Congruence properties of pk(n)

We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

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On the Number of Partition Weights with Kostka Multiplicity One

The Electronic Journal of Combinatorics ◽

10.37236/2574 ◽

2012 ◽

Vol 19 (4) ◽

Author(s):

Zachary Gates ◽

Brian Goldman ◽

C. Ryan Vinroot

Keyword(s):

Positive Integer ◽

Ordered Set ◽

Unitary Groups ◽

Character Theory ◽

Partition Functions ◽

Young Tableaux ◽

Multiplicity One ◽

Restricted Partition ◽

Finite Unitary Groups ◽

Kostka Number

Given a positive integer $n$, and partitions $\lambda$ and $\mu$ of $n$, let $K_{\lambda \mu}$ denote the Kostka number, which is the number of semistandard Young tableaux of shape $\lambda$ and weight $\mu$. Let $J(\lambda)$ denote the number of $\mu$ such that $K_{\lambda \mu} = 1$. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for $J(\lambda)$ in terms of restricted partition functions, which is recursive in the number of distinct part sizes of $\lambda$. We use this to classify all partitions $\lambda$ such that $J(\lambda) = 1$ and all $\lambda$ such that $J(\lambda) = 2$. We then consider signed tableaux, where a semistandard signed tableau of shape $\lambda$ has entries from the ordered set $\{0 < \bar{1} < 1 < \bar{2} < 2 < \cdots \}$, and such that $i$ and $\bar{i}$ contribute equally to the weight. For a weight $(w_0, \mu)$ with $\mu$ a partition, the signed Kostka number $K^{\pm}_{\lambda,(w_0, \mu)}$ is defined as the number of semistandard signed tableaux of shape $\lambda$ and weight $(w_0, \mu)$, and $J^{\pm}(\lambda)$ is then defined to be the number of weights $(w_0, \mu)$ such that $K^{\pm}_{\lambda, (w_0, \mu)} = 1$. Using different methods than in the unsigned case, we find that the only nonzero value which $J^{\pm}(\lambda)$ can take is $1$, and we find all sequences of partitions with this property. We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.

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Anti-integral extensions $ {R[{\alpha}]/R$

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.35.2004.220 ◽

2004 ◽

Vol 35 (1) ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Kiyoshi Baba ◽

Ken-Ichi Yoshida

Keyword(s):

Positive Integer ◽

Integral Domain ◽

Arbitrary Positive Integer ◽

Integral Element ◽

The Ideal

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

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Characterizations for the n-strong Drazin invertibility in a ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498821501413 ◽

2020 ◽

pp. 2150141

Author(s):

Honglin Zou ◽

Jianlong Chen ◽

Huihui Zhu ◽

Yujie Wei

Keyword(s):

Positive Integer ◽

Generalized Inverse ◽

Product Formula ◽

Drazin Inverse ◽

Equivalent Definition ◽

New Type ◽

Definition Of ◽

Existence Criteria

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

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A further result on the complex oscillation theory of periodic second order linear differential equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028959 ◽

1990 ◽

Vol 33 (1) ◽

pp. 143-158 ◽

Cited By ~ 7

Author(s):

Shian Gao

Keyword(s):

Entire Function ◽

Positive Integer ◽

Oscillation Theory ◽

Transcendental Entire Function ◽

Arbitrary Positive Integer ◽

Order Of Growth ◽

Exponent Of Convergence ◽

Complex Oscillation ◽

Zero Sequence ◽

Linear Differential

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

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A Sum Connected with Quadratic Residues

Nagoya Mathematical Journal ◽

10.1017/s0027763000000039 ◽

1956 ◽

Vol 10 ◽

pp. 1-7

Author(s):

L. Carlitz

Keyword(s):

Positive Integer ◽

Legendre Symbol ◽

Arbitrary Positive Integer ◽

Arbitrary Integer ◽

Quadratic Residues

Let p be a prime > 2 and m an arbitrary positive integer; definewhere (r/p) is the Legendre symbol. We consider the problem of finding the highest power of p dividing Sm. A little more generally, if we putwhere a is an arbitrary integer, we seek the highest power of p dividing Sm(a). Clearly Sm = Sm(0), and Sm(a) = Sm(b) when a ≡ b (mod p).

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On rings all of whose factor rings are integral domains

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700034078 ◽

1993 ◽

Vol 55 (3) ◽

pp. 325-333

Author(s):

Yasuyuki Hirano

Keyword(s):

Positive Integer ◽

Zero Divisor ◽

Arbitrary Positive Integer ◽

Integral Domains ◽

Factor Ring ◽

Zero Divisors ◽

Proper Quotient

AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

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