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...     

Partitioning Sets of Triples into Small Planes

We study partitions of the set of all ₃ ^v triples chosen from a v-set intopairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2,2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions)or copies of some planes of each type (mixed partitions).We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in severalcases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We constructsuch partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, andan affine partition for v = 18. Using these as starter partitions, we prove that Fano partitionsexist for v = 7 ⁿ + 1, 13 ⁿ + 1,27 ⁿ + 1, and affine partitions for v = 8 ⁿ + 1,9 ⁿ + 1, 17 ⁿ + 1. In particular, both Fano and affine partitionsexist for v = 3⁶ⁿ + 1. Using properties of 3-wise balanced designs, weextend these results to show that affine partitions also exist for v = 3²ⁿ .Similarly, mixed partitions are shown to exist for v = 8 ⁿ ,9 ⁿ , 11 ⁿ + 1.     

Double perfect partitions

MacMahon [Combinatory Analysis, vols. I and II, Cambridge University Press, Cambridge, 1915, 1916 (reprinted, Chelsea, 1960)] introduced a perfect partition of positive integer n, which is a partition such that every positive integer less than or equal to n can be uniquely represented by the sum of its parts. We generalize perfect partition and find a relation with ordered factorizations.     

About the Notion of Semiring of Sets

We introduce a generalized notion of semiring and prove that all known properties that semirings have according to the old definition are preserved.     

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.     

Analysis of some new partition statistics

We study several statistics for integer partitions: for a random partition of an integer n we consider the average size of the smallest gap (missing part size), the multiplicity of the largest part, and the largest repeated part size. Furthermore, we estimate the number of gap-free partitions of n. 2000 Mathematics Subject Classification Primary—05A17; Secondary—11P82 Dedicated to Helmut Prodinger on the occasion of his 50th birthday P.J. Grabner is supported by the START-project Y96-MAT of the Austrian Science Fund. This material is based upon work supported by the National Research Foundation under grant number 2053740.     

Classification of partitions of all triples on ten points into copies of Fano and affine planes

We continue our study of partitions of the full set of triples chosen from a v-set into copies of the Fano plane PG(2,2) (Fano partitions) or copies of the affine plane AG(2,3) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are v=8 where Fano partitions exist, v=9 where affine partitions exist, and v=10 where both affine and mixed partitions exist. The Fano partitions for v=8 and the affine partitions for v=9 and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for i=1,…,7, and (2) the mixed partitions for v=10 into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of AG(2,3).     

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.     

Partition-Theoretic Interpretations of Certain Modular Equations of Schröter,Russell, and Ramanujan

We show that certain modular equations studied by Schr?oter, Russell, and Ramanujan yield elegant identities for colored partitions. Received May 3, 2006 Bruce C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.     

A Note on Partitions into Distinct Parts and Odd Parts

Bousquet-Mélou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a well-known result by Euler. We give a different graphical interpretation of the bijection by Sylvester on partitions into distinct parts and partitions into odd parts, and show that the bijection implies the above statement.