Boris Y. Rubinstein
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called Sylvester waves) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.
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We find general solutions to the generating-function equation sum c_q^{(X)} z^q = F(z)^X, where X is a complex number and F(z) is a convergent power series with |F(0)| >0. We then use these results to derive finite expressions containing only integers or simple fractions for partition functions and for Euler, Bernoulli, and Stirling numbers.
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed.
On the restricted partition function via determinants with Bernoulli polynomials
本文利用由 Bernoulli 多项式值构成的行列式 $Δ_{r,D}$，证明了在行列式非零条件下，限制分拆函数 $p_{\mathbf a}(n)$ 可通过 Bernoulli 多项式值与 Bernoulli Barnes 数进行计算。
On the restricted partition function via determinants with Bernoulli polynomials. II
本文在 D=1 或 D 为素数的条件下，将限制分拆函数的计算转化为系数为伯努利多项式与伯努利·巴恩斯数的线性方程组求解问题。
Determinants with Bernoulli polynomials and the restricted partition function
本文研究由伯努利多项式构造的两个 rD 阶行列式，并建立其与受限分拆函数之间的内在联系。
Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf s},{\bf D})$, i.e., a number of integer solutions to a linear system ${\bf x} \ge 0, {\bf D x} = {\bf s}$. It is shown that $W({\bf s},{\bf D})$ can be expressed through the vector Bernoulli polynomials of higher order.
By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the paper we present a new infinite series representation of the Riemann zeta function via the Lambert $W$.
A new recursive procedure for calculation of restricted partition function is suggested. An explicit formula for the restricted partition function is found based on this procedure.
We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a gap of 6.
Some restricted partition functions in terms of 2-adic valuation
本文研究受限分拆函数的表示问题，利用整数的2进赋值给出了若干此类函数的表达形式。
We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.
Given an integer base $b\geq 2$, a number $ρ\geq 1$ of colors, and a finite sequence $Λ=(λ_1,\ldots,λ_ρ)$ of positive integers, we introduce the concept of a $Λ$-restricted $ρ$-colored $b$-ary partition of an integer $n\geq 1$. We also define a sequence of polynomials in $λ_1+\cdots+λ_ρ$ variables, and prove that the $n$th polynomial characterizes all $Λ$-restricted $ρ$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.
In this paper, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our extended technique could be a useful tool to prove congruences for certain types of combinatorial functions based on a bounded number of calculations. As applications of our result, we establish new and existing restricted plane partition congruences, restricted plane overpartition congruences and several examples of restricted partition congruences.
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 generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. These techniques are also shown to give the asymptotics of Sylvester waves.
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.
Let Q(4n,d) be the sum of all multiple zeta values of depth d and weight 4n whose arguments are all multiples of 4. In this paper we derive a formula of Q(4n,d) for all d>2 as a finite sum involving binomial coefficients, Bernoulli numbers and the quantities Q(4m,2). The number of terms is about 3d^2/8 which is independent of n.
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.
Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerator…
Integer Partitions With Restricted Distinct Parts
本文研究正整数分拆为互不相同部分且禁止特定模 $t$ 同余类的计数函数 $Q_{t}^{s}(n)$，并利用 $q$-级数与 theta 函数恒等式证明了若干 Ramanujan 型同余式。
On the Sylvester waves in partition function
本文证明了西维斯特波可表示为伯努利多项式有限求和形式，并验证了该表示式的有效性。
Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts
本文通过经典q级数操作和拉马努金θ函数性质，证明了关于特定算术级数中过着色划分函数的模2幂同余猜想。
Overcolored Partition $k$-tuples Restricted by Parity of the Parts
本文研究超着色分拆k元组的计数函数$\bar{b}^k_{r,s}(n)$，扩展Chacon与Sellers的相关结果，建立其模素数的可除性性质及模2幂的新同余式，所用技术基于theta函数恒等式和模形式。
The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Raman…
Log-concavity And The Multiplicative Properties of Restricted Partition Functions
本文证明满足特定初始条件的对数凹序列必满足乘法不等式，从而统一解释分拆函数的相关性质，并说明这些条件充分但非必要。
Andrews, Brietzke, Rødseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$ and thus confirm the conjecture of Andrews, et al.
Congruences for Overcolored Partition $k$-tuples Restricted by Parity of the Parts
本文提出统一框架研究过度着色分拆 $k$-元组 $\bar{b}^k_{r,s}(n)$ 模奇素数的同余性质，并证明特定情形下 $\bar{b}_{4,2}^4(5n)$ 模 5 同余为零。
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".
The difference of the sums of odd and even parts of restricted partitions
本文推导了在部分不重复的限制条件下，分拆的奇部和与偶部和之差的生成函数，并给出了模4的同余式。
Congruences modulo powers of $2$ for three restricted partition functions of Pushpa and Vasuki
本文通过q级数技术，研究了三项受限分割函数对2的幂次的同余关系，并推广了之前发现的特定同余关系。
Connections between colored restricted $b$-ary and ordinary partitions
该研究建立了带色有界普通分拆与带色二元、$b$元分拆间的联系，基于特殊递推关系得到带色$b$元分拆的恒等式及合成分拆相关结果。
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。
We derive the asymptotic formula for $p_n(N,M)$, the number of partitions of integer $n$ with part size at most $N$ and length at most $M$. We consider both $N$ and $M$ are comparable to $\sqrt{n}$. This is an extension of the classical Hardy-Ramanujan formula and Szekeres' formula. The proof relies on the saddle point method.
Overcolored Partition Restricted by Parity of the Parts
本文扩展了Thejitha等人定义的a_{r,s}(n)函数，将其应用于过划分，考虑了部分奇偶性对多重着色划分的影响。
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 …
Restricted divisor functions and half Appell sums in higher-level
本文研究高阶半Appell和相关q-级数系数的值分布，证明其等于限制除数函数且可取任意非负整数，并证实了两个相关猜想。
Given a set of positive integers A = {a_1,...,a_n}, we study the number p_A (t) of nonnegative integer solutions (m_1,...,m_n) to m_1 a_1 + ... m_n a_n = t. We derive an explicit formula for the polynomial part of p_A.
Suppose that $a_1(n),a_2(n),...,a_s(n),m(n)$ are integer-valued polynomials in $n$ with positive leading coefficients. This paper presents Popoviciu type formulas for the generalized restricted partition function $$p_{A(n)}(m(n)):=#\{(x_1,...,x_s)\in \mathbb{Z}^{s}: all x_j\geqslant 0, x_1a_1(n)+...+x_sa_s(n)=m(n) \}$$ when $s=2$ or 3. In either case, the formula implies that the function is an integer-valued quasi-polynomial. The main result is proved by a reciprocity law for a class of fractional part sums and the theory of generalized Euclidean division.
Identities involving partitions with distinct even parts and $4$-regular partitions
本文在已知不同偶数部分分拆数与4-正则分拆数相等的基础上，证明了一类受限的不同偶数部分分拆与受限4-正则分拆之间的恒等式。
Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m})_{n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$. We investigate periodicity of the sequence and lower and upper bounds for the density of the set $\{n\in\mathbb{N}: p_\mathcal{A}(n,k) \equiv i \pmod{m}\}$ for a fixed positive integer $k$ and $i\in\{0,1,\ldots, m-1\}$. In particular, we apply our results t…
Let $p_{o}(n)$ (resp. $p_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim, and Lovejoy proved that $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured that $d_{o}(n)>d_{e}(n)$ for all $n>19$ where $d_{o}(n)$ (resp. $d_{e}(n)$) denote the number of partitions into distinct parts having more odd parts (resp. even parts) than even parts (resp. odd parts). In this paper we provide combinatorial proofs for both the result and the conjecture of Kim, Kim and Lovejoy. In addition, we show that if we restrict the …
Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts
本文研究了受部分奇偶性限制的多色划分数目$a_{r,s}(n)$的算术性质，采用生成函数和$q$-级数技术进行分析。
Restricted partition functions and the $r$-log-concavity of quasi-polynomial-like functions
该研究针对弱递增正整数序列对应的受限分拆函数，推广其对数凹性性质，解决了强对数凹性及类拟多项式函数与受限分拆函数的r-对数凹性问题，给出有效解法。
On Graphical Partitions with Restricted Parts
研究受限图划分的概率，通过Durfee平方建立上界，并证明概率趋于零，结合Nash-Williams条件、乘法方法和Edgeworth展开。
We use the celebrated circle method of Hardy and Ramanujan to develop convergent formulae for counting a restricted class of partitions that arise from the Göllnitz--Gordon identities.
Restricted partition functions and additive complements
本文构造了无限集合A和M，使得受限分拆函数p(n,A,M)对所有正整数n为正且具有多项式增长，从而肯定回答了Dai和Chen在2016年提出的问题。
Partition-Theoretic Results and Recurrence Relations for the Coefficients of Some Mock Theta Functions
本文研究某些模拟 theta 函数系数的分拆理论性质，证明其同余性质，并建立与受限分拆函数之间的递推关系。
A note on congruences modulo 7 and 11 for two restricted partition functions
本文通过q级数技术研究了某些受限划分函数在7和11模下的同余关系，并推广了之前使用模形式方法得到的结果。
We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for o…
Algebraic Structures on Sets of Partitions
受Robert Schneider工作的启发，本文探索分拆上的二元运算，发现受限分拆集合构成阿贝尔群、向量空间和交换环，为分拆分析提供代数工具。
Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset $\{a_1,a_2,\ldots,a_k\}$. We find out a new type of Bessenrodt-Ono inequality for the function $p_\mathcal{A}(n,k)$. Further, we discover when and under what conditions on $k$, $\{a_1,a_2,\ldots,a_k\}$ and $N\in\mathbb{N}_+$, the sequence $\left(p_\mathcal{A}(n,k)\right)_{n=N}^\infty$ is log-concave. Our proofs are based on the asymptotic behavior of $p_\mathcal{A}(n,k)$, i…
The generalized Frobenius problem via restricted partition functions
本文利用受限分拆函数的水平集与拟多项式渐近结构，研究广义 Frobenius 数及可表示整数计数，给出大 $k$ 时的公式、完整渐近性并恢复若干特例结果。
Remarks on a certain restricted partition function of Lin
本文通过q级数和分割公式，给出了林关于b(3n+1)和b(3n+2)生成函数的初等证明，并建立了b(n)对3的无限多个内部同余式。
Restricted Partitions and $SL_2$ Cohomology
本文研究正整数受限分拆数的上界问题，证明一类集合下分拆数不超过第n个斐波那契数，同时给出系数为Weyl模的单代数群$SL_2$上同调界的新结果。
The Li--Wan sieve is extended to multisets when the underlying set is symmetric. The main ingredient of the proof is the Mobius inversion formula on the poset of partitions of $\{1,2,\dots,k\}$ ordered by refinement. As illustrative applications, we investigate the problems of partitions over finite fields and zero-sum multisets over the additive group $\mathbb{Z}/n\mathbb{Z}$. .
In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare our result with similar formulas obtained with different approaches already in the XIX century. Examples are given for Bernoulli, Euler and Fibonacci numbers.
For a set of positive integers $A$, let $p_A(n)$ denote the number of ways to write $n$ as a sum of integers from $A$, and let $p(n)$ denote the usual partition function. In the early 40s, Erdős extended the classical Hardy--Ramanujan formula for $p(n)$ by showing that $A$ has density $α$ if and only if $\log p_A(n) \sim \log p(αn)$. Nathanson asked if Erdős's theorem holds also with respect to $A$'s lower density, namely, whether $A$ has lower-density $α$ if and only if $\log p_A(n) / \log p(αn)$ has lower limit $1$. We answer this question negatively by constructing, for every $α> 0$, a set …
Let $p$ be a prime and let $S$ be a non-empty subset of $\mathbb{F}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ that vanishes on $S^n$, such that the polynomial $P-P_0$ has bounded rank. Our argument uses two black boxes: that a tensor with high partition rank has high analytic rank …
We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: 1) for given $N,k$, finding the total number $λ(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\leq{n_2}...\leq{n_k}$. 2) finding the total number $λ(N)=\sum_{k=1}^Nλ(N|k)$ of partitions of a natural number $N$ We propose an exact analytic expression for $λ(N|k)$ by relating two-point short-distance correlation functions of irregular vertex operators in $c=1$ conformal field theory ( the …
Judicious partitions for restricted self-sumsets in cyclic groups
本文研究循环群着色中受限自和集的极小极大问题，确定素数情形的精确最小值，并给出一般情形的紧界，结合受限Erdős–Heilbronn现象与块/陪集着色证明最优着色结构。
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