured inequalities of Mao.
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm.
A remark on decay rates of odd partitions: An application of spectral asymptotics of the Neumann--Poincaré operators
本研究借助Neumann--Poincaré算子的谱渐近性质，以该算子为紧线性算子工具，证明了一个与分拆问题相关的、此前仅能通过紧算子特征值渐近行为验证的定理。
We obtain an asymptotic formula for Andrews and Paule's broken $k$-diamond partition function $Δ_k(n)$ where $k=1$ or $2$. Based on this asymptotic formula, we derive that $Δ_k(n)$ satisfies the order $d$ Turán inequalities for $d\geq 1$ and for sufficiently large $n$ when $k=1$ and $ 2$ by using a general result of Griffin, Ono, Rolen and Zagier. We also show that Andrews and Paule's broken $k$-diamond partition function $Δ_k(n)$ is log-concave for $n\geq 1$ when $k=1$ and $2$. This leads to $Δ_k(a)Δ_k(b)\geΔ_k(a+b)$ for $a,b\ge 1$ when $k=1$ and $ 2$.
We apply the saddle-point method to derive asymptotic estimates or asymptotic series for the number of partitions of a natural integer into parts chosen from a subset of the positive integers whose associated Dirichlet series satisfies certain analytic properties. This enables grouping in a single statement many cases studied in the literature, as well as a number of new ones.
We study generating functions which count the sizes of $t$-cores of partitions, and, more generally, the sizes of higher rows in $t$-core towers. We then use these results to derive an asymptotic for the average size of the $t$-defect of partitions, as well as some curious congruences.
Let $α>1$ be an irrational number. We establish asymptotic formulas for the number of partitions of $n$ into summands and distinct summands, chosen from the Beatty sequence $(\lfloorαm\rfloor)_{m\in\mathbb{N}}$. This improves some results of Erdös and Richmond established in 1977.
Simple error bounds for an asymptotic expansion of the partition function
本文为整数划分函数 $p(n)$ 的 Poincaré 型简化渐近展开建立了可计算的误差界，该界比近期文献更尖锐且推导更简单。
We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple l…
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