关于乘法分拆计数函数的取值

本文讨论了自然数 n 的乘法分拆的计数函数 g(n)。设 A={1/K;K 是自然数,K≠2}。本文证明了设任给 α∈A,则都存在自然数的子序列 α_n,n=1,2,…使 leg g(α_n)～αlog α_n,n→∞。在 Riemann 假设下,本文证明了设任给 β∈〔0,1/2〕,则都存在自然数的     

自然数乘法分拆个数及因子个数上界估计

我们证明，若n充分大，则其乘法分拆数小于n/lnn，这几乎解决了关于自然数乘法分拆数的一个猜测,也得到了自然数因子个数的一个上界估计。     

自然数乘法分拆数的上界

设 f(n)表示自然数 n 的乘法分拆数.1983年 Hughes 与 shallit 证明了f(n)≤2n~(?),1987年陈小夏证明了 f(n)≤n.本文则得到下面的定理:f(n)≤1/4n+1.     

关于乘法分拆数的一个猜想和上界

本文证明了乘法分拆数的一个上界,由此证明了Hughes-Shallit的第二猜想,同时证明了对任意的正数a,存在一个自然数N,当n≥N时,n的乘法分拆数f(n)0,使这个集合中的自然数的乘法分拆数≤n~a。     

乘法分拆函数的均值估计

将正整数ｎ分拆成正整数的方法数记为ｇ（ｎ），本文对计数函数ｇ（ｎ）进行了均值估计。关于下限我们改进了［３］的结果。证明了对任意正整数ｋ皆有Σｎ≤ｘ１／ｎｇ（ｎ）≥３（４ｌｏｇ２ ｋ！２ｋ（ｋ＋１）／２）＾－１ｘｌｏｇ＾ｋｘ，ｘ≥１还获得了一个关于上限的结果Σｎ≤ｘ１／ｎｇ（ｎ）≤（ｋ－１）！Σ＾ｋ－１ｎ＝０１／ｎ！ｘ＾１／ｋ，ｘ≥１。     

行严格长为偶数的平面分拆的计数

概念非负整数n的平面分拆是指形如下列形式的表:     

不可思议的分拆算法

如果一些东西（例如某项产品或元件）的形状,大小完全一模一样,简直无从区别．那末,把它们分拆（最小的单位是一件,分数或小数都是没有意义的）出来,不同的分法共有多少种呢？显然,这是一个人们日常生活中经常碰到的问题,不是什么偏题、怪题,或者是凭空制造出来的...     

自然数的等比分拆

自然数的等比分拆李建章（陕西华阴黄河工程机械厂中７１４２０２）文［１］讨论了自然数的等差分拆，本文给出自然数分拆成等比数列之和的充要条件，从而得出分拆的一种方法。一、定义把自然数表示成自然数等比数列之和的形式，叫自然数的等比分拆；公比为１的等比分拆称...     

An Application of Polya's Enumeration Theorem to Partitions of Subsets of Positive Integers

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a ₁, a ₂,...,a _r in S with repetitions allowed such that . Here we apply Polya's enumeration theorem to find the number P(n; S) of partitions of n into S, and the number DP(n; S) of distinct partitions of n into S. We also present recursive formulas for computing P(n; S) and DP(n; S).     

Enumeration of M-partitions

An M-partition of a positive integer m is a partition of m with as few parts as possible such that every positive integer less than m can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahon's perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers m, the number of M-partitions of m is equal to the number of binary partitions of 2ⁿ⁺¹-1-m, where . In this note we extend O’Shea's result to cover all numbers m.     

On the Combinatorics of Lecture Hall Partitions

A lecture hall partition of length n is an integer sequence satisfying Bousquet-Mélou and Eriksson showed that the number of lecture hall partitions of length n of a positive integer N whose alternating sum is k equals the number of partitions of N into k odd parts less than 2n. We prove the fact by a natural combinatorial bijection. This bijection, though defined differently, is essentially the same as one of the bijections found by Bousquet-Mélou and Eriksson.     

Jean-Louis Nicolas and the Partitions

A survey of Jean-Louis Nicolas’s papers on partitions is given.Dedicated to Jean-Louis Nicolas on the occasion of his 60th birthdayPartially supported by the Hungarian National Foundation for Scientific Research, Grant No. T 029759.2000 Mathematics Subject Classification: Primary—11P81     

Jagged Partitions

By jagged partitions we refer to an ordered collection of non-negative integers (n₁, n₂,..., n_m) with n_m≥ p for some positive integer p, further subject to some weakly decreasing conditions that prevent them for being genuine partitions. The case analyzed in greater detail here corresponds to p = 1 and the following conditions n_i≥ n_i+1−1 and n_i≥ n_i+2. A number of properties for the corresponding partition function are derived, including rather remarkable congruence relations. An interesting application of jagged partitions concerns the derivation of generating functions for enumerating partitions with special restrictions, a point that is illustrated with various examples. 2000 Mathematics Subject Classification: Primary—05A15, 05A17, 05A19     

Lecture Hall Partitions

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of lecture hall partitions of length n of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Bott's formula for the Poincaré series of the affine Coxeter group , and one direct proof.     

求解半定乘性规划的外部逼近算法

定义了半定乘性规划问题,提出一种求解它的外部逼近算法,并通过具体的实例说明算法的可行性.     

Arithmetic Properties of Summands of Partitions

Let d d, d2 2. We prove that for almost all partitions of an integer the parts are well distributed in residue classes mod d. The limitations of the uniformity of this distribution are also studied.     

Maximal Multiplicative Properties of Partitions

Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any n ≠ 7. The basis for this is an inequality for the partition function which seems not to have been noticed before.