Asymptotics of various partitions
Gert Almkvist
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。
Asymptotic formulas of the number of various partitions are studied, like 3-colored partitions, concave partitions, certain plane partitions, partitions without small parts, the number of p-rings.
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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.
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。
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 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.
Asymptotic formulas for general colored partition functions
1917年，Hardy和Ramanujan得到了经典配分函数$p(n)$的渐近公式。经典配分函数$p(n)$已被广泛研究。最近，Luca和Ralaivaosaona得到了平方根函数的渐近公式。一些特殊的有色分拆函数上的同余关系一直受到数学家的关注。本文研究了一般有色分拆函数。给定正整数$1=s_1<s_2<\dots <s_k$和$\ell_1, \ell_2,\dots , \ell_k$。令$g(\mathbf{s}, \mathbf{l}, n)$为$n$的$\ell$着色分区的数量，其中颜色的$\ell_i$仅出现在$s_i\ (1\le i\le k)$的乘法中，其中$\ell = \ell_1+\cdots +\ell_k$。通过使用初等方法，我们获得了带有显式误差项的配分函数$g(\mathbf{s}, \mathbf{l}, n)$的渐进公式。
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 behavior of odd-even partitions
安德鲁斯研究了一个功能出现在拉马努金的身份。在拉马努金的《失落的笔记》中，有几个公式涉及这个函数，但它们并不像其他类似形状的函数的恒等式那样简单。尽管如此，安德鲁斯发现这个函数具有组合信息、奇偶划分。在本文中，我们提供了这个组合对象的渐近公式。我们还研究了它的同伴奇偶超分割。
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 $N(r,t;n)$.
Asymptotics and congruences for partition functions which arise from finitary permutation groups
在最近的一篇论文中，Bacher和de la Harpe研究了有限置换群的共轭增长级数。在研究与有限交错群有关的级数的系数时，他们引入了广义分拆函数$p(n)_{\textbf{e}}$。群论激发了对这些函数渐近性的研究。此外，巴赫尔和德拉哈普还猜想超过200同余这些职能类似的拉马努金同余的无限制分拆函数$p(n)$。我们得到了所有$p(n)_{\textbf{e}}$的渐近公式，并证明了它们的收敛同余。
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…
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…
Asymptotics for the partial fractions of the restricted partition generating function I
$p_N(n)$的生成函数，即将$n$划分为至多$N$部分的数目，可以被写为$N$因子的乘积。通过鞍点方法，我们发现了这个乘积的部分分式分解中系数的行为为$N \to \infty$，其中我们需要的鞍点与解析连续双对数的零点相关联。我们的主要结果否定了Rademacher的一个猜想。
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.
Asymptotics for the partial fractions of the restricted partition generating function II
$p_N(n)$的生成函数，即将$n$划分为至多$N$部分的数目，可以被写为$N$因子的乘积。在第一部分中，我们利用鞍点方法研究了该乘积作为$N \to \infty$的部分分式分解中系数的性质，得到了主要项的渐近性质。在第二部分中，我们约束了误差项。这涉及到估计正弦的乘积和进一步的鞍点参数。所需的鞍点与解析连双对数的零点有关。
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.
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。
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。
On the Andrews-Zagier asymptotics for partitions without sequences
我们建立了Andrews $G_k(q)$函数的渐近行为为$q\to 1.$
Asymptotic density and the asymptotics of partition functions
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.
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。
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。
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.
Asymptotic inequalities for positive crank and rank moments
Andrews、Chan和Kim最近引入了一个修正的曲柄矩和秩矩的定义，允许研究偶数和奇数矩。在本文中，我们证明了这些时刻在所有情况下的渐近行为，我们的主要结果表明，而这两个家庭的力矩函数是渐近相等的，曲柄时刻总是渐近大于秩矩。Andrews，Chan和Kim主要集中在一种情况下，并证明了更强的结果，即对于所有分区，第一曲柄矩严格大于第一秩矩，通过显示差值等于他们命名为ospt函数的分区上的组合统计量。因此，我们的主要结果也给出了ospt函数的渐近行为，我们进一步确定其模2的行为，通过将其奇偶性与Andrews spt函数联系起来。
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。
For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for $p_{\mathcal A}(n)$, where $\mathcal A$ is the set of $k$-th powers of primes (for fixed $k$). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into $k$-th powers. This new asymptotic formula is an extension of a pattern indicated by several results about …
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。
We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of independent random variables indexed by $t$. The expansion consists of a factor (which is the leading term) times an asymptotic series expansion in inverse powers of a quantity that grows as $\sqrt n$ as $n\to \infty$, and whose coefficients are simple combinatorial expressions. The asymptotic series is obtained by expanding the characteristic functions in terms of…
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。
The Number of Parts in Certain Residue Classes of Integer Partitions
我们使用圆的方法来推导渐近公式的功能有关的部分的分区，特别是剩余类的数量。
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…
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。
Power Partitions
在1918年，哈代和拉马努金发表了开创性的文件，其中包括一个渐近公式的分拆函数。在他们的论文中，他们还没有证明就声称了$p^k(n)$的渐近等价性，即一个数$n$分成$k$次幂的分割数。本文利用Hardy-Littlewood圆方法给出了$p^k(n)$的一个渐近公式。我们还提供了一个公式的差异函数$p^k(n+1)-p^k(n)$。作为证明的必要步骤，我们得到了$\sum_{m=1}^q e(\frac{am^k}{q})$形式的指数和的非平凡界。
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…
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 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。
Partition asymptotics from one-dimensional quantum entropy and energy currents
我们给出了一种替代方法，Hardy-Ramanujan-Rademacher推导出领先的指数项的渐近逼近的配分函数p（n，a），定义为一个正整数“n”到整数被加数的分解的数量，每个被加数出现在最多“a”倍，在一个给定的分解。推导涉及映射到一个等效的物理问题，该问题涉及在连接热储层的一维通道中流动的粒子的量子熵和能流，并且服从Gentile的中间统计，统计参数为“a”。该方法也适用于分区与Halfman的分数排除统计。
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.
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。
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.
On the asymptotic prime partitions of integers
在本文中，我们讨论P（n），一个给定的整数n可以写为素数和的方式的数量。特别地，利用量子统计力学的标准方法，得到了P_as（n）对n向无穷大有效的渐近形式.首先，构造了素数的玻色配分函数，即数论中的无限制素数分拆的生成函数。接着，在大n的极限下，使用用于配分函数的拉普拉斯反演的鞍点方法来获得态密度。这直接给出了素分拆的渐近数P_as（n）。渐近表达式中的首项按sqrt[n/ln（n）]呈指数增长，并与以前的估计一致。我们计算了指数中的次首阶项，与ln[ln（n）]/ln（n）成比例，并证明了文献中关于其系数的早期结果是不正确的。此外，我们还计算下一个高阶校正，与1/ln（n）成比例，并在等式中给出。(43)，这是迄今为止还没有在文献中。最后，我们比较我们的分析结果与精确的数值P（n）的n \sim 8 10 ^6。对于最高值，精确的P（n）和我们的P_as（n）之间的剩余误差仅为首阶（LO）近似所获得的误差的一半。但是我们也证明了，与其他类型的划分不同，即使对于n \sim 10 ^7，素划分的渐近极限仍然远远没有达到。
We prove an exact formula for OEIS A000119, which counts partitions into distinct Fibonacci numbers. We also establish an exact formula for its mean value, and determine the asymptotic behaviour.
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