A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)

[2303.07149] A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version) Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2303.07149 (math) [Submitted on 13 Mar 2023] Title: A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version) Authors: Feihu Liu , Guoce Xin View a PDF of the paper titled A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version), by Feihu Liu and Guoce Xin View PDF HTML (experimental) Abstract: Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $g(A)$ is the greatest integer not belonging to the set $\big\{ \sum_{i=1}^na_ix_i\ |x_i\in \mathbb{N}\big\}$. The general Frobenius problem includes the determination of $g(A)$ and the related Sylvester number $n(A)$ and Sylvester sum $s(A)$. We present a new approach to the Frobenius problem. Basically, we transform the problem into an easier optimization problem. If the new problem can be solved explicitly, then we will be able to obtain a formula of $g(A)$. We illustrate the idea by giving concise proof of some existing formulas and finding some interesting new formulas of $g(A), n(A), s(A)$. Moreover, we find that MacMahon's partition analysis applies to give a new way of calculating $n(A), s(A)$ by using a rational function representation of a polynomial determined by $A$. Subjects: Combinatorics (math.CO) ; Number Theory (math.NT) Cite as: arXiv:2303.07149 [math.CO] (or arXiv:2303.07149v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2303.07149 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.aam.2024.102719 Focus to learn more DOI(s) linking to related resources Submission history From: FeiHu Liu [ view email ] [v1] Mon, 13 Mar 2023 14:20:28 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version), by Feihu Liu and Guoce Xin View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2023-03 Change to browse by: math math.NT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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