Divisibility of Certain Partition Functions by Powers of Primes

Let be the prime factorization of a positive integer k and let b _k (n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let S _k (N; M) be the number of positive integers N for which b _k(n ) 0(mod M). If we prove that, for every positive integer j In other words for every positive integer j, b _k(n) is a multiple of for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupS _n is a multiple of p ^j. We also examine the behavior of b _k(n) (mod ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n (mod t) satisfies b _k(n) 0 (mod ), we show that there are infinitely many non-negative integers n r (mod t) for which b _k(n) 0 (mod ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 .     

Arithmetic properties of overpartition triples

Let p_3(n) be the number of overpartition triples of n. By elementary series manipulations,we establish some congruences for p_3(n) modulo small powers of 2, such as p_3(16 n + 14) ≡ 0(mod 32), p_3(8 n + 7) ≡ 0(mod 64).We also find many arithmetic properties for p_3(n) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α≥ 1 and n ≥ 0, we have p_3 (3~(2α+1)(3n + 2))≡ 0(mod 9 · 2~4), p_3(4~(α-1)(56 n + 49)) ≡ 0(mod 7),p_3 (7~(2α+1)(7 n + 3))≡ p_3 (7~(2α+1)(7 n + 5))≡ p_3 (7~(2α+1)(7 n + 6))≡ 0(mod 7),and for r ∈ {1, 2, 3, 4, 5, 6},p_3(11 · 7~(4α-1)(7 n + r)≡ 0(mod 11).     

On the number of partitions with distinct even parts

Let ped(n) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for ped(n)mod2 and mod4 by the theory of Hecke eigenforms.     

On an Arithmetical Function

In this paper we introduce an arithmetical function (n), the difference between the number of divisors of n congruent to 1 mod 3 and those congruent to –1 mod 3. This function then is related to the classical function (n) which is the sum of the divisors of n. In particular we prove the identity

Elementary Proofs of Infinitely Many Congruences for 8-Cores

Using a very elementary argument, we prove the congruences where a₈(n) is the number of 8-core partitions of n. We also exhibit two infinite families of congruences modulo 2 for 8-cores.     

Parity Results for Certain Partition Functions

Blecksmith, Brillhart and Gerst proved four congruences modulo 2 involving partition generating functions of the following sort.

On the stability of some integer programming algorithms

In the present paper we develop our approach for studying the stability of integer programming problems. We prove that the L-class enumeration method is stable on integer linear programming problems in the case of bounded relaxation sets [9]. The stability of some cutting plane algorithms is discussed.     

On the Asymptotic Behaviour of General Partition Functions

For A = {a₁, a₂,...} N, let p_A(n) denote the number of partitions of n into a's and let q_A(n) denote the number of partitions of n into distinct a's. The asymptotic behaviour of the quotient is studied.     

A generalization of a curious congruence on harmonic sums

Zhao established a curious harmonic congruence for prime :

In this note the authors extend it to the following congruence for any prime and positive integer :

Other improvements on congruences of harmonic sums are also obtained.

     

On the quintuple product identity

In this note we present a new proof of the quintuple product identity which is based on our study of order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.

     

On the Asymptotic Behaviour of General Partition Functions,II

Let = {a ₁, a ₂,...} be a set of positive integers and let p (n) and q (n) denote the number of partitions of n into a's, resp. distinct a's. In an earlier paper the authors studied large values of log(max (2,p (n)))/log(max(2,q (n))). In this paper the small values of the same quotient are studied.     

Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions

Combinatorial proofs are given for certain entries in Ramanujan's lost notebook. Bijections of Sylvester, Franklin, and Wright, and applications of Algorithm Z of Zeilberger are employed. A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujan's fascinating identities for a partial theta function.     

On the explicit formulas for zeta functions

In this paper, we study the properties of a large class of zeta functions that arises in geometric analysis and mathematical physics. They are attached to some elliptic operators. This method can be used to evaluate explicitly the special values of zeta functions of elliptic operators defined on some symmetric spaces.     

三正则连通图的Cordial性

用调整顶点标号的方法确定了3正则连通图的Cordial性．     

On the complexity of constrained VC-classes

Sauer's lemma is extended to classes H_N of binary-valued functions h on [n]={1,…,n} which have a margin less than or equal to N on all x∈[n] with h(x)=1, where the margin μ_h(x) of h at x∈[n] is defined as the largest non-negative integer a such that h is constant on the interval I_a(x)=[x-a,x+a]⊆[n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S⊆[n].     

关于3-正则图的平均亏格

一个图G的2－因子F是一个使得每个点v在F中的度dF(v)=2的G的生成子图。易知F中的每个圈是点不交的。如果F中每个圈的长度为4，我们说G有四边形2－因子F。我们首先在3－正则图上定义了3种扩张运算，然后讨论这些运算对平均亏格的影响。运用扩张运算，我们研究了含有四边形2－因子的3－正则图的平均亏格，得到了3－正则图的平均亏格与最大亏格之间的关系。     

Partition identities arising from theta function identities

The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities.     

关于β-正则与一致β-正则诱导极限

本文举出反例说明文献［７］中关于一致β－正则的充要条件是错误的，并分别给出了诱导极限为β－正则与一致β－正则的正确的充要条件．