Recent zbMATH articles in MSC 20K01https://zbmath.org/atom/cc/20K012022-07-25T18:03:43.254055ZWerkzeugSome algebraic properties of finite binary sequenceshttps://zbmath.org/1487.050362022-07-25T18:03:43.254055Z"Filipczak, Małgorzata"https://zbmath.org/authors/?q=ai:filipczak.malgorzata"Filipczak, Tomasz"https://zbmath.org/authors/?q=ai:filipczak.tomaszSummary: We study properties of differences of finite binary sequences with a fixed number of ones, treated as binary numbers from \(\mathbb{Z}(2^m)\).We show that any binary sequence consisting of \(m\) terms (except of the sequence \((1, 0,\ldots , 0)\)) can be presented as a difference of two sequences having exactly \(n\) ones, whenever.Counting subgroups of fixed order in finite abelian groupshttps://zbmath.org/1487.200202022-07-25T18:03:43.254055Z"Admasu, Fikreab Solomon"https://zbmath.org/authors/?q=ai:admasu.fikreab-solomon"Sehgal, Amit"https://zbmath.org/authors/?q=ai:sehgal.amitLet \(n\) be a positive integer and \(p\) a prime number. The number of finite abelian groups of order \(p^{n}\) is given by the partition function \(p(n)\). The number of subgroups of a fixed order in a finite abelian group of given rank is given by sums of Hall polynomials.
The main result of the paper reads like this.
Theorem 2.1: For every \(0\leq b\leq a_{1}+a_{2}+a_{3}\) the number \( h_{b}^{(a_{3},a_{2},a_{1})}(p)\) of all subgroups of order \(p^{b}\) in the finite abelian \(p\)-group \(\mathbb{Z}/p^{a_{1}}\times \mathbb{Z} /p^{a_{2}}\times \mathbb{Z}/p^{a_{3}}\) where \(1\leq a_{1}\leq a_{2}\leq a_{3} \), is given by one of the following polynomials expressed as rational functions.
There are 10 possible cases whose formulas are too complicated to be reproduced here.
The method used in the proof of the theorem derives from a lemma [Commun. Algebra 45, No. 8, 3365--3376 (2017; Zbl 1400.20015), Lemma 2.3] that \textit{Y. Hironaka} deduces from a recurrence relation of \textit{T. Stehling} [Combinatorica 12, No. 4, 475--479 (1992; Zbl 0769.05009)], as their lemma shows how to express the formulas in higher rank in terms of those of lower rank.
The formulas are also used to derive similar formulas in few cases for rank 4.
Reviewer: Grigore Călugăreanu (Cluj-Napoca)